id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-22200 | Three-Point Form of Equation of Plane/Vector Form | Let $P_1$, $P_2$ and $P_3$ be non-collinear points in a cartesian $3$-space whose position vectors are $\mathbf a$, $\mathbf b$ and $\mathbf c$.
Let $P$ be the plane passing through $P_1$, $P_2$ and $P_3$.
$P$ can be expressed in the form:
:$\mathbf r = \mathbf a + \lambda \paren {\mathbf b - \mathbf a} + \mu \paren {\... | Let $\bsalpha$ be the non-zero cross product:
:$\bsalpha = \paren {\mathbf b - \mathbf a} \times \paren {\mathbf c - \mathbf a}$
If $P_1$, $P_2$, and $P_3$ were collinear, then $\mathbf b - \mathbf a$ and $\mathbf c - \mathbf a$ would be parallel or antiparallel.
Since both the cross product of parallel vectors and the... | Let $P_1$, $P_2$ and $P_3$ be non-[[Definition:Collinear Points|collinear]] [[Definition:Point|points]] in a [[Definition:Cartesian Space|cartesian $3$-space]] whose [[Definition:Position Vector|position vectors]] are $\mathbf a$, $\mathbf b$ and $\mathbf c$.
Let $P$ be the [[Definition:Plane|plane]] passing through $... | Let $\bsalpha$ be the non-[[Definition:Zero Vector|zero]] [[Definition:Vector Cross Product|cross product]]:
:$\bsalpha = \paren {\mathbf b - \mathbf a} \times \paren {\mathbf c - \mathbf a}$
If $P_1$, $P_2$, and $P_3$ were [[Definition:Collinear Points|collinear]], then $\mathbf b - \mathbf a$ and $\mathbf c - \mathb... | Three-Point Form of Equation of Plane/Vector Form | https://proofwiki.org/wiki/Three-Point_Form_of_Equation_of_Plane/Vector_Form | https://proofwiki.org/wiki/Three-Point_Form_of_Equation_of_Plane/Vector_Form | [
"Three-Point Form of Equation of Plane"
] | [
"Definition:Collinear/Points",
"Definition:Point",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Position Vector",
"Definition:Plane Surface",
"Definition:Position Vector",
"Definition:Point",
"Definition:Real Number",
"Definition:Parameter"
] | [
"Definition:Zero Vector",
"Definition:Vector Cross Product",
"Definition:Collinear/Points",
"Definition:Parallel (Geometry)/Lines",
"Definition:Antiparallel Vectors",
"Cross Product of Parallel Vectors",
"Cross Product of Antiparallel Vectors is Zero",
"Definition:Zero Vector",
"Definition:Collinear... |
proofwiki-22201 | Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $I$ be a closed ideal.
Let $x \in I$ be normal.
Let $\map {\sigma_A} x \subseteq \R$ be the spectrum of $x$ in $A$.
Let $K$ be a compact set such that $\map {\sigma_A} x \cup \set 0 \subseteq K$.
Let $f : K \to \R$ be a continuous f... | From Closed Ideal of C*-Algebra is Self-Adjoint, we have $x^\ast \in I$.
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\map \CC K$.
From Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Numbers, the set:
:$\ds \AA = \set {z \m... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $I$ be a [[Definition:Closed Set|closed]] [[Definition:Ideal of Algebra|ideal]].
Let $x \in I$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A... | From [[Closed Ideal of C*-Algebra is Self-Adjoint]], we have $x^\ast \in I$.
Let $\norm {\, \cdot \,}_\infty$ be the [[Definition:Supremum Norm|supremum norm]] on $\map \CC K$.
From [[Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Num... | Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal | https://proofwiki.org/wiki/Continuous_Function_Vanishing_at_Zero_applied_to_Normal_Element_of_Closed_Ideal_of_Unital_C*-Algebra_is_contained_in_Ideal | https://proofwiki.org/wiki/Continuous_Function_Vanishing_at_Zero_applied_to_Normal_Element_of_Closed_Ideal_of_Unital_C*-Algebra_is_contained_in_Ideal | [
"Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal",
"Continuous Functional Calculus"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Closed Set",
"Definition:Ideal of Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Compact Topological Space",
"Definition:Continuous Function",
"Definition... | [
"Closed Ideal of C*-Algebra is Self-Adjoint",
"Definition:Supremum Norm",
"Subalgebra Generated by Inclusion and Conjugate is Everywhere Dense in Space of Continuous Functions on Compact Subset of Complex Numbers",
"Definition:Set",
"Definition:Everywhere Dense",
"Definition:Ideal of Algebra",
"Definiti... |
proofwiki-22202 | Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ be the unitization of $\struct {A, \ast, \norm {\, \cdot \,} }$.
Let $A_0 = \set {\tuple {a, 0} : a \in A}$.
Let $x \in I$ be normal.
Let $\map {\sigma_{A_+} } x \subseteq \R$ be the sp... | From Normed Algebra Embeds into Unitization as Closed Ideal, $A_0$ is a closed ideal of $A$.
We have that $\tuple {x, 0}$ is normal in $A_+$.
The result follows from Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal.
{{qed}}
Category:Continuous Fu... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\struct {A_+, \ast, \norm {\, \cdot \,}_\ast}$ be the [[Definition:Unitization of C*-Algebra|unitization]] of $\struct {A, \ast, \norm {\, \cdot \,} }$.
Let $A_0 = \set... | From [[Normed Algebra Embeds into Unitization as Closed Ideal]], $A_0$ is a [[Definition:Closed Set|closed]] [[Definition:Ideal of Algebra|ideal]] of $A$.
We have that $\tuple {x, 0}$ is [[Definition:Normal Element of *-Algebra|normal]] in $A_+$.
The result follows from [[Continuous Function Vanishing at Zero applied... | Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal/Corollary | https://proofwiki.org/wiki/Continuous_Function_Vanishing_at_Zero_applied_to_Normal_Element_of_Closed_Ideal_of_Unital_C*-Algebra_is_contained_in_Ideal/Corollary | https://proofwiki.org/wiki/Continuous_Function_Vanishing_at_Zero_applied_to_Normal_Element_of_Closed_Ideal_of_Unital_C*-Algebra_is_contained_in_Ideal/Corollary | [
"Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Unitization of C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Non-Unital Algebra",
"Definition:Continuous Function",
"Definition:Continuous Functional Calculus"
] | [
"Normed Algebra Embeds into Unitization as Closed Ideal",
"Definition:Closed Set",
"Definition:Ideal of Algebra",
"Definition:Normal Element of *-Algebra",
"Continuous Function Vanishing at Zero applied to Normal Element of Closed Ideal of Unital C*-Algebra is contained in Ideal",
"Category:Continuous Fun... |
proofwiki-22203 | Inverse Image Mapping of Bijection is Inverse of Direct Image Mapping | Let $S$ and $T$ be sets.
Let $f :S \to T$ be a bijection.
Let $f^\to$ be the direct image mapping of $f$.
Let $f^\gets$ be the inverse image mapping of $f$.
Then:
:$\paren {f^\to}^{-1} = f^\gets$
where $\paren {f^\to}^{-1}$ denotes the inverse of $f^\to$. | From Mapping is Bijection iff Direct Image Mapping is Bijection:
:$f^\to$ is a bijection
Let $\paren {f^\to}^{-1}$ denote the inverse of $f^\to$.
We have:
{{begin-eqn}}
{{eqn | l = f^\gets \circ f^\to
| r = \operatorname{id}_{\powerset S} \circ \paren{f^\gets \circ f^\to}
| c = Identity Mapping is Left Iden... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f :S \to T$ be a [[Definition:Bijection|bijection]].
Let $f^\to$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Let $f^\gets$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$.
Then:
:$\paren {f^\... | From [[Mapping is Bijection iff Direct Image Mapping is Bijection]]:
:$f^\to$ is a [[Definition:Bijection|bijection]]
Let $\paren {f^\to}^{-1}$ denote the [[Definition:Inverse Mapping|inverse]] of $f^\to$.
We have:
{{begin-eqn}}
{{eqn | l = f^\gets \circ f^\to
| r = \operatorname{id}_{\powerset S} \circ \paren... | Inverse Image Mapping of Bijection is Inverse of Direct Image Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Bijection_is_Inverse_of_Direct_Image_Mapping | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Bijection_is_Inverse_of_Direct_Image_Mapping | [
"Direct Image Mappings",
"Inverse Image Mappings"
] | [
"Definition:Set",
"Definition:Bijection",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Inverse of Mapping"
] | [
"Mapping is Bijection iff Direct Image Mapping is Bijection",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Identity Mapping is Left Identity",
"Composition of Mappings is Associative",
"Direct Image of Inverse Image of Direct Image equals Direct Image Mapping",
"Identity Mapping is Right Ident... |
proofwiki-22204 | Generalized Polar Decomposition in C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $x \in A$.
Let $\cmod x$ be the modulus of $x$.
Let $0 < \beta < 1$ be a real number.
Then there exists $u \in A$ such that:
:$x = u \cmod x^\beta$
where $\cmod x^\beta$ is obtained from the continuous functional calculus. | Let $\le_A$ be the canonical preordering of $A$.
From Product of Element of C*-Algebra with its Star is Positive, we have ${\mathbf 0}_A \le_A x^\ast x$.
Hence since $x^\ast x \le_A x^\ast x$ and $0 < \beta/2 < 1/2$, there exists $u \in A$ such that:
:$x = u \paren {x^\ast x}^{\beta/2}$
from Factorization Theorem for C... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $x \in A$.
Let $\cmod x$ be the [[Definition:Modulus of Element of C*-Algebra|modulus]] of $x$.
Let $0 < \beta < 1$ be a [[Definition:Real Number|real number]].
Then there exists $u \in A$ such that:
:$x = u ... | Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
From [[Product of Element of C*-Algebra with its Star is Positive]], we have ${\mathbf 0}_A \le_A x^\ast x$.
Hence since $x^\ast x \le_A x^\ast x$ and $0 < \beta/2 < 1/2$, there exists $u \in A$ such that:
:$x = u \par... | Generalized Polar Decomposition in C*-Algebra | https://proofwiki.org/wiki/Generalized_Polar_Decomposition_in_C*-Algebra | https://proofwiki.org/wiki/Generalized_Polar_Decomposition_in_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Modulus of Element of C*-Algebra",
"Definition:Real Number",
"Definition:Continuous Functional Calculus"
] | [
"Definition:Canonical Preordering of C*-Algebra",
"Product of Element of C*-Algebra with its Star is Positive",
"Factorization Theorem for C*-Algebra in terms of Bound on Modulus",
"Power of Power of Positive Element of Unital C*-Algebra"
] |
proofwiki-22205 | Hilbert Space Projections with Zero Product are Pointwise Orthogonal | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $P, Q : \HH \to \HH$ be projections (in the Hilbert space sense) such that:
:$P Q = 0$
Then:
:$\innerprod {Q x} {P y} = 0$ for each $x, y \in \HH$. | We have, for each $x \in \HH$:
:$\map {\paren {P Q} } x = {\mathbf 0}_\HH$
Hence:
:$\map P {Q x} = {\mathbf 0}_\HH$
So we have $Q x \in \ker P$.
Since $x \in \HH$ was arbitrary we have $\Rng Q \subseteq \ker P$.
From the definition of a projections (in the Hilbert space sense), we have:
:$\ker P = \paren {\Rng Q}^\bot$... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $P, Q : \HH \to \HH$ be [[Definition:Projection (Hilbert Spaces)|projections (in the Hilbert space sense)]] such that:
:$P Q = 0$
Then:
:$\innerprod {Q x} {P y} = 0$ for each $x, y \in \HH$. | We have, for each $x \in \HH$:
:$\map {\paren {P Q} } x = {\mathbf 0}_\HH$
Hence:
:$\map P {Q x} = {\mathbf 0}_\HH$
So we have $Q x \in \ker P$.
Since $x \in \HH$ was arbitrary we have $\Rng Q \subseteq \ker P$.
From the definition of a [[Definition:Projection (Hilbert Spaces)|projections (in the Hilbert space sens... | Hilbert Space Projections with Zero Product are Pointwise Orthogonal | https://proofwiki.org/wiki/Hilbert_Space_Projections_with_Zero_Product_are_Pointwise_Orthogonal | https://proofwiki.org/wiki/Hilbert_Space_Projections_with_Zero_Product_are_Pointwise_Orthogonal | [
"Orthogonal Projections"
] | [
"Definition:Hilbert Space",
"Definition:Projection (Hilbert Spaces)"
] | [
"Definition:Projection (Hilbert Spaces)",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement",
"Category:Orthogonal Projections"
] |
proofwiki-22206 | Orthogonality of Resolution of the Identity evaluated at Disjoint Sets | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.
Let $A, B ... | From $(2)$ in the definition of a resolution of the identity, we have:
:$\map \EE A \map \EE B = \map \EE {A \cap B} = \map \EE \O$
From $(1)$ in the definition of a resolution of the identity, we then have:
:$\map \EE A \map \EE B = 0$
From Hilbert Space Projections with Zero Product are Pointwise Orthogonal, we there... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B {\HH}$ be the [[Definition:Space of Bounded Lin... | From $(2)$ in the definition of a [[Definition:Resolution of the Identity|resolution of the identity]], we have:
:$\map \EE A \map \EE B = \map \EE {A \cap B} = \map \EE \O$
From $(1)$ in the definition of a [[Definition:Resolution of the Identity|resolution of the identity]], we then have:
:$\map \EE A \map \EE B = 0... | Orthogonality of Resolution of the Identity evaluated at Disjoint Sets | https://proofwiki.org/wiki/Orthogonality_of_Resolution_of_the_Identity_evaluated_at_Disjoint_Sets | https://proofwiki.org/wiki/Orthogonality_of_Resolution_of_the_Identity_evaluated_at_Disjoint_Sets | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity"
] | [
"Definition:Resolution of the Identity",
"Definition:Resolution of the Identity",
"Hilbert Space Projections with Zero Product are Pointwise Orthogonal",
"Category:Resolutions of the Identity"
] |
proofwiki-22207 | Scalar-Valued Measure associated with Resolution of Identity is Positive Measure if Equal Vectors | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.
For each $... | Let $A \in \map \BB X$.
We then have:
{{begin-eqn}}
{{eqn | l = \map {\EE_x} A
| r = \innerprod {\map \EE A x} x
}}
{{eqn | r = \innerprod {\map \EE A^2 x} x
| c = Orthogonal Projection is Projection
}}
{{eqn | r = \innerprod {\map \EE A x} {\map \EE A x}
| c = from Characterization of Projections, $\map \EE A$... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B {\HH}$ be the [[Definition:Space of Bounded Lin... | Let $A \in \map \BB X$.
We then have:
{{begin-eqn}}
{{eqn | l = \map {\EE_x} A
| r = \innerprod {\map \EE A x} x
}}
{{eqn | r = \innerprod {\map \EE A^2 x} x
| c = [[Orthogonal Projection is Projection]]
}}
{{eqn | r = \innerprod {\map \EE A x} {\map \EE A x}
| c = from [[Characterization of Projections]], $\m... | Scalar-Valued Measure associated with Resolution of Identity is Positive Measure if Equal Vectors | https://proofwiki.org/wiki/Scalar-Valued_Measure_associated_with_Resolution_of_Identity_is_Positive_Measure_if_Equal_Vectors | https://proofwiki.org/wiki/Scalar-Valued_Measure_associated_with_Resolution_of_Identity_is_Positive_Measure_if_Equal_Vectors | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity",
"Definition:Measure (Measure Theory)",
"Definition:Total Variation"
] | [
"Orthogonal Projection is Projection",
"Characterization of Projections",
"Definition:Hermitian Operator",
"Definition:Measure (Measure Theory)",
"Definition:Total Variation/Measure Theory",
"Category:Resolutions of the Identity"
] |
proofwiki-22208 | Bound on Total Variation of Scalar-Valued Measure associated with Resolution of the Identity | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.
For each $... | Let $\map P X$ be the set of finite partitions of $X$ into $\map \BB X$-measurable sets.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\EE_{x, y} }
| r = \map {\cmod {\EE_{x, y} } } X
| c = {{Defof|Total Variation/Measure Theory/Complex Measure|Total Variation: Measure Theory: Complex Measure}}
}}
{{eqn | r = \sup \set... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B {\HH}$ be the [[Definition:Space of Bounded Lin... | Let $\map P X$ be the set of [[Definition:Finite Set|finite]] [[Definition:Set Partition|partitions]] of $X$ into [[Definition:Measurable Set|$\map \BB X$-measurable sets]].
We have:
{{begin-eqn}}
{{eqn | l = \norm {\EE_{x, y} }
| r = \map {\cmod {\EE_{x, y} } } X
| c = {{Defof|Total Variation/Measure Theory/Compl... | Bound on Total Variation of Scalar-Valued Measure associated with Resolution of the Identity | https://proofwiki.org/wiki/Bound_on_Total_Variation_of_Scalar-Valued_Measure_associated_with_Resolution_of_the_Identity | https://proofwiki.org/wiki/Bound_on_Total_Variation_of_Scalar-Valued_Measure_associated_with_Resolution_of_the_Identity | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity",
"Definition:Total Variation/Measure Theory/Complex Measure"
] | [
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Measurable Set",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces",
"Definition:Set Partition",
"Orthogonality of Resolution of the Identity evaluated at Disjoint Sets",
"Pythagoras's Theorem (Inner Product Space)",
"Definitio... |
proofwiki-22209 | Null Sets Closed under Countable Union/Resolution of the Identity | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.
Let $\set ... | Let $x \in \HH$.
Define $\EE_{x, x} : \map \BB X \to \C$ by:
:$\map {\EE_{x, x} } A = \innerprod {\map \EE A x} x$
for each $A \in \map \BB X$.
Since $\map \EE {A_j} = 0$ for each $j \in \N$, we have $\map {\EE_{x, x} } {A_j} = 0$ for each $j \in \N$.
Since $\EE_{x, x}$ is a complex measure, we have:
:$\map {\EE_{x, ... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B {\HH}$ be the [[Definition:Space of Bounded Lin... | Let $x \in \HH$.
Define $\EE_{x, x} : \map \BB X \to \C$ by:
:$\map {\EE_{x, x} } A = \innerprod {\map \EE A x} x$
for each $A \in \map \BB X$.
Since $\map \EE {A_j} = 0$ for each $j \in \N$, we have $\map {\EE_{x, x} } {A_j} = 0$ for each $j \in \N$.
Since $\EE_{x, x}$ is a [[Definition:Complex Measure|complex me... | Null Sets Closed under Countable Union/Resolution of the Identity | https://proofwiki.org/wiki/Null_Sets_Closed_under_Countable_Union/Resolution_of_the_Identity | https://proofwiki.org/wiki/Null_Sets_Closed_under_Countable_Union/Resolution_of_the_Identity | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity"
] | [
"Definition:Complex Measure",
"Definition:Countably Additive Function",
"Scalar-Valued Measure associated with Resolution of Identity is Positive Measure if Equal Vectors"
] |
proofwiki-22210 | Poincaré Duality Theorem | Let $M$ be an $n$-manifold.
Let its $n$th homology group $\map {H_n} M$ be infinite and cyclic (that is, that $M$ is an orientable manifold).
Then $\map {H_r} M$ is homeomorphic to the $\paren {n - r}$th cohomology group $\map {H^{n - r} } M$ for all $r$. | {{ProofWanted}}
{{Namedfor|Jules Henri Poincaré|cat = Poincaré}} | Let $M$ be an [[Definition:Topological Manifold|$n$-manifold]].
Let its [[Definition:Homology Group|$n$th homology group]] $\map {H_n} M$ be [[Definition:Infinite Group|infinite]] and [[Definition:Cyclic Group|cyclic]] (that is, that $M$ is an [[Definition:Orientable Manifold|orientable manifold]]).
Then $\map {H_r}... | {{ProofWanted}}
{{Namedfor|Jules Henri Poincaré|cat = Poincaré}} | Poincaré Duality Theorem | https://proofwiki.org/wiki/Poincaré_Duality_Theorem | https://proofwiki.org/wiki/Poincaré_Duality_Theorem | [
"Poincaré Duality Theorem",
"Riemannian Geometry"
] | [
"Definition:Topological Manifold",
"Definition:Homology Group",
"Definition:Infinite Group",
"Definition:Cyclic Group",
"Definition:Orientable Manifold",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Cohomology Group"
] | [] |
proofwiki-22211 | Null Sets Closed under Subset/Resolution of the Identity | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.
Let $E, E'... | For each $x, y \in \HH$, define $\EE_{x, y} : \map \BB X \to \C$ by:
:$\map {\EE_{x, y} } A = \innerprod {\map \EE A x} y$
for each $A \in \map \BB X$.
We then have:
:$\map {\EE_{x, y} } {E'} = 0$
for each $x, y \in \HH$.
From Null Sets Closed under Subset, we have $\map {\EE_{x, y} } E = 0$ for each $x, y \in \HH$.
Th... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B {\HH}$ be the [[Definition:Space of Bounded Lin... | For each $x, y \in \HH$, define $\EE_{x, y} : \map \BB X \to \C$ by:
:$\map {\EE_{x, y} } A = \innerprod {\map \EE A x} y$
for each $A \in \map \BB X$.
We then have:
:$\map {\EE_{x, y} } {E'} = 0$
for each $x, y \in \HH$.
From [[Null Sets Closed under Subset]], we have $\map {\EE_{x, y} } E = 0$ for each $x, y \in \H... | Null Sets Closed under Subset/Resolution of the Identity/Proof 1 | https://proofwiki.org/wiki/Null_Sets_Closed_under_Subset/Resolution_of_the_Identity | https://proofwiki.org/wiki/Null_Sets_Closed_under_Subset/Resolution_of_the_Identity/Proof_1 | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity"
] | [
"Null Sets Closed under Subset",
"Linear Subspace Dense iff Zero Orthocomplement"
] |
proofwiki-22212 | Null Sets Closed under Subset/Resolution of the Identity | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to \map B {\HH}$ be a resolution of the identity.
Let $E, E'... | Let $\le_{\map B \HH}$ be the canonical preordering on $\map B \HH$.
From Measure is Monotone: Resolution of the Identity, we have:
:$\map \EE E \le_{\map B \HH} \map \EE {E'} = {\mathbf 0}_{\map B \HH}$
On the other hand from Bounds on Projection in Unital C*-Algebra, we have:
:${\mathbf 0}_{\map B \HH} \le_{\map B \H... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B {\HH}$ be the [[Definition:Space of Bounded Lin... | Let $\le_{\map B \HH}$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] on $\map B \HH$.
From [[Measure is Monotone/Resolution of the Identity|Measure is Monotone: Resolution of the Identity]], we have:
:$\map \EE E \le_{\map B \HH} \map \EE {E'} = {\mathbf 0}_{\map B \HH}$
On the other... | Null Sets Closed under Subset/Resolution of the Identity/Proof 2 | https://proofwiki.org/wiki/Null_Sets_Closed_under_Subset/Resolution_of_the_Identity | https://proofwiki.org/wiki/Null_Sets_Closed_under_Subset/Resolution_of_the_Identity/Proof_2 | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity"
] | [
"Definition:Canonical Preordering of C*-Algebra",
"Measure is Monotone/Resolution of the Identity",
"Bounds on Projection in Unital C*-Algebra",
"Canonical Preordering of C*-Algebra is Antisymmetric"
] |
proofwiki-22213 | Essential Image is Well-Defined and Smallest Closed Set containing Almost All Function Values/Resolution of the Identity | Let $X$ be a topological space.
Let $\map \BB X$ be the Borel $\sigma$-algebra of $X$.
Let $\map \BB \C$ be the Borel $\sigma$-algebra of $\C$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B {\HH}$ be the space of bounded linear transformations on $\HH$.
Let $\EE : \map \BB X \to ... | We first show that $\map {\operatorname {EssIm} } f$ is well-defined.
Let $\sequence {D_n}_{n \mathop \in \N}$ and $\sequence {D_n'}_{n \mathop \in \N}$ be analytic bases of $\C$ consisting of open balls.
Let $V$ be the union of those $D_i$ such that:
:$\map \EE {f^{-1} \sqbrk {D_n} } = 0$ for each $n \in \N$.
Similarl... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\map \BB X$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $X$.
Let $\map \BB \C$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of $\C$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert ... | We first show that $\map {\operatorname {EssIm} } f$ is well-defined.
Let $\sequence {D_n}_{n \mathop \in \N}$ and $\sequence {D_n'}_{n \mathop \in \N}$ be [[Definition:Analytic Basis|analytic bases]] of $\C$ consisting of [[Definition:Open Ball|open balls]].
Let $V$ be the [[Definition:Set Union|union]] of those $D_... | Essential Image is Well-Defined and Smallest Closed Set containing Almost All Function Values/Resolution of the Identity | https://proofwiki.org/wiki/Essential_Image_is_Well-Defined_and_Smallest_Closed_Set_containing_Almost_All_Function_Values/Resolution_of_the_Identity | https://proofwiki.org/wiki/Essential_Image_is_Well-Defined_and_Smallest_Closed_Set_containing_Almost_All_Function_Values/Resolution_of_the_Identity | [
"Resolutions of the Identity"
] | [
"Definition:Topological Space",
"Definition:Borel Sigma-Algebra",
"Definition:Borel Sigma-Algebra",
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Resolution of the Identity",
"Definition:Measurable Function",
"Definition:Essential Image/Resolution of the... | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Open Ball",
"Definition:Set Union",
"Definition:Set Union",
"Null Sets Closed under Countable Union/Resolution of the Identity",
"Preimage of Union under Mapping/General Result",
"Definition:Basis (Topology)/Analytic Basis",
"Null Sets Closed u... |
proofwiki-22214 | Canonical Preordering of C*-Algebra is Antisymmetric | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\le_A$ be the canonical preordering of $A$.
Let $a, b \in A$ such that:
:$a \le_A b$
and:
:$b \le_A a$
Then $a = b$.
That is, $\le_A$ is antisymmetric and hence $\le_A$ is a partial ordering. | From the definition of $\le_A$ we have:
:$b - a$ is positive.
Hence $b - a$ is Hermitian and:
:$\map {\sigma_A} {b - a} \subseteq \hointr 0 \infty$
From the Spectral Mapping Theorem for Polynomials, we also have:
:$\map {\sigma_A} {a - b} = \set {-t : t \in \map {\sigma_A} {a - b} } \subseteq \hointl {-\infty} 0$
Agai... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $a, b \in A$ such that:
:$a \le_A b$
and:
:$b \le_A a$
Then $a = b$.
That is, $\le_A$ is [[Definition:Antisymm... | From the definition of $\le_A$ we have:
:$b - a$ is [[Definition:Positive Element of C*-Algebra|positive]].
Hence $b - a$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]] and:
:$\map {\sigma_A} {b - a} \subseteq \hointr 0 \infty$
From the [[Spectral Mapping Theorem for Polynomials]], we also have:
:$\map {\... | Canonical Preordering of C*-Algebra is Antisymmetric | https://proofwiki.org/wiki/Canonical_Preordering_of_C*-Algebra_is_Antisymmetric | https://proofwiki.org/wiki/Canonical_Preordering_of_C*-Algebra_is_Antisymmetric | [
"Canonical Preorderings of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Antisymmetric Relation",
"Definition:Partial Ordering"
] | [
"Definition:Positive Element of C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Spectral Mapping Theorem for Polynomials",
"Definition:Positive Element of C*-Algebra",
"Spectrum of Element of Banach Algebra is Non-Empty",
"Definition:Spectral Radius",
"Spectral Radius of Normal Element of C*-... |
proofwiki-22215 | Characterization of Injective Linear Transformations with Closed Image | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be Banach spaces.
Let $T : X \to Y$ be a bounded linear transformation.
Then $T$ is injective and $\Img T$ is closed {{iff}}:
:there exists $c > 0$ such that $\norm {T x}_Y \ge c \norm x_X$ for each $x \in X$. | === Sufficient Condition ===
Suppose that:
:there exists $c > 0$ such that $\norm {T x}_Y \ge c \norm x_X$ for each $x \in X$.
First, if $T x = {\mathbf 0}_Y$ for some $x \in X$, then we have:
:$c \norm x_X \le 0$
Since $c > 0$ and $\norm x_X \ge 0$, we have $\norm x_X = 0$.
Hence by {{NormAxiomVector|1}}, we have $x =... | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Banach Space|Banach spaces]].
Let $T : X \to Y$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Then $T$ is [[Definition:Injection|injective]] and $\Img T$ is [[Definition:Closed Set|cl... | === Sufficient Condition ===
Suppose that:
:there exists $c > 0$ such that $\norm {T x}_Y \ge c \norm x_X$ for each $x \in X$.
First, if $T x = {\mathbf 0}_Y$ for some $x \in X$, then we have:
:$c \norm x_X \le 0$
Since $c > 0$ and $\norm x_X \ge 0$, we have $\norm x_X = 0$.
Hence by {{NormAxiomVector|1}}, we have ... | Characterization of Injective Linear Transformations with Closed Image | https://proofwiki.org/wiki/Characterization_of_Injective_Linear_Transformations_with_Closed_Image | https://proofwiki.org/wiki/Characterization_of_Injective_Linear_Transformations_with_Closed_Image | [
"Characterization of Injective Linear Transformations with Closed Image",
"Bounded Linear Transformations",
"Banach Spaces",
"Characterization of Injective Linear Transformations with Closed Image"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Transformation",
"Definition:Injection",
"Definition:Closed Set"
] | [
"Definition:Injection",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Linear Transformation",
"Definition:Convergent Sequence",
"Definition:Cauchy Sequence",
"Convergent Sequence is Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Banach Space",... |
proofwiki-22216 | Characterization of Spectrum of Bounded Linear Operator in Hilbert Space in terms of Approximate Eigenvalues | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded linear operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then $\lambda \in \map \sigma T$ {{iff}} $\lambda$ is an approximate eigenvalue of $T$ or $\overline \lambda$ is an eigenvalue of $T^\ast$. | === Sufficient Condition ===
Suppose that $\lambda$ is an approximate eigenvalue of $T$.
Then there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\HH$ such that:
:$\norm {x_n} = 1$ for each $n \in \N$
and:
:$\norm {T x_n - \lambda x_n} \to 0$ as $n \to \infty$.
By Characterization of Injective Linear Tra... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map \sigma T$ be the [[Definition:Spectrum (Spectral Theory)/Bounded Linear Operator|spectrum]] of $T$.
Then $\lambd... | === Sufficient Condition ===
Suppose that $\lambda$ is an [[Definition:Approximate Eigenvalue of Bounded Linear Operator|approximate eigenvalue]] of $T$.
Then there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ in $\HH$ such that:
:$\norm {x_n} = 1$ for each $n \in \N$
and:
:$\norm {T... | Characterization of Spectrum of Bounded Linear Operator in Hilbert Space in terms of Approximate Eigenvalues | https://proofwiki.org/wiki/Characterization_of_Spectrum_of_Bounded_Linear_Operator_in_Hilbert_Space_in_terms_of_Approximate_Eigenvalues | https://proofwiki.org/wiki/Characterization_of_Spectrum_of_Bounded_Linear_Operator_in_Hilbert_Space_in_terms_of_Approximate_Eigenvalues | [
"Approximate Eigenvalues (Bounded Linear Operators)",
"Characterization of Spectrum of Bounded Linear Operator in Hilbert Space in terms of Approximate Eigenvalues"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Approximate Eigenvalue/Bounded Linear Operator",
"Definition:Eigenvalue/Linear Operator"
] | [
"Definition:Approximate Eigenvalue/Bounded Linear Operator",
"Definition:Sequence",
"Characterization of Injective Linear Transformations with Closed Range/Corollary",
"Definition:Injection",
"Definition:Closed Set",
"Definition:Closed Set",
"Definition:Surjection",
"Definition:Eigenvalue/Linear Opera... |
proofwiki-22217 | Eigenvalue of Bounded Linear Operator is Approximate Eigenvalue | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $T : \HH \to \HH$ be a bounded linear operator.
Let $\lambda \in \C$ be an eigenvalue of $T$.
Then $\lambda$ is an approximate eigenvalue of $T$. | Since $\lambda$ is an eigenvalue of $T$, there exists $y \in \HH$ with $y \ne {\mathbf 0}_\HH$ such that:
:$T y = \lambda y$
Setting:
:$\ds x_n = \frac y {\norm y}$
we have $\norm {x_n} = 1$ by {{NormAxiomVector|2}} and:
:$T x_n = \lambda x_n$
We then have:
:$\norm {T x_n - \lambda x_n} = 0$ for each $n \in \N$.
So:
:... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $T : \HH \to \HH$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\lambda \in \C$ be an [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $T$.
Then $\lambda$ is an [[Definiti... | Since $\lambda$ is an [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $T$, there exists $y \in \HH$ with $y \ne {\mathbf 0}_\HH$ such that:
:$T y = \lambda y$
Setting:
:$\ds x_n = \frac y {\norm y}$
we have $\norm {x_n} = 1$ by {{NormAxiomVector|2}} and:
:$T x_n = \lambda x_n$
We then have:
:$\norm {T x_n... | Eigenvalue of Bounded Linear Operator is Approximate Eigenvalue | https://proofwiki.org/wiki/Eigenvalue_of_Bounded_Linear_Operator_is_Approximate_Eigenvalue | https://proofwiki.org/wiki/Eigenvalue_of_Bounded_Linear_Operator_is_Approximate_Eigenvalue | [
"Approximate Eigenvalues",
"Approximate Eigenvalues (Bounded Linear Operators)",
"Approximate Eigenvalues (Bounded Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Eigenvalue/Linear Operator",
"Definition:Approximate Eigenvalue/Bounded Linear Operator"
] | [
"Definition:Eigenvalue/Linear Operator",
"Definition:Approximate Eigenvalue/Bounded Linear Operator",
"Category:Approximate Eigenvalues (Bounded Linear Operators)"
] |
proofwiki-22218 | Space of Bounded Linear Operators on Hilbert Space is Unital C*-Algebra | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\map B \HH$ be the space of bounded linear operators.
Let $\norm {\, \cdot \,}_{\map B \HH}$ be the operator norm on $\map B \HH$.
Let $\ast : \map B \HH \to \map B \HH$ denote the adjoint operation.
Then $\struct {\map B \HH, \ast, \norm {\... | From Space of Bounded Linear Transformations is Unital Banach Algebra, $\struct {\map B \HH, \norm {\, \cdot \,} }$ is a unital Banach algebra. | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\map B \HH$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear operators]].
Let $\norm {\, \cdot \,}_{\map B \HH}$ be the [[Definition:Operator Norm|operator norm]] on $\map B \... | From [[Space of Bounded Linear Transformations is Unital Banach Algebra]], $\struct {\map B \HH, \norm {\, \cdot \,} }$ is a [[Definition:Unital Banach Algebra|unital Banach algebra]]. | Space of Bounded Linear Operators on Hilbert Space is Unital C*-Algebra | https://proofwiki.org/wiki/Space_of_Bounded_Linear_Operators_on_Hilbert_Space_is_Unital_C*-Algebra | https://proofwiki.org/wiki/Space_of_Bounded_Linear_Operators_on_Hilbert_Space_is_Unital_C*-Algebra | [
"C*-Algebras"
] | [
"Definition:Hilbert Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Operator Norm",
"Definition:Adjoint Linear Transformation",
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra"
] | [
"Space of Bounded Linear Transformations is Unital Banach Algebra",
"Definition:Unital Banach Algebra",
"Definition:Unital Banach Algebra"
] |
proofwiki-22219 | Characterization of Homeomorphic Topological Spaces/Necessary Condition | Let $T_1 = \struct {S_1, \tau_1}$ be topological space.
Let $S_2$ be a set.
Let $\tau_2$ be a subset of the powerset $\powerset {S_2}$.
Let $\struct {S_2, \tau_2}$ be a topological space homeomorphic to $T_1$.
Then:
:there exists a mapping $f : S_1 \to S_2$:
::$(1): \quad f$ is a bijection
::$(2): \quad f^\to \restrict... | Let $f: S_1 \to S_2$ be a homeomorphism.
By definition of a homeomorphism:
:$f$ is a bijection
:$f$ is an open mapping
:$f$ is a continuous mapping
By definition of an open mapping:
:$\forall U \in \tau_1 : f \sqbrk U \in \tau_2$
By definition of direct image mapping:
:$\forall U \in \tau_1 : \map {f^\to} U \in \tau_2$... | Let $T_1 = \struct {S_1, \tau_1}$ be [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $\tau_2$ be a [[Definition:Subset|subset]] of the [[Definition:Powerset|powerset]] $\powerset {S_2}$.
Let $\struct {S_2, \tau_2}$ be a [[Definition:Topological Space|topological space... | Let $f: S_1 \to S_2$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
By definition of a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]]:
:$f$ is a [[Definition:Bijection|bijection]]
:$f$ is an [[Definition:Open Mapping|open mapping]]
:$f$ is a [[Definition:Continuous Mapping (Top... | Characterization of Homeomorphic Topological Spaces/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Homeomorphic_Topological_Spaces/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Homeomorphic_Topological_Spaces/Necessary_Condition | [
"Characterization of Homeomorphic Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Restriction/Mapping",
"Definition:Direct... | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Bijection",
"Definition:Open Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Mapping",
"Definition:Direct Image Mapping",
"Definition:Continuous Mapping (Topology)",
"Imag... |
proofwiki-22220 | Characterization of Homeomorphic Topological Spaces/Sufficient Condition | Let $T_1 = \struct{S_1, \tau_1}$ be topological space.
Let $S_2$ be a set.
Let $\tau_2$ be a subset of the powerset $\powerset {S_2}$.
Let $f : S_1 \to S_2$ be a mapping such that:
:$(1)\quad f$ is a bijection
:$(2)\quad f^\to \restriction_{\tau_1}$ is a surjection from $\tau_1$ onto $\tau_2$
where
:$f^\to \restriction... | From Direct Image Mapping is Bijection iff Mapping is Bijection
:$f^\to$ is a bijection
From Restriction of Injection is Injection:
:$f^\to \restriction_{\Sigma_L}$ is an injection
Hence $f^\to \restriction_{\Sigma_L}$ is a bijection onto $\Sigma'_L$.
==== $\tau_2$ satisfies Open Set Axiom $(\text O 1)$ ====
Let $\set{... | Let $T_1 = \struct{S_1, \tau_1}$ be [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $\tau_2$ be a [[Definition:Subset|subset]] of the [[Definition:Powerset|powerset]] $\powerset {S_2}$.
Let $f : S_1 \to S_2$ be a [[Definition:Mapping|mapping]] such that:
:$(1)\quad f$... | From [[Direct Image Mapping is Bijection iff Mapping is Bijection]]
:$f^\to$ is a [[Definition:Bijection|bijection]]
From [[Restriction of Injection is Injection]]:
:$f^\to \restriction_{\Sigma_L}$ is an [[Definition:Injection|injection]]
Hence $f^\to \restriction_{\Sigma_L}$ is a [[Definition:Bijection|bijection]] ... | Characterization of Homeomorphic Topological Spaces/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Homeomorphic_Topological_Spaces/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Homeomorphic_Topological_Spaces/Sufficient_Condition | [
"Characterization of Homeomorphic Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Surjection",
"Definition:Restriction/Mapping",
"Definition:Direct Image Mapping",
"Definition:Topological Space",
"Definition:Homeomorphism/Top... | [
"Direct Image Mapping is Bijection iff Mapping is Bijection",
"Definition:Bijection",
"Restriction of Injection is Injection",
"Definition:Injection",
"Definition:Bijection",
"Definition:Indexing Set/Family of Sets",
"Definition:Bijection",
"Image of Union under Mapping",
"Definition:Bijection",
"... |
proofwiki-22221 | Spectrum of Projection in *-Algebra | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $p$ be a projection on $A$.
Let $\map {\sigma_A} p$ be the spectrum of $p$ in $A$.
Then $\map {\sigma_A} p \subseteq \set {0, 1}$. | From the definition of a projection, we have $p^2 = p$.
Hence $p^2 - p = {\mathbf 0}_A$.
Hence from Spectrum of Zero Vector in Algebra, we have $\map {\sigma_A} {p^2 - p} = \map {\sigma_A} { {\mathbf 0}_A} = \set 0$.
From the Spectral Mapping Theorem for Polynomials, we have:
:$\set 0 = \set {z^2 - z : z \in \map {\sig... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $p$ be a [[Definition:Projection (*-Algebras)|projection]] on $A$.
Let $\map {\sigma_A} p$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $p$ in $A$.
Then $\map {\sigma_A} p \subseteq \set {0, 1}$. | From the definition of a [[Definition:Projection (*-Algebras)|projection]], we have $p^2 = p$.
Hence $p^2 - p = {\mathbf 0}_A$.
Hence from [[Spectrum of Zero Vector in Algebra]], we have $\map {\sigma_A} {p^2 - p} = \map {\sigma_A} { {\mathbf 0}_A} = \set 0$.
From the [[Spectral Mapping Theorem for Polynomials]], we... | Spectrum of Projection in *-Algebra | https://proofwiki.org/wiki/Spectrum_of_Projection_in_*-Algebra | https://proofwiki.org/wiki/Spectrum_of_Projection_in_*-Algebra | [
"Projections (*-Algebras)",
"Spectra (Spectral Theory)",
"Spectrum of Projection in *-Algebra"
] | [
"Definition:*-Algebra",
"Definition:Projection (*-Algebras)",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Projection (*-Algebras)",
"Spectrum of Zero Vector in Algebra",
"Spectral Mapping Theorem for Polynomials",
"Category:Projections (*-Algebras)",
"Category:Spectra (Spectral Theory)",
"Category:Spectrum of Projection in *-Algebra"
] |
proofwiki-22222 | Spectrum of Zero Vector in Algebra | Let $A$ be an algebra over $\C$.
Let ${\mathbf 0}_A$ be the zero vector of $A$.
Let $\map {\sigma_A} { {\mathbf 0}_A}$ be the spectrum of ${\mathbf 0}_A$ in $A$.
Then:
:$\map {\sigma_A} { {\mathbf 0}_A} = \set 0$ if $A \ne \set { {\mathbf 0}_A}$
:$\map {\sigma_A} { {\mathbf 0}_A} = \O$ if $A = \set { {\mathbf 0}_A}$. | First suppose that $A$ is unital with identity element ${\mathbf 1}_A$.
For $\lambda \in \C \setminus \set 0$, we have:
:$\lambda^{-1} {\mathbf 1}_A \paren {\lambda {\mathbf 1}_A - {\mathbf 0}_A} = {\mathbf 1}_A = \paren {\lambda {\mathbf 1}_A - {\mathbf 0}_A} \lambda^{-1} {\mathbf 1}_A$
Hence $\lambda \not \in \map {\... | Let $A$ be an [[Definition:Algebra over Field|algebra]] over $\C$.
Let ${\mathbf 0}_A$ be the [[Definition:Zero Vector|zero vector]] of $A$.
Let $\map {\sigma_A} { {\mathbf 0}_A}$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of ${\mathbf 0}_A$ in $A$.
Then:
:$\map {\sigma_A} { {\mathbf 0}_A} = \set 0$... | First suppose that $A$ is [[Definition:Unital Algebra|unital]] with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A$.
For $\lambda \in \C \setminus \set 0$, we have:
:$\lambda^{-1} {\mathbf 1}_A \paren {\lambda {\mathbf 1}_A - {\mathbf 0}_A} = {\mathbf 1}_A = \paren {\lambda {\mathbf 1}_A - {\mathbf 0}... | Spectrum of Zero Vector in Algebra | https://proofwiki.org/wiki/Spectrum_of_Zero_Vector_in_Algebra | https://proofwiki.org/wiki/Spectrum_of_Zero_Vector_in_Algebra | [
"Spectra (Spectral Theory)"
] | [
"Definition:Algebra over Field",
"Definition:Zero Vector",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Unital Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unital Algebra",
"Definition:Unitization of Algebra over Field",
"Definition:Unital Algebra",
"Category:Spectra (Spectral Theory)"
] |
proofwiki-22223 | Spectrum of Projection in *-Algebra/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $p$ be a projection on $A$.
Let $\map {\sigma_A} p$ be the spectrum of $p$ in $A$.
Then:
:$p = {\mathbf 0}_A$ {{iff}} $\map {\sigma_A} p = \set 0$
:$p = {\mathbf 1}_A$ {{iff}} $\map {\sigma_A} p = \set 1$
:$p \not \in \set { {\mathbf ... | Note first that since $p$ is Hermitian, it is normal. | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $p$ be a [[Definition:Projection (*-Algebras)|projection]] on $A$.
Let $\map {\sigma_A} p$ be the [[Definition:Spectrum (Spectral Theory)|spectrum]] of $p$ in $A$.
The... | Note first that since $p$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]], it is [[Definition:Normal Element of *-Algebra|normal]]. | Spectrum of Projection in *-Algebra/Corollary | https://proofwiki.org/wiki/Spectrum_of_Projection_in_*-Algebra/Corollary | https://proofwiki.org/wiki/Spectrum_of_Projection_in_*-Algebra/Corollary | [
"Spectrum of Projection in *-Algebra"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Projection (*-Algebras)",
"Definition:Spectrum (Spectral Theory)"
] | [
"Definition:Hermitian Element of *-Algebra",
"Definition:Normal Element of *-Algebra"
] |
proofwiki-22224 | Normal Element of C*-Algebra is Projection iff Spectrum contains only Zero and One | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $p \in A$ be normal.
Let $\map {\sigma_A} p$ denote the spectrum of $p$ in $A$.
Then $p$ is a projection {{iff}}:
:$\map {\sigma_A} p \subseteq \set {0, 1}$ | We first take $A$ unital. | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $p \in A$ be [[Definition:Normal Element of *-Algebra|normal]].
Let $\map {\sigma_A} p$ denote the [[Definition:Spectrum (Spectral Theory)/Unital Algebra|spectrum]] of ... | We first take $A$ [[Definition:Unital Banach Algebra|unital]]. | Normal Element of C*-Algebra is Projection iff Spectrum contains only Zero and One | https://proofwiki.org/wiki/Normal_Element_of_C*-Algebra_is_Projection_iff_Spectrum_contains_only_Zero_and_One | https://proofwiki.org/wiki/Normal_Element_of_C*-Algebra_is_Projection_iff_Spectrum_contains_only_Zero_and_One | [
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Normal Element of *-Algebra",
"Definition:Spectrum (Spectral Theory)/Unital Algebra",
"Definition:Projection (*-Algebras)"
] | [
"Definition:Unital Banach Algebra"
] |
proofwiki-22225 | Platonic Solid is Isogonal | Let $P$ be a Platonic solid.
Then $P$ is also isogonal. | {{ProofWanted|Various symmetry group realizations, or just rotational arguments needed}}
Category:Platonic Solids
Category:Isogonal Polyhedra
b2hvdb6lu9ewgzshyaal0klbiipx5so | Let $P$ be a [[Definition:Platonic Solid|Platonic solid]].
Then $P$ is also [[Definition:Isogonal Polyhedron|isogonal]]. | {{ProofWanted|Various symmetry group realizations, or just rotational arguments needed}}
[[Category:Platonic Solids]]
[[Category:Isogonal Polyhedra]]
b2hvdb6lu9ewgzshyaal0klbiipx5so | Platonic Solid is Isogonal | https://proofwiki.org/wiki/Platonic_Solid_is_Isogonal | https://proofwiki.org/wiki/Platonic_Solid_is_Isogonal | [
"Platonic Solids",
"Isogonal Polyhedra",
"Platonic Solids",
"Isogonal Polyhedra"
] | [
"Definition:Platonic Solid",
"Definition:Isogonal Polyhedron"
] | [
"Category:Platonic Solids",
"Category:Isogonal Polyhedra"
] |
proofwiki-22226 | Platonic Solid is Uniform Polyhedron | Let $P$ be a Platonic solid.
Then $P$ is also uniform. | {{Recall|Uniform Polyhedron}}
{{:Definition:Uniform Polyhedron}}
{{Recall|Platonic Solid}}
{{:Definition:Platonic Solid}}
From Platonic Solid is Isogonal, $P$ is isogonal.
That all the faces are regular polygons follows {{afortiori}}.
Hence the result.
{{qed}} | Let $P$ be a [[Definition:Platonic Solid|Platonic solid]].
Then $P$ is also [[Definition:Uniform Polyhedron|uniform]]. | {{Recall|Uniform Polyhedron}}
{{:Definition:Uniform Polyhedron}}
{{Recall|Platonic Solid}}
{{:Definition:Platonic Solid}}
From [[Platonic Solid is Isogonal]], $P$ is [[Definition:Isogonal Polyhedron|isogonal]].
That all the [[Definition:Face of Polyhedron|faces]] are [[Definition:Regular Polygon|regular polygons]] f... | Platonic Solid is Uniform Polyhedron | https://proofwiki.org/wiki/Platonic_Solid_is_Uniform_Polyhedron | https://proofwiki.org/wiki/Platonic_Solid_is_Uniform_Polyhedron | [
"Platonic Solids",
"Uniform Polyhedra"
] | [
"Definition:Platonic Solid",
"Definition:Uniform Polyhedron"
] | [
"Platonic Solid is Isogonal",
"Definition:Isogonal Polyhedron",
"Definition:Polyhedron/Face",
"Definition:Polygon/Regular"
] |
proofwiki-22227 | Characterization of Partial Isometries | Let $\struct {\HH_1, \innerprod \cdot \cdot_1}$ and $\struct {\HH_2, \innerprod \cdot \cdot_2}$ be Hilbert spaces.
Let $T : \HH_1 \to \HH_2$ be a bounded linear transformation.
{{TFAE}}
:$(1) \quad$ $T$ is a partial isometry
:$(2) \quad$ $T = T T^\ast T$
:$(3) \quad$ $T^\ast T$ is a Hilbert space projection
:$(4) \qua... | === $(2)$ implies $(3)$ ===
Suppose that $T = T T^\ast T$.
We then have $T^\ast T = T^\ast \paren {T T^\ast T} = \paren {T^\ast T}^2$.
From Product of Element in *-Star Algebra with its Star is Hermitian, $T^\ast T$ is Hermitian.
Hence from Characterization of Projections, $T^\ast T$ is a Hilbert space projection.
{{qe... | Let $\struct {\HH_1, \innerprod \cdot \cdot_1}$ and $\struct {\HH_2, \innerprod \cdot \cdot_2}$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $T : \HH_1 \to \HH_2$ be a [[Definition:Bounded Linear Transformation|bounded linear transformation]].
{{TFAE}}
:$(1) \quad$ $T$ is a [[Definition:Partial Isometry|part... | === $(2)$ implies $(3)$ ===
Suppose that $T = T T^\ast T$.
We then have $T^\ast T = T^\ast \paren {T T^\ast T} = \paren {T^\ast T}^2$.
From [[Product of Element in *-Star Algebra with its Star is Hermitian]], $T^\ast T$ is [[Definition:Hermitian Operator|Hermitian]].
Hence from [[Characterization of Projections]], ... | Characterization of Partial Isometries | https://proofwiki.org/wiki/Characterization_of_Partial_Isometries | https://proofwiki.org/wiki/Characterization_of_Partial_Isometries | [
"Partial Isometries"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Transformation",
"Definition:Partial Isometry",
"Definition:Projection (Hilbert Spaces)",
"Definition:Projection (Hilbert Spaces)"
] | [
"Product of Element in *-Star Algebra with its Star is Hermitian",
"Definition:Hermitian Operator",
"Characterization of Projections",
"Definition:Projection (Hilbert Spaces)",
"Definition:Projection (Hilbert Spaces)",
"Definition:Projection (Hilbert Spaces)",
"Product of Element in *-Star Algebra with ... |
proofwiki-22228 | Uniform Prism is Uniform Polyhedron | Let $P$ be a uniform prism.
Then $P$ is a uniform polyhedron. | {{Recall|Uniform Polyhedron}}
{{:Definition:Uniform Polyhedron}}
{{Recall|Uniform Prism}}
{{:Definition:Uniform Prism}}
From Uniform Prism is Isogonal, $P$ is isogonal.
That all the faces of $P$ are regular polygons follows {{afortiori}} from the definition of regular prism.
Hence the result.
{{qed}} | Let $P$ be a [[Definition:Uniform Prism|uniform prism]].
Then $P$ is a [[Definition:Uniform Polyhedron|uniform polyhedron]]. | {{Recall|Uniform Polyhedron}}
{{:Definition:Uniform Polyhedron}}
{{Recall|Uniform Prism}}
{{:Definition:Uniform Prism}}
From [[Uniform Prism is Isogonal]], $P$ is [[Definition:Isogonal Polyhedron|isogonal]].
That all the [[Definition:Face of Polyhedron|faces]] of $P$ are [[Definition:Regular Polygon|regular polygons... | Uniform Prism is Uniform Polyhedron | https://proofwiki.org/wiki/Uniform_Prism_is_Uniform_Polyhedron | https://proofwiki.org/wiki/Uniform_Prism_is_Uniform_Polyhedron | [
"Uniform Prisms",
"Uniform Polyhedra"
] | [
"Definition:Uniform Prism",
"Definition:Uniform Polyhedron"
] | [
"Uniform Prism is Isogonal",
"Definition:Isogonal Polyhedron",
"Definition:Polyhedron/Face",
"Definition:Polygon/Regular",
"Definition:Regular Prism"
] |
proofwiki-22229 | Uniform Antiprism is Uniform Polyhedron | Let $P$ be a uniform antiprism.
Then $P$ is a uniform polyhedron. | {{Recall|Uniform Polyhedron}}
{{:Definition:Uniform Polyhedron}}
{{Recall|Uniform Antiprism}}
{{:Definition:Uniform Antiprism}}
From Uniform Antiprism is Isogonal, $P$ is isogonal.
Hence the result.
{{qed}} | Let $P$ be a [[Definition:Uniform Antiprism|uniform antiprism]].
Then $P$ is a [[Definition:Uniform Polyhedron|uniform polyhedron]]. | {{Recall|Uniform Polyhedron}}
{{:Definition:Uniform Polyhedron}}
{{Recall|Uniform Antiprism}}
{{:Definition:Uniform Antiprism}}
From [[Uniform Antiprism is Isogonal]], $P$ is [[Definition:Isogonal Polyhedron|isogonal]].
Hence the result.
{{qed}} | Uniform Antiprism is Uniform Polyhedron | https://proofwiki.org/wiki/Uniform_Antiprism_is_Uniform_Polyhedron | https://proofwiki.org/wiki/Uniform_Antiprism_is_Uniform_Polyhedron | [
"Uniform Antiprisms",
"Uniform Polyhedra"
] | [
"Definition:Uniform Antiprism",
"Definition:Uniform Polyhedron"
] | [
"Uniform Antiprism is Isogonal",
"Definition:Isogonal Polyhedron"
] |
proofwiki-22230 | Extension of Bounded Linear Transformation from Closed Subspace of Hilbert Space to Whole Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH_1, \innerprod \cdot \cdot_1}$ and $\struct {\HH_2, \innerprod \cdot \cdot_2}$ be Hilbert spaces over $\GF$.
Let $\KK_1$ and $\KK_2$ be closed linear subspaces of $\HH_1$ and $\HH_2$ respectively.
Let $T_1 : \KK_1 \to \KK_2$ be a bounded linear transformation.
Let $\KK_1^\b... | We first show that $T$ is well-defined.
Let $u_1, u_2 \in \KK_1$ and $v_1, v_2 \in \KK_2$.
From Direct Sum of Subspace and Orthocomplement, we have $\KK_1 \oplus \KK_1^\bot = \HH_1$ as an internal direct sum.
Hence $\KK_1 \cap \KK_1^\bot = \set { {\mathbf 0}_\HH}$, and if $u_1 + v_1 = u_2 + v_2$, then $u_1 = u_2$ and ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH_1, \innerprod \cdot \cdot_1}$ and $\struct {\HH_2, \innerprod \cdot \cdot_2}$ be [[Definition:Hilbert Space|Hilbert spaces]] over $\GF$.
Let $\KK_1$ and $\KK_2$ be [[Definition:Closed Linear Subspace|closed linear subspaces]] of $\HH_1$ and $\HH_2$ respectively.
Let $T_... | We first show that $T$ is well-defined.
Let $u_1, u_2 \in \KK_1$ and $v_1, v_2 \in \KK_2$.
From [[Direct Sum of Subspace and Orthocomplement]], we have $\KK_1 \oplus \KK_1^\bot = \HH_1$ as an [[Definition:Internal Direct Sum of Modules|internal direct sum]].
Hence $\KK_1 \cap \KK_1^\bot = \set { {\mathbf 0}_\HH}$, ... | Extension of Bounded Linear Transformation from Closed Subspace of Hilbert Space to Whole Space | https://proofwiki.org/wiki/Extension_of_Bounded_Linear_Transformation_from_Closed_Subspace_of_Hilbert_Space_to_Whole_Space | https://proofwiki.org/wiki/Extension_of_Bounded_Linear_Transformation_from_Closed_Subspace_of_Hilbert_Space_to_Whole_Space | [
"Bounded Linear Transformations",
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Closed Linear Subspace",
"Definition:Bounded Linear Transformation",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Extension of Mapping"
] | [
"Direct Sum of Subspace and Orthocomplement",
"Definition:Internal Direct Sum of Modules",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation",
"Pythagoras's Theorem (Inner Product Space)",
"Pythagoras's... |
proofwiki-22231 | Polar Decomposition for Bounded Linear Operator on Hilbert Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\GF$.
Let $T : \HH \to \HH$.
Then there exists a unique partial isometry $U : \HH \to \HH$ such that:
:$T = U \cmod T$ and $\map \ker T = \map \ker U$
where $\cmod T$ is the modulus of $T$.
Further, $U^\ast T = \cmod T$. | === Existence ===
From the definition of the modulus, $\cmod T$ is positive and hence Hermitian.
We have, for each $x \in \HH$:
{{begin-eqn}}
{{eqn | l = \norm {\cmod T x}^2
| r = \innerprod {\cmod T x} {\cmod T x}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \innerprod {\cmod T^2 x} x
| c = {{Defof|Adjoint ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\GF$.
Let $T : \HH \to \HH$.
Then there exists a unique [[Definition:Partial Isometry|partial isometry]] $U : \HH \to \HH$ such that:
:$T = U \cmod T$ and $\map \ker T = \map \ker U$
where... | === Existence ===
From the definition of the [[Definition:Modulus of Element of C*-Algebra|modulus]], $\cmod T$ is [[Definition:Positive Element of C*-Algebra|positive]] and hence [[Definition:Hermitian Operator|Hermitian]].
We have, for each $x \in \HH$:
{{begin-eqn}}
{{eqn | l = \norm {\cmod T x}^2
| r = \innerpr... | Polar Decomposition for Bounded Linear Operator on Hilbert Space | https://proofwiki.org/wiki/Polar_Decomposition_for_Bounded_Linear_Operator_on_Hilbert_Space | https://proofwiki.org/wiki/Polar_Decomposition_for_Bounded_Linear_Operator_on_Hilbert_Space | [
"Polar Decompositions",
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Partial Isometry",
"Definition:Modulus of Element of C*-Algebra"
] | [
"Definition:Modulus of Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Hermitian Operator",
"Adjoint is Involutive",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Isometry",
"Bounded Lin... |
proofwiki-22232 | Orthocomplement of Closure | Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $S \subseteq V$ be non-empty.
Then:
:$S^\bot = \map \cl S^\bot$
where $\bot$ denotes orthocomplementation. | From Orthocomplement Reverses Subset, we have:
:$\map \cl S^\bot \subseteq S^\bot$
Conversely, let $y \in S^\bot$.
Let $x \in \map \cl S$.
Then there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that $x_n \to x$.
Since $y \in S^\bot$, we have $\innerprod {x_n} y = 0$ for each $n \in \N$.
From Inne... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]].
Let $S \subseteq V$ be [[Definition:Non-Empty Set|non-empty]].
Then:
:$S^\bot = \map \cl S^\bot$
where $\bot$ denotes [[Definition:Orthocomplement|orthocomplementation]]. | From [[Orthocomplement Reverses Subset]], we have:
:$\map \cl S^\bot \subseteq S^\bot$
Conversely, let $y \in S^\bot$.
Let $x \in \map \cl S$.
Then there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that $x_n \to x$.
Since $y \in S^\bot$, we have $\innerprod {x_n} y = 0... | Orthocomplement of Closure | https://proofwiki.org/wiki/Orthocomplement_of_Closure | https://proofwiki.org/wiki/Orthocomplement_of_Closure | [
"Orthocomplements"
] | [
"Definition:Inner Product Space",
"Definition:Non-Empty Set",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement"
] | [
"Orthocomplement Reverses Subset",
"Definition:Sequence",
"Inner Product is Continuous",
"Category:Orthocomplements"
] |
proofwiki-22233 | *-Algebra Homomorphism between C*-Algebras is Norm-Decreasing | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ and $\struct {B, \dagger, \norm {\, \cdot \,}_B}$ be $\text C^\ast$-algebras.
Let $\phi : A \to B$ be a $\ast$-algebra homomorphism.
Then:
:$\norm {\map \phi x}_B \le \norm x_A$ for each $x \in A$. | Let $x \in A$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\map \phi x}^2_B
| r = \norm {\map \phi x \paren {\map \phi x}^\dagger}_B
| c = {{Defof|C*-Algebra}}
}}
{{eqn | r = \map {r_B} {\map \phi x \paren {\map \phi x}^\dagger}
| c = Spectral Radius of Normal Element of C*-Algebra Equal to Norm
}}
{{eqn | r = \map {... | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ and $\struct {B, \dagger, \norm {\, \cdot \,}_B}$ be [[Definition:C*-Algebra|$\text C^\ast$-algebras]].
Let $\phi : A \to B$ be a [[Definition:*-Algebra Homomorphism|$\ast$-algebra homomorphism]].
Then:
:$\norm {\map \phi x}_B \le \norm x_A$ for each $x \in A$. | Let $x \in A$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\map \phi x}^2_B
| r = \norm {\map \phi x \paren {\map \phi x}^\dagger}_B
| c = {{Defof|C*-Algebra}}
}}
{{eqn | r = \map {r_B} {\map \phi x \paren {\map \phi x}^\dagger}
| c = [[Spectral Radius of Normal Element of C*-Algebra Equal to Norm]]
}}
{{eqn | r = \... | *-Algebra Homomorphism between C*-Algebras is Norm-Decreasing | https://proofwiki.org/wiki/*-Algebra_Homomorphism_between_C*-Algebras_is_Norm-Decreasing | https://proofwiki.org/wiki/*-Algebra_Homomorphism_between_C*-Algebras_is_Norm-Decreasing | [
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:*-Algebra Homomorphism"
] | [
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Spectrum of Image of Element of Unital Algebra under Unital Algebra Homomorphism/Corollary",
"Spectral Radius of Normal Element of C*-Algebra Equal to Norm",
"Category:C*-Algebras"
] |
proofwiki-22234 | Linear Functional on *-Algebra Real-Valued at Hermitian Elements preserves Star | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $f : A \to \C$ be a linear functional such that:
:for each $x \in A$ Hermitian, we have $\map \phi x \in \R$.
Then:
:$\overline {\map \phi x} = \map \phi {x^\ast}$ for each $x \in A$. | Let $x \in A$.
From Element of *-Algebra Uniquely Decomposes into Hermitian Elements, there exists Hermitian $b, c \in A$ such that:
:$x = b + i c$
Then:
:$x^\ast = b - i c$
by $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in the definition of an involution.
By hypothesis we have $\map \phi b, \map \phi c \in \R$.
Then we ... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $f : A \to \C$ be a [[Definition:Linear Functional|linear functional]] such that:
:for each $x \in A$ [[Definition:Hermitian Element of *-Algebra|Hermitian]], we have $\map \phi x \in \R$.
Then:
:$\overline {\map \phi x} = \map \phi ... | Let $x \in A$.
From [[Element of *-Algebra Uniquely Decomposes into Hermitian Elements]], there exists [[Definition:Hermitian Element of *-Algebra|Hermitian]] $b, c \in A$ such that:
:$x = b + i c$
Then:
:$x^\ast = b - i c$
by $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in the definition of an [[Definition:Involution o... | Linear Functional on *-Algebra Real-Valued at Hermitian Elements preserves Star | https://proofwiki.org/wiki/Linear_Functional_on_*-Algebra_Real-Valued_at_Hermitian_Elements_preserves_Star | https://proofwiki.org/wiki/Linear_Functional_on_*-Algebra_Real-Valued_at_Hermitian_Elements_preserves_Star | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Linear Functional",
"Definition:Hermitian Element of *-Algebra"
] | [
"Element of *-Algebra Uniquely Decomposes into Hermitian Elements",
"Definition:Hermitian Element of *-Algebra",
"Definition:Involution on Algebra",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Category:*-Algebras"
] |
proofwiki-22235 | Positive Linear Functional on C*-Algebra preserves Star | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\phi : A \to \C$ be a positive linear functional.
Then:
:$\overline {\map \phi x} = \map \phi {x^\ast}$ | From Linear Functional on *-Algebra Real-Valued at Hermitian Elements preserves Star, it is enough to show that:
:$\map \phi x \in \R$ whenever $x \in A$ is Hermitian.
This is precisely Positive Linear Functional on C*-Algebra is Real on Hermitian Elements.
{{qed}}
Category:Positive Linear Functionals
Category:C*-Algeb... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\phi : A \to \C$ be a [[Definition:Positive Linear Functional/C*-Algebra|positive linear functional]].
Then:
:$\overline {\map \phi x} = \map \phi {x^\ast}$ | From [[Linear Functional on *-Algebra Real-Valued at Hermitian Elements preserves Star]], it is enough to show that:
:$\map \phi x \in \R$ whenever $x \in A$ is [[Definition:Hermitian Element of *-Algebra|Hermitian]].
This is precisely [[Positive Linear Functional on C*-Algebra is Real on Hermitian Elements]].
{{qed}}... | Positive Linear Functional on C*-Algebra preserves Star | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_preserves_Star | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_preserves_Star | [
"Positive Linear Functionals",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Positive Linear Functional/C*-Algebra"
] | [
"Linear Functional on *-Algebra Real-Valued at Hermitian Elements preserves Star",
"Definition:Hermitian Element of *-Algebra",
"Positive Linear Functional on C*-Algebra is Real on Hermitian Elements",
"Category:Positive Linear Functionals",
"Category:C*-Algebras"
] |
proofwiki-22236 | Bound on Norm of Real and Imaginary Parts of Element of Banach *-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a Banach $\ast$-algebra.
Let $x \in A$.
Let $\map \Re x$ and $\map \Im x$ be the real and imaginary parts of $x$ respectively.
Then:
:$\norm {\map \Re x} \le \norm x$
and:
:$\norm {\map \Im x} \le \norm x$
for each $x \in A$. | Let $x \in A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {\frac 1 2 \paren {x + x^\ast} }
| r = \frac 1 2 \norm {x + x^\ast}
| c = {{NormAxiomVector|2}}
}}
{{eqn | o = \le
| r = \frac 1 2 \paren {\norm x + \norm {x^\ast} }
| c = {{NormAxiomVector|3}}
}}
{{eqn | r = \frac 1 2 \paren {\norm x + \norm x}
| ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Banach *-Algebra|Banach $\ast$-algebra]].
Let $x \in A$.
Let $\map \Re x$ and $\map \Im x$ be the [[Definition:Real Part of Element of *-Algebra|real]] and [[Definition:Imaginary Part of Element of *-Algebra|imaginary parts]] of $x$ respectively.
Then:... | Let $x \in A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {\frac 1 2 \paren {x + x^\ast} }
| r = \frac 1 2 \norm {x + x^\ast}
| c = {{NormAxiomVector|2}}
}}
{{eqn | o = \le
| r = \frac 1 2 \paren {\norm x + \norm {x^\ast} }
| c = {{NormAxiomVector|3}}
}}
{{eqn | r = \frac 1 2 \paren {\norm x + \norm x}
|... | Bound on Norm of Real and Imaginary Parts of Element of Banach *-Algebra | https://proofwiki.org/wiki/Bound_on_Norm_of_Real_and_Imaginary_Parts_of_Element_of_Banach_*-Algebra | https://proofwiki.org/wiki/Bound_on_Norm_of_Real_and_Imaginary_Parts_of_Element_of_Banach_*-Algebra | [
"Banach *-Algebras"
] | [
"Definition:Banach *-Algebra",
"Definition:Real Part of Element of *-Algebra",
"Definition:Imaginary Part of Element of *-Algebra"
] | [
"Category:Banach *-Algebras"
] |
proofwiki-22237 | Linear Transformation from Banach *-Algebra is Bounded if Bounded on Hermitian Elements | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ be a Banach $\ast$-algebra.
Let $\struct {B, \norm {\, \cdot \,}_B}$ be a normed vector space over $\C$.
Let $T : A \to B$ be a linear transformation such that there exists $M > 0$ with:
:$\norm {T x}_B \le M \norm x_A$ for each $x \in A$ Hermitian.
Then $T$ is bounded. | Let $x \in A$.
From Element of *-Algebra Uniquely Decomposes into Hermitian Elements, we have:
:$x = \map \Re x + i \map \Im x$
where $\map \Re x$ and $\map \Im x$ are Hermitian.
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {T x}_B
| r = \norm {\map T {\map \Re x} + i \map T {\map \Im x} }_B
| c = {{Defof|Linear T... | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ be a [[Definition:Banach *-Algebra|Banach $\ast$-algebra]].
Let $\struct {B, \norm {\, \cdot \,}_B}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\C$.
Let $T : A \to B$ be a [[Definition:Linear Transformation|linear transformation]] such that there e... | Let $x \in A$.
From [[Element of *-Algebra Uniquely Decomposes into Hermitian Elements]], we have:
:$x = \map \Re x + i \map \Im x$
where $\map \Re x$ and $\map \Im x$ are [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {T x}_B
| r = \norm {\map T {\map \Re x} ... | Linear Transformation from Banach *-Algebra is Bounded if Bounded on Hermitian Elements | https://proofwiki.org/wiki/Linear_Transformation_from_Banach_*-Algebra_is_Bounded_if_Bounded_on_Hermitian_Elements | https://proofwiki.org/wiki/Linear_Transformation_from_Banach_*-Algebra_is_Bounded_if_Bounded_on_Hermitian_Elements | [
"Banach *-Algebras"
] | [
"Definition:Banach *-Algebra",
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Hermitian Element of *-Algebra",
"Definition:Bounded Linear Transformation"
] | [
"Element of *-Algebra Uniquely Decomposes into Hermitian Elements",
"Definition:Hermitian Element of *-Algebra",
"Bound on Norm of Real and Imaginary Parts of Element of Banach *-Algebra",
"Definition:Bounded Linear Transformation",
"Category:Banach *-Algebras"
] |
proofwiki-22238 | Sum of Two Positive Elements of C*-Algebra is Positive/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $a_1, a_2, \ldots, a_n \in A$ be positive.
Then:
:$\ds \sum_{j \mathop = 1}^n a_j$ is positive. | Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:for all $a_1, a_2, \ldots, a_n \in A$ positive, $\ds \sum_{j \mathop = 1}^n a_j$ is positive. | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $a_1, a_2, \ldots, a_n \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Then:
:$\ds \sum_{j \mathop = 1}^n a_j$ is [[Definition:Positive Element of C*-Algebra|positive]]. | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:for all $a_1, a_2, \ldots, a_n \in A$ [[Definition:Positive Element of C*-Algebra|positive]], $\ds \sum_{j \mathop = 1}^n a_j$ is [[Definition:Positive Element of C*-Algebra|po... | Sum of Two Positive Elements of C*-Algebra is Positive/Corollary | https://proofwiki.org/wiki/Sum_of_Two_Positive_Elements_of_C*-Algebra_is_Positive/Corollary | https://proofwiki.org/wiki/Sum_of_Two_Positive_Elements_of_C*-Algebra_is_Positive/Corollary | [
"Sum of Two Positive Elements of C*-Algebra is Positive"
] | [
"Definition:C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:... |
proofwiki-22239 | Positive Linear Functional on C*-Algebra is Bounded | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ be a $\text C^\ast$-algebra.
Let $f : A \to \C$ be a positive linear functional.
Then $f$ is bounded. | From Linear Transformation from C*-Algebra is Bounded if Bounded on Positive Elements, it is enough to show that:
:there exists $c > 0$ such that for all positive $x \in A$ we have $\cmod {\map f x} \le c \norm x$.
{{AimForCont}} suppose that:
:there does not exist $c > 0$ such that for all positive $x \in A$ we have $... | Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $f : A \to \C$ be a [[Definition:Positive Linear Functional|positive linear functional]].
Then $f$ is [[Definition:Bounded Linear Functional|bounded]]. | From [[Linear Transformation from C*-Algebra is Bounded if Bounded on Positive Elements]], it is enough to show that:
:there exists $c > 0$ such that for all [[Definition:Positive Element of C*-Algebra|positive]] $x \in A$ we have $\cmod {\map f x} \le c \norm x$.
{{AimForCont}} suppose that:
:there does not exist $c ... | Positive Linear Functional on C*-Algebra is Bounded | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_is_Bounded | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_is_Bounded | [
"Positive Linear Functionals",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Positive Linear Functional",
"Definition:Bounded Linear Functional"
] | [
"Linear Transformation from C*-Algebra is Bounded if Bounded on Positive Elements",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Non-Negative Multiple of Positive Element... |
proofwiki-22240 | Norm of Positive Linear Functional on Unital C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra with identity element ${\mathbf 1}_A$.
Let $f : A \to \C$ be a positive linear functional.
Let $\norm {\, \cdot \,}_{A^\ast}$ be the norm of a bounded linear functional.
Then $f$ is bounded and:
:$\norm f_{A^\ast} = \map f { {\mathbf 1}_A... | From Positive Linear Functional on C*-Algebra is Bounded, $f$ is bounded.
We show that:
:$\cmod {\map f x} \le \map f { {\mathbf 1}_A}$
for each $x \in A$ with $\norm x \le 1$.
Since equality is attained for $x = {\mathbf 1}_A$, we will then have $\norm f_{A^\ast} = \map f { {\mathbf 1}_A}$.
We have, for all $u \in A$... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]] with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A$.
Let $f : A \to \C$ be a [[Definition:Positive Linear Functional|positive linear functional]].
Let $\no... | From [[Positive Linear Functional on C*-Algebra is Bounded]], $f$ is [[Definition:Bounded Linear Functional|bounded]].
We show that:
:$\cmod {\map f x} \le \map f { {\mathbf 1}_A}$
for each $x \in A$ with $\norm x \le 1$.
Since equality is attained for $x = {\mathbf 1}_A$, we will then have $\norm f_{A^\ast} = \map ... | Norm of Positive Linear Functional on Unital C*-Algebra | https://proofwiki.org/wiki/Norm_of_Positive_Linear_Functional_on_Unital_C*-Algebra | https://proofwiki.org/wiki/Norm_of_Positive_Linear_Functional_on_Unital_C*-Algebra | [
"Positive Linear Functionals",
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Positive Linear Functional",
"Definition:Norm/Bounded Linear Functional",
"Definition:Bounded Linear Functional"
] | [
"Positive Linear Functional on C*-Algebra is Bounded",
"Definition:Bounded Linear Functional",
"Positive Linear Functional on C*-Algebra preserves Star",
"Definition:Linear Functional",
"Element of *-Algebra Uniquely Decomposes into Hermitian Elements",
"Definition:Hermitian Element of *-Algebra",
"Boun... |
proofwiki-22241 | Positive Linear Functional on C*-Algebra is Real on Hermitian Elements | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\phi : A \to \C$ be a positive linear functional.
Let $x \in A$ be Hermitian.
Then $\map \phi x \in \R$. | From Hermitian Element of C*-Algebra Decomposes into Positive Elements, there exists positive elements $x^+, x^- \in A$ such that:
:$x = x^+ - x^-$
Since $f$ is positive, we have $\map \phi {x^+} \ge 0$ and $\map \phi {x^-} \ge 0$.
Hence from linearity:
:$\map \phi x = \map \phi {x^+} - \map \phi {x^-} \in \R$
{{qed}... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\phi : A \to \C$ be a [[Definition:Positive Linear Functional/C*-Algebra|positive linear functional]].
Let $x \in A$ be [[Definition:Hermitian Element of *-Algebra|Hermitian]].
Then $\map \phi x \in \R$. | From [[Hermitian Element of C*-Algebra Decomposes into Positive Elements]], there exists [[Definition:Positive Element of C*-Algebra|positive elements]] $x^+, x^- \in A$ such that:
:$x = x^+ - x^-$
Since $f$ is [[Definition:Positive Linear Functional/C*-Algebra|positive]], we have $\map \phi {x^+} \ge 0$ and $\map \ph... | Positive Linear Functional on C*-Algebra is Real on Hermitian Elements | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_is_Real_on_Hermitian_Elements | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_is_Real_on_Hermitian_Elements | [
"Positive Linear Functionals",
"C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Positive Linear Functional/C*-Algebra",
"Definition:Hermitian Element of *-Algebra"
] | [
"Hermitian Element of C*-Algebra Decomposes into Positive Elements",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Linear Functional/C*-Algebra",
"Definition:Linear Functional",
"Category:Positive Linear Functionals",
"Category:C*-Algebras"
] |
proofwiki-22242 | Positive Linear Functional on C*-Algebra is Increasing on Hermitian Elements | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra with identity element ${\mathbf 1}_A$.
Let $f : A \to \C$ be a positive linear functional.
Let $\le_A$ be the canonical preordering of $A$.
Let $x, y \in A$ be Hermitian such that $x \le_A y$.
Then $\map f x \le \map f y$. | From the definition of the canonical preordering, we have:
:$y - x$ is positive.
From the definition of a positive linear functional we have:
:$\map f {y - x} \ge 0$
Since $f$ is linear, we have:
:$\map f y - \map f x \ge 0$
From Positive Linear Functional on C*-Algebra is Real on Hermitian Elements, we have $\map f x... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]] with [[Definition:Identity Element|identity element]] ${\mathbf 1}_A$.
Let $f : A \to \C$ be a [[Definition:Positive Linear Functional|positive linear functional]].
Let $\le... | From the definition of the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]], we have:
:$y - x$ is [[Definition:Positive Element of C*-Algebra|positive]].
From the definition of a [[Definition:Positive Linear Functional|positive linear functional]] we have:
:$\map f {y - x} \ge 0$
Since $f$ is ... | Positive Linear Functional on C*-Algebra is Increasing on Hermitian Elements | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_is_Increasing_on_Hermitian_Elements | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_is_Increasing_on_Hermitian_Elements | [
"Positive Linear Functionals",
"C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Positive Linear Functional",
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Hermitian Element of *-Algebra"
] | [
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Linear Functional",
"Definition:Linear Functional",
"Positive Linear Functional on C*-Algebra is Real on Hermitian Elements",
"Category:Positive Linear Functionals",
"Category:C*-Algebras"... |
proofwiki-22243 | Positive Linear Functional on C*-Algebra induces Semi-Inner Product | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $f : A \to \C$ be a positive linear functional.
Define $\innerprod \cdot \cdot : A^2 \to A$ by:
:$\innerprod x y = \map f {y^\ast x}$
for each $x, y \in A$.
Then $\innerprod \cdot \cdot$ is a semi-inner product on $A$. | === Proof of Conjugate Symmetry ===
Let $x, y \in A$ we have:
{{begin-eqn}}
{{eqn | l = \overline {\innerprod x y}
| r = \overline {\map f {y^\ast x} }
}}
{{eqn | r = \map f {\paren {y^\ast x}^\ast}
| c = Positive Linear Functional on C*-Algebra preserves Star
}}
{{eqn | r = \map f {x^\ast y}
| c = $(\text C^\ast... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $f : A \to \C$ be a [[Definition:Positive Linear Functional/C*-Algebra|positive linear functional]].
Define $\innerprod \cdot \cdot : A^2 \to A$ by:
:$\innerprod x y = \map f {y^\ast x}$
for each $x, y \in A$.
... | === Proof of Conjugate Symmetry ===
Let $x, y \in A$ we have:
{{begin-eqn}}
{{eqn | l = \overline {\innerprod x y}
| r = \overline {\map f {y^\ast x} }
}}
{{eqn | r = \map f {\paren {y^\ast x}^\ast}
| c = [[Positive Linear Functional on C*-Algebra preserves Star]]
}}
{{eqn | r = \map f {x^\ast y}
| c = $(\text C... | Positive Linear Functional on C*-Algebra induces Semi-Inner Product | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_induces_Semi-Inner_Product | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_induces_Semi-Inner_Product | [
"Positive Linear Functional on C*-Algebra induces Semi-Inner Product",
"Semi-Inner Product Spaces",
"C*-Algebras",
"Positive Linear Functionals",
"Positive Linear Functional on C*-Algebra induces Semi-Inner Product"
] | [
"Definition:C*-Algebra",
"Definition:Positive Linear Functional/C*-Algebra",
"Definition:Semi-Inner Product"
] | [
"Positive Linear Functional on C*-Algebra preserves Star"
] |
proofwiki-22244 | Positive Linear Functional on C*-Algebra induces Semi-Inner Product/Corollary | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $f : A \to \C$ be a positive linear functional.
Let $x, y \in A$.
Then:
:$\cmod {\map f {y^\ast x} }^2 \le \map f {y^\ast y} \map f {x^\ast x}$ | Define $\innerprod \cdot \cdot : A^2 \to A$ by:
:$\innerprod x y = \map f {y^\ast x}$
for each $x, y \in A$.
By Positive Linear Functional on C*-Algebra induces Semi-Inner Product, $\innerprod \cdot \cdot$ is a semi-inner product.
Hence from Cauchy-Bunyakovsky-Schwarz Inequality: Inner Product Spaces, we have:
:$\cmod... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $f : A \to \C$ be a [[Definition:Positive Linear Functional/C*-Algebra|positive linear functional]].
Let $x, y \in A$.
Then:
:$\cmod {\map f {y^\ast x} }^2 \le \map f {y^\ast y} \map f {x^\ast x}$ | Define $\innerprod \cdot \cdot : A^2 \to A$ by:
:$\innerprod x y = \map f {y^\ast x}$
for each $x, y \in A$.
By [[Positive Linear Functional on C*-Algebra induces Semi-Inner Product]], $\innerprod \cdot \cdot$ is a [[Definition:Semi-Inner Product Space|semi-inner product]].
Hence from [[Cauchy-Bunyakovsky-Schwarz In... | Positive Linear Functional on C*-Algebra induces Semi-Inner Product/Corollary | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_induces_Semi-Inner_Product/Corollary | https://proofwiki.org/wiki/Positive_Linear_Functional_on_C*-Algebra_induces_Semi-Inner_Product/Corollary | [
"Positive Linear Functional on C*-Algebra induces Semi-Inner Product"
] | [
"Definition:C*-Algebra",
"Definition:Positive Linear Functional/C*-Algebra"
] | [
"Positive Linear Functional on C*-Algebra induces Semi-Inner Product",
"Definition:Semi-Inner Product Space",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces"
] |
proofwiki-22245 | Canonical Mapping of Locale to Powerset of Points is Frame Homomorphism | Let $\struct{L, \preceq}$ be a locale.
Let $\map {\operatorname{pt}} L$ denote the points of $L$ as completely prime filters.
For each $a \in L$, let:
:$\Sigma_a = \set{p \in \map {\operatorname{pt}} L : a \in p}$
Let $\Sigma : L \to \powerset {\map {\operatorname{pt}} L}$ be the mapping defined by:
:$\forall a \in L :... | === $\Sigma$ Preserves Arbitrary Supremums ===
Let $\set{a_i : i \in I}$ be an indexed family of elements of $L$.
Let $\ds \bigvee_{i \in I} a_i$ denote the supremum of $\set{a_i : i \in I}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall p \in \map {\operatorname{pt} } L
| l = p
| o = \in
| r = \map \Si... | Let $\struct{L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\map {\operatorname{pt}} L$ denote the [[Definition:Point of Locale as Completely Prime Filter|points of $L$ as completely prime filters]].
For each $a \in L$, let:
:$\Sigma_a = \set{p \in \map {\operatorname{pt}} L : a \in p}$
Let... | === $\Sigma$ Preserves Arbitrary Supremums ===
Let $\set{a_i : i \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of elements]] of $L$.
Let $\ds \bigvee_{i \in I} a_i$ denote the [[Definition:Supremum|supremum]] of $\set{a_i : i \in I}$.
We have:
{{begin-eqn}}
{{eqn | q = \forall p \in \map {\opera... | Canonical Mapping of Locale to Powerset of Points is Frame Homomorphism | https://proofwiki.org/wiki/Canonical_Mapping_of_Locale_to_Powerset_of_Points_is_Frame_Homomorphism | https://proofwiki.org/wiki/Canonical_Mapping_of_Locale_to_Powerset_of_Points_is_Frame_Homomorphism | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Point of Locale/Completely Prime Filter",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Frame Homomorphism"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Supremum",
"Characterization of Completely Prime Filter in Complete Lattice",
"Definition:Set Equality",
"Definition:Arbitrary Join Preserving Mapping",
"Definition:Indexing Set/Family of Sets",
"Characterization of Completely Prime Filter in Complet... |
proofwiki-22246 | Frame Homomorphism Preserves Greatest Element | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be frames.
Let $\phi: L_1 \to L_2$ be a frame homomophism.
Let $\top_1$ and $\top_2$ denote the greatest elelment of $L_1$ and $L_2$ respectively.
Then:
:$\map \phi \top_1 = \top_2$ | we have:
{{begin-eqn}}
{{eqn | l = \map \phi \top_1
| r = \map \phi {\bigwedge \O}
| c = {{Defof|Frame}} and Infimum of Empty Set is Greatest Element
}}
{{eqn | r = \bigwedge \set{\map \phi x : x \in \O}
| c = Frame homomorphism is finite meet preserving
}}
{{eqn | r = \bigwedge \O
| c = {{Defof... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Frame Homomorphism|frame homomophism]].
Let $\top_1$ and $\top_2$ denote the [[Definition:Greatest Element|greatest elelment]] of $L_1$ and $L_2$ respecti... | we have:
{{begin-eqn}}
{{eqn | l = \map \phi \top_1
| r = \map \phi {\bigwedge \O}
| c = {{Defof|Frame}} and [[Infimum of Empty Set is Greatest Element]]
}}
{{eqn | r = \bigwedge \set{\map \phi x : x \in \O}
| c = [[Definition:Frame Homomorphism|Frame homomorphism]] is [[Definition:Finite Meet Preserv... | Frame Homomorphism Preserves Greatest Element | https://proofwiki.org/wiki/Frame_Homomorphism_Preserves_Greatest_Element | https://proofwiki.org/wiki/Frame_Homomorphism_Preserves_Greatest_Element | [
"Frame Homomorphisms",
"Greatest Elements"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Greatest Element"
] | [
"Infimum of Empty Set is Greatest Element",
"Definition:Frame Homomorphism",
"Definition:Finite Meet Preserving Mapping",
"Infimum of Empty Set is Greatest Element"
] |
proofwiki-22247 | Frame Homomorphism Preserves Smallest Element | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be frames.
Let $\phi: L_1 \to L_2$ be a frame homomophism.
Let $\bot_1$ and $\bot_2$ denote the smallest elelment of $L_1$ and $L_2$ respectively.
Then:
:$\map \phi \bot_1 = \bot_2$ | We have:
{{begin-eqn}}
{{eqn | l = \map \phi \bot_1
| r = \map \phi {\bigvee \O}
| c = {{Defof|Frame}} and Supremum of Empty Set is Smallest Element
}}
{{eqn | r = \bigvee \set{\map \phi x : x \in \O}
| c = Frame homomorphism is arbitrary join preserving
}}
{{eqn | r = \bigvee \O
| c = {{Defof|E... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi: L_1 \to L_2$ be a [[Definition:Frame Homomorphism|frame homomophism]].
Let $\bot_1$ and $\bot_2$ denote the [[Definition:Smallest Element|smallest elelment]] of $L_1$ and $L_2$ respecti... | We have:
{{begin-eqn}}
{{eqn | l = \map \phi \bot_1
| r = \map \phi {\bigvee \O}
| c = {{Defof|Frame}} and [[Supremum of Empty Set is Smallest Element]]
}}
{{eqn | r = \bigvee \set{\map \phi x : x \in \O}
| c = [[Definition:Frame Homomorphism|Frame homomorphism]] is [[Definition:Arbitrary Join Preserv... | Frame Homomorphism Preserves Smallest Element | https://proofwiki.org/wiki/Frame_Homomorphism_Preserves_Smallest_Element | https://proofwiki.org/wiki/Frame_Homomorphism_Preserves_Smallest_Element | [
"Frame Homomorphisms",
"Smallest Elements"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Smallest Element"
] | [
"Supremum of Empty Set is Smallest Element",
"Definition:Frame Homomorphism",
"Definition:Arbitrary Join Preserving Mapping",
"Supremum of Empty Set is Smallest Element"
] |
proofwiki-22248 | Zero Set of Semi-Inner Product is Vector Subspace | Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot}$ be a semi-inner product space over $\GF$.
Let:
:$N = \set {x \in V : \innerprod x x = 0}$
Then $N$ is a vector subspace of $V$. | Let $x, y \in N$ and
Then by Cauchy-Bunyakovsky-Schwarz Inequality: Inner Product Spaces we have:
:$\cmod {\innerprod x y}^2 \le \innerprod x x \innerprod y y = 0$
and:
:$\cmod {\innerprod y x}^2 \le \innerprod x x \innerprod y y = 0$
Hence $\innerprod x y = \innerprod y x = 0$.
Then we have:
{{begin-eqn}}
{{eqn | l ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot}$ be a [[Definition:Semi-Inner Product Space|semi-inner product space]] over $\GF$.
Let:
:$N = \set {x \in V : \innerprod x x = 0}$
Then $N$ is a [[Definition:Vector Subspace|vector subspace]] of $V$. | Let $x, y \in N$ and
Then by [[Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces|Cauchy-Bunyakovsky-Schwarz Inequality: Inner Product Spaces]] we have:
:$\cmod {\innerprod x y}^2 \le \innerprod x x \innerprod y y = 0$
and:
:$\cmod {\innerprod y x}^2 \le \innerprod x x \innerprod y y = 0$
Hence $\innerprod ... | Zero Set of Semi-Inner Product is Vector Subspace | https://proofwiki.org/wiki/Zero_Set_of_Semi-Inner_Product_is_Vector_Subspace | https://proofwiki.org/wiki/Zero_Set_of_Semi-Inner_Product_is_Vector_Subspace | [
"Semi-Inner Product Spaces"
] | [
"Definition:Semi-Inner Product Space",
"Definition:Vector Subspace"
] | [
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces",
"Definition:Sesquilinear Form",
"Category:Semi-Inner Product Spaces"
] |
proofwiki-22249 | Semi-Inner Product induces Inner Product on Quotient | Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot}$ be a semi-inner product space over $\GF$.
Let:
:$N = \set {x \in V : \innerprod x x = 0}$
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Define:
:$\innerprod {x + N} {y + N}_{X/N} = \innerprod x y$
for each $x, y \in X$.
Then $\innerpr... | First note that Zero Set of Semi-Inner Product is Vector Subspace shows that $N$ is a vector subspace and hence $X/N$ is well-defined.
We show that $\innerprod \cdot \cdot_{X/N}$ is well-defined.
We need to show that if $x, y, x', y' \in X$ are such that $x - x' \in N$ and $y - y' \in N$, then $\innerprod x y = \inner... | Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot}$ be a [[Definition:Semi-Inner Product Space|semi-inner product space]] over $\GF$.
Let:
:$N = \set {x \in V : \innerprod x x = 0}$
Let $X/N$ be the [[Definition:Quotient Vector Space|quotient vector space of $X$ modulo $N$]].
Define:
:$\innerpr... | First note that [[Zero Set of Semi-Inner Product is Vector Subspace]] shows that $N$ is a [[Definition:Vector Subspace|vector subspace]] and hence $X/N$ is well-defined.
We show that $\innerprod \cdot \cdot_{X/N}$ is well-defined.
We need to show that if $x, y, x', y' \in X$ are such that $x - x' \in N$ and $y - y' ... | Semi-Inner Product induces Inner Product on Quotient | https://proofwiki.org/wiki/Semi-Inner_Product_induces_Inner_Product_on_Quotient | https://proofwiki.org/wiki/Semi-Inner_Product_induces_Inner_Product_on_Quotient | [
"Semi-Inner Product Spaces",
"Quotient Vector Spaces"
] | [
"Definition:Semi-Inner Product Space",
"Definition:Quotient Vector Space",
"Definition:Inner Product"
] | [
"Zero Set of Semi-Inner Product is Vector Subspace",
"Definition:Vector Subspace",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces",
"Definition:Inner Product"
] |
proofwiki-22250 | Generalized Sum Commutes with Inner Product | Let $\struct {\HH, \innerprod \cdot \cdot}$ be an inner product space.
Let $\Lambda$ be a set.
Let $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a $\Lambda$-indexed family in $\HH$ such that the generalized sum $\ds \sum \set {x_\lambda : \lambda \in \Lambda}$ converges.
Let $y \in \HH$.
Then:
:$\ds \sum \s... | Let $\struct {\FF, \subseteq}$ be the set of finite subsets of $\Lambda$ ordered by inclusion.
Define, for $F \in \FF$:
:$\ds \map {s_1} F = \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda}$
and:
:$\ds \map {s_2} F = \innerprod {\sum \set {x_\lambda : \lambda \in \Lambda} } y$
Noting that $\ds \family {\sum_{... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]].
Let $\Lambda$ be a [[Definition:Set|set]].
Let $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Indexed Family|$\Lambda$-indexed family]] in $\HH$ such that the [[Definition:Generalized Su... | Let $\struct {\FF, \subseteq}$ be the [[Definition:Set|set]] of [[Definition:Finite Subset|finite subsets]] of $\Lambda$ ordered by [[Definition:Set Inclusion|inclusion]].
Define, for $F \in \FF$:
:$\ds \map {s_1} F = \sum \set {\innerprod {x_\lambda} y : \lambda \in \Lambda}$
and:
:$\ds \map {s_2} F = \innerprod {\su... | Generalized Sum Commutes with Inner Product | https://proofwiki.org/wiki/Generalized_Sum_Commutes_with_Inner_Product | https://proofwiki.org/wiki/Generalized_Sum_Commutes_with_Inner_Product | [
"Generalized Sums"
] | [
"Definition:Inner Product Space",
"Definition:Set",
"Definition:Indexing Set/Family",
"Definition:Generalized Sum",
"Definition:Generalized Sum/Net Convergence",
"Definition:Generalized Sum/Net Convergence"
] | [
"Definition:Set",
"Definition:Finite Subset",
"Definition:Subset",
"Definition:Convergent Net",
"Definition:Convergent Net",
"Characterization of Continuity in terms of Nets",
"Definition:Linear Transformation",
"Definition:Convergent Net",
"Definition:Generalized Sum/Net Convergence",
"Category:G... |
proofwiki-22251 | Equivalence of Definitions of Positive Definite Matrix | Let $\mathbf A$ be a symmetric square matrix of order $n$.
{{TFAE|def = Positive Definite Matrix}} | {{MissingLinks}} | Let $\mathbf A$ be a [[Definition:Symmetric Matrix|symmetric]] [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]].
{{TFAE|def = Positive Definite Matrix}} | {{MissingLinks}} | Equivalence of Definitions of Positive Definite Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Positive_Definite_Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Positive_Definite_Matrix | [
"Positive Definite Matrices"
] | [
"Definition:Symmetric Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [] |
proofwiki-22252 | Orthocomplement equal to Orthocomplement of Linear Span | Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\GF$.
Let $S \subseteq V$ be non-empty.
Then:
:$S^\bot = \paren {\map \span S}^\bot$
where $\bot$ denotes orthocomplement. | From Orthocomplement Reverses Subset, we have:
:$\paren {\map \span S}^\bot \subseteq S^\bot$
Conversely let $y \in S^\bot$.
Let $x \in \map \span S$.
From the definition of the linear span there exists $x_1, \ldots, x_n \in S$ and $\alpha_1, \ldots, \alpha_n \in \GF$ such that:
:$\ds x = \sum_{j \mathop = 1}^n \alph... | Let $\GF \in \set {\R, \C}$.
Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]] over $\GF$.
Let $S \subseteq V$ be [[Definition:Non-Empty Set|non-empty]].
Then:
:$S^\bot = \paren {\map \span S}^\bot$
where $\bot$ denotes [[Definition:Orthocomplement|orthocomplem... | From [[Orthocomplement Reverses Subset]], we have:
:$\paren {\map \span S}^\bot \subseteq S^\bot$
Conversely let $y \in S^\bot$.
Let $x \in \map \span S$.
From the definition of the [[Definition:Linear Span|linear span]] there exists $x_1, \ldots, x_n \in S$ and $\alpha_1, \ldots, \alpha_n \in \GF$ such that:
:$\d... | Orthocomplement equal to Orthocomplement of Linear Span | https://proofwiki.org/wiki/Orthocomplement_equal_to_Orthocomplement_of_Linear_Span | https://proofwiki.org/wiki/Orthocomplement_equal_to_Orthocomplement_of_Linear_Span | [
"Orthocomplements"
] | [
"Definition:Inner Product Space",
"Definition:Non-Empty Set",
"Definition:Orthogonal (Linear Algebra)/Orthogonal Complement"
] | [
"Orthocomplement Reverses Subset",
"Definition:Generated Submodule/Linear Span",
"Definition:Linear Transformation",
"Category:Orthocomplements"
] |
proofwiki-22253 | Adjoint of Direct Sum of Bounded Linear Operators on Hilbert Space | Let $\GF \in \set {\R, \C}$.
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ be a bounded linear operator.
Suppose that:
:$\ds \sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } < \inf... | From Norm of Adjoint, we have:
:$\norm {T_i}_{\map B {\HH_i} } = \norm {T_i^\ast}_{\map B {\HH_i} }$
and hence:
:$\ds \sup_{i \mathop \in I} \norm {T_i^\ast}_{\map B {\HH_i} } < \infty$
so we can indeed define $\ds \bigoplus_{i \mathop \in I} T_i^\ast$.
Let $f, g \in \HH$.
We then have:
{{begin-eqn}}
{{eqn | l = \inne... | Let $\GF \in \set {\R, \C}$.
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Hilbert Space|Hilbert spaces]] over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ be a [[Definition:Bounded Linear Operator|bounde... | From [[Norm of Adjoint]], we have:
:$\norm {T_i}_{\map B {\HH_i} } = \norm {T_i^\ast}_{\map B {\HH_i} }$
and hence:
:$\ds \sup_{i \mathop \in I} \norm {T_i^\ast}_{\map B {\HH_i} } < \infty$
so we can indeed define $\ds \bigoplus_{i \mathop \in I} T_i^\ast$.
Let $f, g \in \HH$.
We then have:
{{begin-eqn}}
{{eqn | l =... | Adjoint of Direct Sum of Bounded Linear Operators on Hilbert Space | https://proofwiki.org/wiki/Adjoint_of_Direct_Sum_of_Bounded_Linear_Operators_on_Hilbert_Space | https://proofwiki.org/wiki/Adjoint_of_Direct_Sum_of_Bounded_Linear_Operators_on_Hilbert_Space | [
"Direct Sums of Hilbert Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Hilbert Space Direct Sum",
"Definition:Inner Product",
"Definition:Inner Product Norm",
"Definition:Direct Sum of Bounded Linear Operators on... | [
"Norm of Adjoint",
"Category:Direct Sums of Hilbert Spaces"
] |
proofwiki-22254 | Composition of Direct Sums of Bounded Linear Operators on Hilbert Space | Let $\GF \in \set {\R, \C}$.
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ and $S_i : \HH_i \to \HH_i$ be bounded linear operators.
Suppose that:
:$\ds \sup_{i \mathop \in I} \norm {T_... | From Norm on Bounded Linear Transformation is Submultiplicative we have:
{{begin-eqn}}
{{eqn | l = \norm {T_i S_i}_{\map B {\HH_i} }
| o = \le
| r = \norm {T_i}_{\map B {\HH_i} } \norm {S_i}_{\map B {\HH_i} }
}}
{{eqn | o = \le
| r = \paren {\sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } } \paren {\sup_{i \... | Let $\GF \in \set {\R, \C}$.
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Hilbert Space|Hilbert spaces]] over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ and $S_i : \HH_i \to \HH_i$ be [[Definition:Boun... | From [[Norm on Bounded Linear Transformation is Submultiplicative]] we have:
{{begin-eqn}}
{{eqn | l = \norm {T_i S_i}_{\map B {\HH_i} }
| o = \le
| r = \norm {T_i}_{\map B {\HH_i} } \norm {S_i}_{\map B {\HH_i} }
}}
{{eqn | o = \le
| r = \paren {\sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } } \paren {\sup_... | Composition of Direct Sums of Bounded Linear Operators on Hilbert Space | https://proofwiki.org/wiki/Composition_of_Direct_Sums_of_Bounded_Linear_Operators_on_Hilbert_Space | https://proofwiki.org/wiki/Composition_of_Direct_Sums_of_Bounded_Linear_Operators_on_Hilbert_Space | [
"Hilbert Space Direct Sums"
] | [
"Definition:Indexing Set/Family",
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Hilbert Space Direct Sum",
"Definition:Inner Product",
"Definition:Inner Product Norm",
"Definition:Direct Sum of Bounded Linear Operators on... | [
"Norm on Bounded Linear Transformation is Submultiplicative",
"Definition:Direct Sum of Bounded Linear Operators on Hilbert Space",
"Category:Hilbert Space Direct Sums"
] |
proofwiki-22255 | Linear Combination of Direct Sums of Bounded Linear Operators on Hilbert Space | Let $\GF \in \set {\R, \C}$.
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a $I$-indexed family of Hilbert spaces over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ and $S_i : \HH_i \to \HH_i$ be bounded linear operators.
Let $\lambda \in \GF$.
Suppose that:
:$\ds \sup_{i \... | First, from {{NormAxiomVector|3}} and {{NormAxiomVector|2}}, we have:
{{begin-eqn}}
{{eqn | l = \norm {T_i + \lambda S_i}_{\map B {\HH_i} }
| o = \le
| r = \norm {T_i}_{\map B {\HH_i} } + \cmod \lambda \norm {S_i}_{\map B {\HH_i} }
}}
{{eqn | o = \le
| r = \sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } + \c... | Let $\GF \in \set {\R, \C}$.
Let $\sequence {\family {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ be a [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Hilbert Space|Hilbert spaces]] over $\GF$.
For each $i \in I$, let $T_i : \HH_i \to \HH_i$ and $S_i : \HH_i \to \HH_i$ be [[Definition:Boun... | First, from {{NormAxiomVector|3}} and {{NormAxiomVector|2}}, we have:
{{begin-eqn}}
{{eqn | l = \norm {T_i + \lambda S_i}_{\map B {\HH_i} }
| o = \le
| r = \norm {T_i}_{\map B {\HH_i} } + \cmod \lambda \norm {S_i}_{\map B {\HH_i} }
}}
{{eqn | o = \le
| r = \sup_{i \mathop \in I} \norm {T_i}_{\map B {\HH_i} } + \c... | Linear Combination of Direct Sums of Bounded Linear Operators on Hilbert Space | https://proofwiki.org/wiki/Linear_Combination_of_Direct_Sums_of_Bounded_Linear_Operators_on_Hilbert_Space | https://proofwiki.org/wiki/Linear_Combination_of_Direct_Sums_of_Bounded_Linear_Operators_on_Hilbert_Space | [
"Hilbert Space Direct Sums"
] | [
"Definition:Indexing Set/Family",
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Hilbert Space Direct Sum",
"Definition:Inner Product",
"Definition:Inner Product Norm",
"Definition:Direct Sum of Bounded Linear Operators on... | [
"Definition:Direct Sum of Bounded Linear Operators on Hilbert Space",
"Category:Hilbert Space Direct Sums"
] |
proofwiki-22256 | Hermitian Elements of *-Algebra form Real Vector Subspace | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $A_{\mathbf{SA} }$ be the set of Hermitian elements of $A$.
Then $A_{\mathbf{SA} }$ is a $\R$-vector subspace of $A$. | We use the One-Step Vector Subspace Test.
From Zero Vector in *-Algebra is Hermitian, we have ${\mathbf 0}_A \in A_{\mathbf{SA} }$.
In particular $A_{\mathbf{SA} } \ne \O$.
Let $a, b \in A_{\mathbf{SA} }$ and $\lambda \in \R$.
We have, from $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in the definition of an involution:
:... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $A_{\mathbf{SA} }$ be the set of [[Definition:Hermitian Element of *-Algebra|Hermitian elements]] of $A$.
Then $A_{\mathbf{SA} }$ is a [[Definition:Vector Subspace|$\R$-vector subspace]] of $A$. | We use the [[One-Step Vector Subspace Test]].
From [[Zero Vector in *-Algebra is Hermitian]], we have ${\mathbf 0}_A \in A_{\mathbf{SA} }$.
In particular $A_{\mathbf{SA} } \ne \O$.
Let $a, b \in A_{\mathbf{SA} }$ and $\lambda \in \R$.
We have, from $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in the definition of an [... | Hermitian Elements of *-Algebra form Real Vector Subspace | https://proofwiki.org/wiki/Hermitian_Elements_of_*-Algebra_form_Real_Vector_Subspace | https://proofwiki.org/wiki/Hermitian_Elements_of_*-Algebra_form_Real_Vector_Subspace | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Definition:Vector Subspace"
] | [
"One-Step Vector Subspace Test",
"Zero Vector in *-Algebra is Hermitian",
"Definition:Involution on Algebra",
"Definition:Hermitian Element of *-Algebra",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Category:*-Algebras"
] |
proofwiki-22257 | State Space of C*-Algebra is Weak-* Compact | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $S_A$ be the set of states on $A$.
Let $w^\ast$ be the weak-$\ast$ topology.
Then $\struct {S_A, w^\ast}$ is compact. | By definition, we have:
:$S_A \subseteq B_{A^\ast}$
where $B_{A^\ast}$ is the closed unit ball of $A^\ast$.
From Banach-Alaoglu Theorem, $\struct {B_{A^\ast}, w^\ast}$ is compact.
From Closed Subspace of Compact Space is Compact, it is enough to show that $S_A$ is closed in $\struct {B_{A^\ast}, w^\ast}$.
Let $\family ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $S_A$ be the [[Definition:Set|set]] of [[Definition:State on C*-Algebra|states]] on $A$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]].
Then $\struct {S_A, w^\ast}$ is [[Definition:Com... | By definition, we have:
:$S_A \subseteq B_{A^\ast}$
where $B_{A^\ast}$ is the [[Definition:Closed Unit Ball|closed unit ball]] of $A^\ast$.
From [[Banach-Alaoglu Theorem]], $\struct {B_{A^\ast}, w^\ast}$ is [[Definition:Compact Topological Space|compact]].
From [[Closed Subspace of Compact Space is Compact]], it is e... | State Space of C*-Algebra is Weak-* Compact | https://proofwiki.org/wiki/State_Space_of_C*-Algebra_is_Weak-*_Compact | https://proofwiki.org/wiki/State_Space_of_C*-Algebra_is_Weak-*_Compact | [
"State Spaces of C*-Algebras",
"Weak-* Topologies"
] | [
"Definition:C*-Algebra",
"Definition:Set",
"Definition:State on C*-Algebra",
"Definition:Weak-* Topology",
"Definition:Compact Topological Space"
] | [
"Definition:Closed Unit Ball",
"Banach-Alaoglu Theorem",
"Definition:Compact Topological Space",
"Closed Subspace of Compact Space is Compact",
"Definition:Closed Set",
"Definition:Net (Set Theory)",
"Definition:Convergent Net",
"Characterization of Convergent Net in Weak-* Topology",
"Definition:Co... |
proofwiki-22258 | State Space of Unital C*-Algebra is Convex | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $S_A$ be the set of states on $A$.
Then $S_A$ is convex. | Let $t \in \closedint 0 1$ and $f, g \in S_A$.
Then $t f + \paren {1 - t} g$ is a linear functional.
Further, for each $a \in A$ positive, we have:
:$\map f a \ge 0$
and:
:$\map g a \ge 0$
Hence:
:$t \map f a + \paren {1 - t} \map g a \ge 0$
Hence $t f + \paren {1 - t} g$ is a positive linear functional.
From Norm of P... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $S_A$ be the [[Definition:Set|set]] of [[Definition:State on C*-Algebra|states]] on $A$.
Then $S_A$ is [[Definition:Convex Set (Vector Space)|convex]]. | Let $t \in \closedint 0 1$ and $f, g \in S_A$.
Then $t f + \paren {1 - t} g$ is a [[Definition:Linear Functional|linear functional]].
Further, for each $a \in A$ [[Definition:Positive Element of C*-Algebra|positive]], we have:
:$\map f a \ge 0$
and:
:$\map g a \ge 0$
Hence:
:$t \map f a + \paren {1 - t} \map g a \ge... | State Space of Unital C*-Algebra is Convex | https://proofwiki.org/wiki/State_Space_of_Unital_C*-Algebra_is_Convex | https://proofwiki.org/wiki/State_Space_of_Unital_C*-Algebra_is_Convex | [
"State Spaces of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Set",
"Definition:State on C*-Algebra",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Linear Functional",
"Definition:Positive Element of C*-Algebra",
"Definition:Positive Linear Functional",
"Norm of Positive Linear Functional on Unital C*-Algebra",
"Definition:Convex Set (Vector Space)"
] |
proofwiki-22259 | Real Part of Imaginary Unit times Element of *-Algebra | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $x \in A$.
Then:
:$\map \Re {i x} = -\map \Im x$
where $\Re$ and $\Im$ denote the real and imaginary parts of $A$. | We have:
{{begin-eqn}}
{{eqn | l = \map \Re {i x}
| r = \frac {\paren {i x} + \paren {i x}^\ast} 2
| c = {{Defof|Real Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {i x - i x^\ast} 2
| c = $(\text C^\ast 4)$ from {{Defof|Involution on Algebra}}
}}
{{eqn | r = i^2 \frac {x - x^\ast} {2 i}
}}
{{eqn | r = -\fr... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $x \in A$.
Then:
:$\map \Re {i x} = -\map \Im x$
where $\Re$ and $\Im$ denote the [[Definition:Real Part of Element of *-Algebra|real]] and [[Definition:Imaginary Part of Element of *-Algebra|imaginary parts]] of $A$. | We have:
{{begin-eqn}}
{{eqn | l = \map \Re {i x}
| r = \frac {\paren {i x} + \paren {i x}^\ast} 2
| c = {{Defof|Real Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {i x - i x^\ast} 2
| c = $(\text C^\ast 4)$ from {{Defof|Involution on Algebra}}
}}
{{eqn | r = i^2 \frac {x - x^\ast} {2 i}
}}
{{eqn | r = -\fr... | Real Part of Imaginary Unit times Element of *-Algebra | https://proofwiki.org/wiki/Real_Part_of_Imaginary_Unit_times_Element_of_*-Algebra | https://proofwiki.org/wiki/Real_Part_of_Imaginary_Unit_times_Element_of_*-Algebra | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Real Part of Element of *-Algebra",
"Definition:Imaginary Part of Element of *-Algebra"
] | [
"Category:*-Algebras"
] |
proofwiki-22260 | Imaginary Part of Imaginary Unit times Element of *-Algebra | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $x \in A$.
Then:
:$\map \Im {i x} = \map \Re x$
where $\Re$ and $\Im$ denote the real and imaginary parts of $A$. | We have:
{{begin-eqn}}
{{eqn | l = \map \Im {i x}
| r = \frac {i x - \paren {i x}^\ast} {2 i}
| c = {{Defof|Imaginary Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {i x + i x^\ast} {2 i}
| c = $(\text C^\ast 4)$ from {{Defof|Involution on Algebra}}
}}
{{eqn | r = \frac {x + x^\ast} 2
}}
{{eqn | r = \map \Re... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $x \in A$.
Then:
:$\map \Im {i x} = \map \Re x$
where $\Re$ and $\Im$ denote the [[Definition:Real Part of Element of *-Algebra|real]] and [[Definition:Imaginary Part of Element of *-Algebra|imaginary parts]] of $A$. | We have:
{{begin-eqn}}
{{eqn | l = \map \Im {i x}
| r = \frac {i x - \paren {i x}^\ast} {2 i}
| c = {{Defof|Imaginary Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {i x + i x^\ast} {2 i}
| c = $(\text C^\ast 4)$ from {{Defof|Involution on Algebra}}
}}
{{eqn | r = \frac {x + x^\ast} 2
}}
{{eqn | r = \map \Re... | Imaginary Part of Imaginary Unit times Element of *-Algebra | https://proofwiki.org/wiki/Imaginary_Part_of_Imaginary_Unit_times_Element_of_*-Algebra | https://proofwiki.org/wiki/Imaginary_Part_of_Imaginary_Unit_times_Element_of_*-Algebra | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Real Part of Element of *-Algebra",
"Definition:Imaginary Part of Element of *-Algebra"
] | [
"Category:*-Algebras"
] |
proofwiki-22261 | Real Part of Element of *-Algebra is Real Linear | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $x, y \in A$ and $\lambda \in \R$.
Then we have:
:$\map \Re {x + \lambda y} = \map \Re x + \lambda \map \Re y$
where $\Re$ denotes real part. | We have:
{{begin-eqn}}
{{eqn | l = \map \Re {x + \lambda y}
| r = \frac {x + \lambda y + \paren {x + \lambda y}^\ast} 2
| c = {{Defof|Real Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {x + \lambda y + x^\ast + \lambda y^\ast} 2
| c = $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in {{Defof|Involution on Algebr... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $x, y \in A$ and $\lambda \in \R$.
Then we have:
:$\map \Re {x + \lambda y} = \map \Re x + \lambda \map \Re y$
where $\Re$ denotes [[Definition:Real Part of Element of *-Algebra|real part]]. | We have:
{{begin-eqn}}
{{eqn | l = \map \Re {x + \lambda y}
| r = \frac {x + \lambda y + \paren {x + \lambda y}^\ast} 2
| c = {{Defof|Real Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {x + \lambda y + x^\ast + \lambda y^\ast} 2
| c = $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in {{Defof|Involution on Algebr... | Real Part of Element of *-Algebra is Real Linear | https://proofwiki.org/wiki/Real_Part_of_Element_of_*-Algebra_is_Real_Linear | https://proofwiki.org/wiki/Real_Part_of_Element_of_*-Algebra_is_Real_Linear | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Real Part of Element of *-Algebra"
] | [
"Complex Number equals Conjugate iff Wholly Real",
"Category:*-Algebras"
] |
proofwiki-22262 | Imaginary Part of Element of *-Algebra is Real Linear | Let $\struct {A, \ast}$ be a $\ast$-algebra over $\C$.
Let $x, y \in A$ and $\lambda \in \R$.
Then we have:
:$\map \Im {x + \lambda y} = \map \Im x + \lambda \map \Im y$
where $\Im$ denotes real part. | We have:
{{begin-eqn}}
{{eqn | l = \map \Im {x + \lambda y}
| r = \frac {x + \lambda y - \paren {x + \lambda y}^\ast} {2 i}
| c = {{Defof|Real Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {x + \lambda y - x^\ast - \lambda y^\ast} {2 i}
| c = $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in {{Defof|Involution o... | Let $\struct {A, \ast}$ be a [[Definition:*-Algebra|$\ast$-algebra]] over $\C$.
Let $x, y \in A$ and $\lambda \in \R$.
Then we have:
:$\map \Im {x + \lambda y} = \map \Im x + \lambda \map \Im y$
where $\Im$ denotes [[Definition:Real Part of Element of *-Algebra|real part]]. | We have:
{{begin-eqn}}
{{eqn | l = \map \Im {x + \lambda y}
| r = \frac {x + \lambda y - \paren {x + \lambda y}^\ast} {2 i}
| c = {{Defof|Real Part of Element of *-Algebra}}
}}
{{eqn | r = \frac {x + \lambda y - x^\ast - \lambda y^\ast} {2 i}
| c = $(\text C^\ast 2)$ and $(\text C^\ast 4)$ in {{Defof|Involution o... | Imaginary Part of Element of *-Algebra is Real Linear | https://proofwiki.org/wiki/Imaginary_Part_of_Element_of_*-Algebra_is_Real_Linear | https://proofwiki.org/wiki/Imaginary_Part_of_Element_of_*-Algebra_is_Real_Linear | [
"*-Algebras"
] | [
"Definition:*-Algebra",
"Definition:Real Part of Element of *-Algebra"
] | [
"Complex Number equals Conjugate iff Wholly Real",
"Category:*-Algebras"
] |
proofwiki-22263 | State on Unital C*-Subalgebra extends to whole C*-Algebra | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $B$ be a unital $\text C^\ast$-subalgebra of $A$.
Let $f_1 : B \to \C$ be a state.
Then there exists a state $f : A \to \C$ extending $f_1$. | Let $\le_A$ be the canonical preordering of $A$.
Let $B_{\mathbf {SA} }$ and $A_{\mathbf {SA} }$ be the set of Hermitian elements of $B$ and $A$ respectively.
From Hermitian Elements of *-Algebra form Real Vector Subspace, these are vector spaces over $\R$.
From Bounds on Hermitian Element of Unital C*-Algebra in term... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $B$ be a [[Definition:Unital Subalgebra|unital]] [[Definition:C*-Subalgebra|$\text C^\ast$-subalgebra]] of $A$.
Let $f_1 : B \to \C$ be a [[Definition:State on C*-Algeb... | Let $\le_A$ be the [[Definition:Canonical Preordering of C*-Algebra|canonical preordering]] of $A$.
Let $B_{\mathbf {SA} }$ and $A_{\mathbf {SA} }$ be the set of [[Definition:Hermitian Element of *-Algebra|Hermitian elements]] of $B$ and $A$ respectively.
From [[Hermitian Elements of *-Algebra form Real Vector Subspa... | State on Unital C*-Subalgebra extends to whole C*-Algebra | https://proofwiki.org/wiki/State_on_Unital_C*-Subalgebra_extends_to_whole_C*-Algebra | https://proofwiki.org/wiki/State_on_Unital_C*-Subalgebra_extends_to_whole_C*-Algebra | [
"State Spaces on C*-Algebras",
"State Spaces of C*-Algebras",
"State Spaces of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Unital Subalgebra",
"Definition:C*-Subalgebra",
"Definition:State on C*-Algebra",
"Definition:State on C*-Algebra",
"Definition:Extension of Mapping"
] | [
"Definition:Canonical Preordering of C*-Algebra",
"Definition:Hermitian Element of *-Algebra",
"Hermitian Elements of *-Algebra form Real Vector Subspace",
"Definition:Vector Space",
"Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum",
"Definition:Cofinal Subset/Preordered Ve... |
proofwiki-22264 | Canonical Bijection from Completely Prime Filters to Frame Homomorphisms | Let $\struct{L, \preceq}$ be a locale.
Let $\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$ be the spectrum of $L$ as completely prime filters where:
:*$\quad\map {\operatorname{pt}} L$ denotes the set of points as completely prime filters of $L$.
:*$\quad$for each $a \in L$,... | === $f$ is a Bijection ===
From Frame Homomorphism Onto Two Induced by Completely Prime Filter:
:$f$ is a well-defined mapping
Let $g : \map {\operatorname{pt}'} L \to \map {\operatorname{pt}} L$ be the mapping defined by:
:$\forall \phi \in \map {\operatorname{pt}'} L : \map g \phi = \map {\phi^{-1}} \top$
where:
:$\m... | Let $\struct{L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$ be the [[Definition:Spectrum of Locale as Completely Prime Filters|spectrum of $L$ as completely prime filters]] where:
:*$\quad\map {\operator... | === $f$ is a Bijection ===
From [[Frame Homomorphism Onto Two Induced by Completely Prime Filter]]:
:$f$ is a [[Definition:Well-Defined|well-defined]] [[Definition:Mapping|mapping]]
Let $g : \map {\operatorname{pt}'} L \to \map {\operatorname{pt}} L$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall \phi \... | Canonical Bijection from Completely Prime Filters to Frame Homomorphisms | https://proofwiki.org/wiki/Canonical_Bijection_from_Completely_Prime_Filters_to_Frame_Homomorphisms | https://proofwiki.org/wiki/Canonical_Bijection_from_Completely_Prime_Filters_to_Frame_Homomorphisms | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Set",
"Definition:Point of Locale/Completely Prime Filter",
"Definition:Spectrum of Locale/Frame Homomorphisms",
"Definition:Set",
"Definition:Point of Locale/Frame Homomorphism",
"Definition:Ma... | [
"Frame Homomorphism Onto Two Induced by Completely Prime Filter",
"Definition:Well-Defined",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Preimage/Mapping/Element",
"Definition:Mapping",
"Completely Prime Filter Induced by Frame Homomorphism Onto Two",
"Definition:Well-Defined",
"Definiti... |
proofwiki-22265 | Canonical Bijection from Completely Prime Filters to Meet Irreducible Elements | Let $\struct{L, \preceq}$ be a locale.
Let $\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$ be the spectrum of $L$ as completely prime filters where:
:*$\quad\map {\operatorname{pt}} L$ denotes the set of points as completely prime filters of $L$.
:*$\quad$for each $a \in L$,... | === $f$ is a Bijection ===
From Meet Irreducible Element Induced by Completely Prime Filter:
:$f$ is a well-defined mapping
Let $g : \map {\operatorname{pt}'} L \to \map {\operatorname{pt}} L$ be the mapping defined by:
:$\forall m \in \map {\operatorname{pt}'} L : \map g m = \set{a \in L : a \npreceq m}$
From Complete... | Let $\struct{L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$ be the [[Definition:Spectrum of Locale as Completely Prime Filters|spectrum of $L$ as completely prime filters]] where:
:*$\quad\map {\operator... | === $f$ is a Bijection ===
From [[Meet Irreducible Element Induced by Completely Prime Filter]]:
:$f$ is a [[Definition:Well-Defined|well-defined]] [[Definition:Mapping|mapping]]
Let $g : \map {\operatorname{pt}'} L \to \map {\operatorname{pt}} L$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall m \in \ma... | Canonical Bijection from Completely Prime Filters to Meet Irreducible Elements | https://proofwiki.org/wiki/Canonical_Bijection_from_Completely_Prime_Filters_to_Meet_Irreducible_Elements | https://proofwiki.org/wiki/Canonical_Bijection_from_Completely_Prime_Filters_to_Meet_Irreducible_Elements | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Spectrum of Locale/Completely Prime Filters",
"Definition:Set",
"Definition:Point of Locale/Completely Prime Filter",
"Definition:Spectrum of Locale/Meet-Irreducibles",
"Definition:Set",
"Definition:Point of Locale/Meet-Irreducible",
"Definition:Mappin... | [
"Meet Irreducible Element Induced by Completely Prime Filter",
"Definition:Well-Defined",
"Definition:Mapping",
"Definition:Mapping",
"Completely Prime Filter Induced by Meet Irreducible Element",
"Definition:Well-Defined",
"Definition:Mapping",
"Completely Prime Filter Induced by Meet Irreducible Ind... |
proofwiki-22266 | Canonical Bijection from Frame Homomorphisms to Continuous Maps | Let $\struct {L, \preceq}$ be a locale.
Let $\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$ be the spectrum of $L$ as frame homomorphisms where:
{{begin-itemize}}
{{item|*|$\map {\operatorname{pt} } L$ denotes the set of points as frame homomorphisms of $L$.}}
{{item|*|for e... | === $f$ is a Bijection ===
From Frame Homomorphism is Lower Adjoint of Galois Connection and Galois Connection is Unique for Given Lower Adjoint:
:$\forall \phi \in \map {\operatorname{pt}} L: \exists !$ Galois connection $\tuple{\upperadjoint \phi, \phi}$
By definition of continuous map:
:$\forall \phi \in \map {\oper... | Let $\struct {L, \preceq}$ be a [[Definition:Locale (Lattice Theory)|locale]].
Let $\map {\operatorname{Sp}} L = \struct{\map {\operatorname{pt}} L, \set{\Sigma_a : a \in L}}$ be the [[Definition:Spectrum of Locale as Frame Homomorphisms|spectrum of $L$ as frame homomorphisms]] where:
{{begin-itemize}}
{{item|*|$\map... | === $f$ is a Bijection ===
From [[Frame Homomorphism is Lower Adjoint of Galois Connection]] and [[Galois Connection is Unique for Given Lower Adjoint]]:
:$\forall \phi \in \map {\operatorname{pt}} L: \exists !$ [[Definition:Galois Connection|Galois connection]] $\tuple{\upperadjoint \phi, \phi}$
By definition of [[D... | Canonical Bijection from Frame Homomorphisms to Continuous Maps | https://proofwiki.org/wiki/Canonical_Bijection_from_Frame_Homomorphisms_to_Continuous_Maps | https://proofwiki.org/wiki/Canonical_Bijection_from_Frame_Homomorphisms_to_Continuous_Maps | [
"Spectra of Locales"
] | [
"Definition:Locale (Lattice Theory)",
"Definition:Spectrum of Locale/Frame Homomorphisms",
"Definition:Set",
"Definition:Point of Locale/Frame Homomorphism",
"Definition:Spectrum of Locale/Continuous Maps",
"Definition:Set",
"Definition:Point of Locale/Continuous Map",
"Definition:Frame Homomorphism",... | [
"Frame Homomorphism is Lower Adjoint of Galois Connection",
"Galois Connection is Unique for Given Lower Adjoint",
"Definition:Galois Connection",
"Definition:Continuous Map (Locale)",
"Definition:Continuous Map (Locale)",
"Definition:Galois Connection",
"Definition:Well-Defined",
"Definition:Mapping"... |
proofwiki-22267 | Gelfand-Naimark-Segal Construction | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $f : A \to \C$ be a positive linear functional.
Then there exists a cyclic representation $\tuple {\pi_f, \struct {\HH_f, \innerprod \cdot \cdot_f} }$ with cyclic vector $e_f$ such that:
:$\map f a = \innerprod {\map {\pi_f} a e_f} {e... | From Positive Linear Functional on C*-Algebra is Bounded, $f$ is bounded.
Define:
:$\LL_f = \set {x \in A : \map f {x^\ast x} = 0}$
From Positive Linear Functional on C*-Algebra induces Semi-Inner Product, $\LL_f$ is a vector subspace of $A$.
Construct the quotient vector space $A/\LL_f$.
Define $\widetilde {\innerprod... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $f : A \to \C$ be a [[Definition:Positive Linear Functional|positive linear functional]].
Then there exists a [[Definition:Cyclic Representation of C*-Algebra|cyclic re... | From [[Positive Linear Functional on C*-Algebra is Bounded]], $f$ is [[Definition:Bounded Linear Functional|bounded]].
Define:
:$\LL_f = \set {x \in A : \map f {x^\ast x} = 0}$
From [[Positive Linear Functional on C*-Algebra induces Semi-Inner Product]], $\LL_f$ is a [[Definition:Vector Subspace|vector subspace]] of ... | Gelfand-Naimark-Segal Construction | https://proofwiki.org/wiki/Gelfand-Naimark-Segal_Construction | https://proofwiki.org/wiki/Gelfand-Naimark-Segal_Construction | [
"C*-Algebras",
"Representations of C*-Algebras",
"Representations of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Positive Linear Functional",
"Definition:Cyclic Representation of C*-Algebra",
"Definition:Cyclic Representation of C*-Algebra"
] | [
"Positive Linear Functional on C*-Algebra is Bounded",
"Definition:Bounded Linear Functional",
"Positive Linear Functional on C*-Algebra induces Semi-Inner Product",
"Definition:Vector Subspace",
"Definition:Quotient Vector Space",
"Positive Linear Functional on C*-Algebra induces Semi-Inner Product",
"... |
proofwiki-22268 | Everywhere Dense Set determines Adjoint Linear Transformation | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \innerprod \cdot \cdot_X}$ and $\struct {Y, \innerprod \cdot \cdot_Y}$ be Hilbert spaces.
Let $\DD$ be an everywhere dense subset of $Y$.
Let $T : X \to Y$ and $S : Y \to X$ be bounded linear transformations such that:
:$\innerprod {T x} y_Y = \innerprod x {S y}_X$ for ea... | By the definition of the adjoint, we have:
:$\innerprod x {S y}_X = \innerprod {T x} y_Y = \innerprod x {T^\ast y}_X$ for each $x \in X$ and $y \in \DD$.
Let $y \in Y$.
Since $\DD$ is everywhere dense in $Y$, there exists a sequence $\sequence {y_n}_{n \mathop \in \N}$ in $\DD$ such that $y_n \to y$.
Then:
:$\innerprod... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \innerprod \cdot \cdot_X}$ and $\struct {Y, \innerprod \cdot \cdot_Y}$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $\DD$ be an [[Definition:Everywhere Dense|everywhere dense subset]] of $Y$.
Let $T : X \to Y$ and $S : Y \to X$ be [[Definition:Bounded Linear Tra... | By the definition of the [[Definition:Adjoint Linear Transformation|adjoint]], we have:
:$\innerprod x {S y}_X = \innerprod {T x} y_Y = \innerprod x {T^\ast y}_X$ for each $x \in X$ and $y \in \DD$.
Let $y \in Y$.
Since $\DD$ is [[Definition:Everywhere Dense|everywhere dense]] in $Y$, there exists a [[Definition:Sequ... | Everywhere Dense Set determines Adjoint Linear Transformation | https://proofwiki.org/wiki/Everywhere_Dense_Set_determines_Adjoint_Linear_Transformation | https://proofwiki.org/wiki/Everywhere_Dense_Set_determines_Adjoint_Linear_Transformation | [
"Adjoints"
] | [
"Definition:Hilbert Space",
"Definition:Everywhere Dense",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Adjoint Linear Transformation",
"Definition:Everywhere Dense",
"Definition:Sequence",
"Inner Product is Continuous",
"Linear Subspace Dense iff Zero Orthocomplement",
"Category:Adjoints"
] |
proofwiki-22269 | Supremum of Continuous Bounded Real-Valued Function on Everywhere Dense Subset | Let $\struct {X, \tau}$ be a topological space.
Let $\DD$ be an everywhere dense subset of $\struct {X, \tau}$.
Let $f : X \to \R$ be a continuous bounded real-valued function.
Then:
:$\ds \sup_{x \mathop \in \DD} \map f x = \sup_{x \mathop \in X} \map f x$ | Since $\DD \subseteq X$ we have:
:$\ds \sup_{x \mathop \in \DD} \map f x \le \sup_{x \mathop \in X} \map f x$
Conversely, from the definition of supremum, for each $\epsilon > 0$ there exists $y \in X$ such that:
:$\ds \sup_{x \mathop \in X} \map f x - \frac \epsilon 2 \le \map f y \le \sup_{x \mathop \in X} \map f x$
... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\DD$ be an [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Subset|subset]] of $\struct {X, \tau}$.
Let $f : X \to \R$ be a [[Definition:Continuous Function|continuous]] [[Definition:Bounded Real-Valued Function|bounde... | Since $\DD \subseteq X$ we have:
:$\ds \sup_{x \mathop \in \DD} \map f x \le \sup_{x \mathop \in X} \map f x$
Conversely, from the definition of [[Definition:Supremum of Real-Valued Function|supremum]], for each $\epsilon > 0$ there exists $y \in X$ such that:
:$\ds \sup_{x \mathop \in X} \map f x - \frac \epsilon 2 \... | Supremum of Continuous Bounded Real-Valued Function on Everywhere Dense Subset | https://proofwiki.org/wiki/Supremum_of_Continuous_Bounded_Real-Valued_Function_on_Everywhere_Dense_Subset | https://proofwiki.org/wiki/Supremum_of_Continuous_Bounded_Real-Valued_Function_on_Everywhere_Dense_Subset | [
"Continuous Real-Valued Functions",
"Everywhere Dense"
] | [
"Definition:Topological Space",
"Definition:Everywhere Dense",
"Definition:Subset",
"Definition:Continuous Function",
"Definition:Bounded Mapping/Real-Valued"
] | [
"Definition:Supremum of Mapping/Real-Valued Function",
"Definition:Continuous Function",
"Definition:Open Set",
"Definition:Everywhere Dense",
"Category:Continuous Real-Valued Functions",
"Category:Everywhere Dense"
] |
proofwiki-22270 | Norm of Positive Element of Unital C*-Algebra in terms of State Space | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $S_A$ be the set of states on $A$.
Let $a \in A$ be positive.
Then:
:$\norm a = \sup \set {\map \phi a : \phi \in S_A}$ | From Evaluation Linear Transformation on Normed Vector Space is Linear Isometry, we have:
:$\norm a = \sup \set {\map \phi a : \phi \in B_{A^\ast} }$
where $B_{A^\ast}$ is the closed unit ball of $A^\ast$.
We have $S_A \subseteq B_{A^\ast}$.
Hence we have:
:$\norm a \ge \sup \set {\map \phi a : \phi \in S_A}$
It remai... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:Unital Banach Algebra|unital]] [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $S_A$ be the [[Definition:Set|set]] of [[Definition:State on C*-Algebra|states]] on $A$.
Let $a \in A$ be [[Definition:Positive Element of C*-Algebra|positive]].
Then... | From [[Evaluation Linear Transformation on Normed Vector Space is Linear Isometry]], we have:
:$\norm a = \sup \set {\map \phi a : \phi \in B_{A^\ast} }$
where $B_{A^\ast}$ is the [[Definition:Closed Unit Ball|closed unit ball]] of $A^\ast$.
We have $S_A \subseteq B_{A^\ast}$.
Hence we have:
:$\norm a \ge \sup \set ... | Norm of Positive Element of Unital C*-Algebra in terms of State Space | https://proofwiki.org/wiki/Norm_of_Positive_Element_of_Unital_C*-Algebra_in_terms_of_State_Space | https://proofwiki.org/wiki/Norm_of_Positive_Element_of_Unital_C*-Algebra_in_terms_of_State_Space | [
"State Spaces of C*-Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:C*-Algebra",
"Definition:Set",
"Definition:State on C*-Algebra",
"Definition:Positive Element of C*-Algebra"
] | [
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Closed Unit Ball",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Positive Element of C*-Algebra",
"Canonical Preordering of C*-Algebra of Continuous Functions Vanishing at Infinity",
"P... |
proofwiki-22271 | Gelfand-Naimark Theorem/General Case | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Then there exists a representation $\struct {\pi, \HH}$ of $A$ such that $\pi$ is an isometry.
In other words, there exists a Hilbert space over $\C$ such that $A$ is isometrically $\ast$-algebra isomorphic to a $\text C^\ast$-subalgebra of $\ma... | First take $A$ to be unital.
Let $S_A$ be the state space of $A$.
Let $F \subseteq S_A$ be everywhere dense in the weak-$\ast$ topology.
By the Gelfand-Naimark-Segal Construction:
:for each positive linear functional $f : A \to \C$ there exists a cyclic representation $\tuple {\pi_f, \struct {\HH_f, \innerprod \cdot \c... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Then there exists a [[Definition:Representation of C*-Algebra|representation]] $\struct {\pi, \HH}$ of $A$ such that $\pi$ is an [[Definition:Linear Isometry|isometry]].
In other words, there exists a [[Definition:Hi... | First take $A$ to be [[Definition:Unital Banach Algebra|unital]].
Let $S_A$ be the [[Definition:State Space of C*-Algebra|state space]] of $A$.
Let $F \subseteq S_A$ be [[Definition:Everywhere Dense|everywhere dense]] in the [[Definition:Weak-* Topology|weak-$\ast$ topology]].
By the [[Gelfand-Naimark-Segal Construc... | Gelfand-Naimark Theorem/General Case | https://proofwiki.org/wiki/Gelfand-Naimark_Theorem/General_Case | https://proofwiki.org/wiki/Gelfand-Naimark_Theorem/General_Case | [
"Gelfand-Naimark Theorem"
] | [
"Definition:C*-Algebra",
"Definition:Representation of C*-Algebra",
"Definition:Linear Isometry",
"Definition:Hilbert Space",
"Definition:Isometric Isomorphism",
"Definition:*-Algebra Isomorphism",
"Definition:C*-Subalgebra",
"Definition:Space of Bounded Linear Transformations",
"Definition:Separabl... | [
"Definition:Unital Banach Algebra",
"Definition:State Space of C*-Algebra",
"Definition:Everywhere Dense",
"Definition:Weak-* Topology",
"Gelfand-Naimark-Segal Construction",
"Definition:Positive Linear Functional",
"Definition:Cyclic Representation of C*-Algebra",
"Definition:Cyclic Representation of... |
proofwiki-22272 | Hilbert Space Direct Sum of Countably Many Separable Hilbert Spaces is Separable | Let $\GF \in \set {\R, \C}$.
For each $n \in \N$, let $\struct {\HH_n, \innerprod \cdot \cdot_n}$ be a separable Hilbert space over $\GF$.
Let:
:$\ds \HH = \bigoplus_{n \mathop \in \N} \HH_n$
be the direct sum of $\sequence {\HH_n}_{n \mathop \in \N}$.
Then $\HH$ is separable. | For each $n \in \N$, let $\sequence {e_k^{(n)} }_{k \mathop \in \N}$ be a countable everywhere dense subset of $\HH_n$.
Let $\SS$ be the set of $f \in \HH$ for which there exists $N \in \N$ such that:
:$\map f n = e_{\map k n}^{(n)} \in \HH_n$ for some $\map k n \in \N$ if $n \le N$
:$\map f n = {\mathbf 0}_{\HH_n}$ if... | Let $\GF \in \set {\R, \C}$.
For each $n \in \N$, let $\struct {\HH_n, \innerprod \cdot \cdot_n}$ be a [[Definition:Separable Space|separable]] [[Definition:Hilbert Space|Hilbert space]] over $\GF$.
Let:
:$\ds \HH = \bigoplus_{n \mathop \in \N} \HH_n$
be the [[Definition:Hilbert Space Direct Sum|direct sum]] of $\s... | For each $n \in \N$, let $\sequence {e_k^{(n)} }_{k \mathop \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $\HH_n$.
Let $\SS$ be the [[Definition:Set|set]] of $f \in \HH$ for which there exists $N \in \N$ such that:
:$\map f n = e_{\map k n}^{(n)} \in \H... | Hilbert Space Direct Sum of Countably Many Separable Hilbert Spaces is Separable | https://proofwiki.org/wiki/Hilbert_Space_Direct_Sum_of_Countably_Many_Separable_Hilbert_Spaces_is_Separable | https://proofwiki.org/wiki/Hilbert_Space_Direct_Sum_of_Countably_Many_Separable_Hilbert_Spaces_is_Separable | [
"Hilbert Space Direct Sums"
] | [
"Definition:Separable Space",
"Definition:Hilbert Space",
"Definition:Hilbert Space Direct Sum",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Set",
"Definition:Function",
"Set of Finitely Supported Functions on Integers is Countable",
"Subset of Countable Set is Countable",
"Definition:Countable Set",
"Definition:Countable Set",
"Everywhere Dense Subset of Countable Hi... |
proofwiki-22273 | Set of Finitely Supported Functions on Integers is Countable | Let $\SS$ be the set of functions $f : \Z \to \Z$ for which there exists $N \in \N$ such that:
:$\map f n = 0$ for all $n \in \Z$ with $\size n > N$.
Then $\SS$ is countable. | Let $\sequence {p_n}_{n \mathop \in \N}$ be an enumeration of the prime numbers.
For each $f \in \SS$ let $N_f \in \N$ be least such that:
:$\map f n = 0$ for all $n \in \Z$ with $\size n > N_f$.
Then for each $f \in \SS$, define $\map \phi f$ by:
:$\ds \map \phi f = 2^{\map f 0} \prod_{j \mathop = 1}^{N_f} p_{2 j + 1}... | Let $\SS$ be the [[Definition:Set|set]] of [[Definition:Function|functions]] $f : \Z \to \Z$ for which there exists $N \in \N$ such that:
:$\map f n = 0$ for all $n \in \Z$ with $\size n > N$.
Then $\SS$ is [[Definition:Countable Set|countable]]. | Let $\sequence {p_n}_{n \mathop \in \N}$ be an [[Definition:Enumeration|enumeration]] of the [[Definition:Prime Number|prime numbers]].
For each $f \in \SS$ let $N_f \in \N$ be least such that:
:$\map f n = 0$ for all $n \in \Z$ with $\size n > N_f$.
Then for each $f \in \SS$, define $\map \phi f$ by:
:$\ds \map \phi... | Set of Finitely Supported Functions on Integers is Countable | https://proofwiki.org/wiki/Set_of_Finitely_Supported_Functions_on_Integers_is_Countable | https://proofwiki.org/wiki/Set_of_Finitely_Supported_Functions_on_Integers_is_Countable | [
"Countable Sets"
] | [
"Definition:Set",
"Definition:Function",
"Definition:Countable Set"
] | [
"Definition:Enumeration",
"Definition:Prime Number",
"Definition:Injection",
"Fundamental Theorem of Arithmetic",
"Definition:Injection",
"Definition:Countable Set",
"Category:Countable Sets"
] |
proofwiki-22274 | Direct Sum of Representations of C*-Algebra is Representation | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.
Let $\family {\tuple {\pi_i, \HH_i} }_{i \mathop \in I}$ be an $I$-index family of representations of $\struct {A, \ast, \norm {\, \cdot \,} }$.
Let:
:$\ds \HH = \bigoplus_{i \mathop \in I} \HH_i$
be the Hilbert space direct sum of $\sequence {... | Let $a, b \in A$ and $\lambda \in \C$.
{{begin-eqn}}
{{eqn | l = \map \pi {a + \lambda b}
| r = \bigoplus_{i \mathop \in I} \map {\pi_i} {a + \lambda b}
}}
{{eqn | r = \bigoplus_{i \mathop \in I} \paren {\map {\pi_i} a + \lambda \map {\pi_i} b}
| c = $\pi_i$ is a linear transformation for each $i \in I$
}}
{{eqn ... | Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a [[Definition:C*-Algebra|$\text C^\ast$-algebra]].
Let $\family {\tuple {\pi_i, \HH_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|$I$-index family]] of [[Definition:Representation of C*-Algebra|representations]] of $\struct {A, \ast, \norm {\, \cdot \,} }$... | Let $a, b \in A$ and $\lambda \in \C$.
{{begin-eqn}}
{{eqn | l = \map \pi {a + \lambda b}
| r = \bigoplus_{i \mathop \in I} \map {\pi_i} {a + \lambda b}
}}
{{eqn | r = \bigoplus_{i \mathop \in I} \paren {\map {\pi_i} a + \lambda \map {\pi_i} b}
| c = $\pi_i$ is a [[Definition:Linear Transformation|linear transform... | Direct Sum of Representations of C*-Algebra is Representation | https://proofwiki.org/wiki/Direct_Sum_of_Representations_of_C*-Algebra_is_Representation | https://proofwiki.org/wiki/Direct_Sum_of_Representations_of_C*-Algebra_is_Representation | [
"Hilbert Space Direct Sums",
"Representations of C*-Algebras"
] | [
"Definition:C*-Algebra",
"Definition:Indexing Set/Family",
"Definition:Representation of C*-Algebra",
"Definition:Hilbert Space Direct Sum",
"Definition:Norm/Vector Space",
"Definition:Direct Sum of Bounded Linear Operators on Hilbert Space",
"Definition:Representation of C*-Algebra"
] | [
"Definition:Linear Transformation",
"Linear Combination of Direct Sums of Bounded Linear Operators on Hilbert Space",
"Definition:Linear Transformation",
"Definition:Algebra Homomorphism",
"Composition of Direct Sums of Bounded Linear Operators on Hilbert Space",
"Definition:Algebra Homomorphism",
"Adjo... |
proofwiki-22275 | Inverse Image Mapping of Frame Homomorphism Preserves Completely Prime Filter | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be frames.
Let $\phi : L_1 \to L_2$ be a frame homomorphism.
Let $\phi^\gets$ denote the inverse image mapping of $\phi$.
Let $F$ be a completely prime filter of $L_2$.
Then:
:$\map {\phi^\gets} F$ is a completely prime filter of $L_1$ | From Characterization of Completely Prime Filter in Complete Lattice it is sufficient to show:
:$(1)\quad\forall A \subseteq L : \bigvee A \in \map {\phi^\gets} F \iff \paren{\exists a \in A : a \in \map {\phi^\gets} F}$
:$(2)\quad\forall $ finite $A \subseteq L : \bigwedge A \in \map {\phi^\gets} F \iff \paren{\forall... | Let $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$ be [[Definition:Frame (Lattice Theory)|frames]].
Let $\phi : L_1 \to L_2$ be a [[Definition:Frame Homomorphism|frame homomorphism]].
Let $\phi^\gets$ denote the [[Definition:Inverse Image Mapping|inverse image mapping]] of $\phi$.
Let $F$ be a... | From [[Characterization of Completely Prime Filter in Complete Lattice]] it is sufficient to show:
:$(1)\quad\forall A \subseteq L : \bigvee A \in \map {\phi^\gets} F \iff \paren{\exists a \in A : a \in \map {\phi^\gets} F}$
:$(2)\quad\forall $ [[Definition:Finite|finite]] $A \subseteq L : \bigwedge A \in \map {\phi^\g... | Inverse Image Mapping of Frame Homomorphism Preserves Completely Prime Filter | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Frame_Homomorphism_Preserves_Completely_Prime_Filter | https://proofwiki.org/wiki/Inverse_Image_Mapping_of_Frame_Homomorphism_Preserves_Completely_Prime_Filter | [
"Locales",
"Frame Homomorphisms",
"Inverse Image Mappings",
"Completely Prime Filters"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Frame Homomorphism",
"Definition:Inverse Image Mapping",
"Definition:Completely Prime Filter",
"Definition:Completely Prime Filter"
] | [
"Characterization of Completely Prime Filter in Complete Lattice",
"Definition:Finite",
"Characterization of Completely Prime Filter in Complete Lattice",
"Characterization of Completely Prime Filter in Complete Lattice",
"Characterization of Completely Prime Filter in Complete Lattice"
] |
proofwiki-22276 | Existence of Primitive Element for every Prime | Let $p$ be a prime number.
Then there exists a primitive element of the multiplicative group of reduced residues $\Z'_p$. | {{ProofWanted|It needs to be demonstrated that $\Z'_p$ is cyclic, which Nelson assures us is "not elementary, and determining the smallest primitive element can be difficult."}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Then there exists a [[Definition:Primitive Element of Cyclic Modulo Group|primitive element]] of the [[Definition:Multiplicative Group of Reduced Residues|multiplicative group of reduced residues]] $\Z'_p$. | {{ProofWanted|It needs to be demonstrated that $\Z'_p$ is [[Definition:Cyclic Group|cyclic]], which Nelson assures us is "not elementary, and determining the smallest primitive element can be difficult."}} | Existence of Primitive Element for every Prime | https://proofwiki.org/wiki/Existence_of_Primitive_Element_for_every_Prime | https://proofwiki.org/wiki/Existence_of_Primitive_Element_for_every_Prime | [
"Primitive Elements of Cyclic Modulo Groups"
] | [
"Definition:Prime Number",
"Definition:Primitive Element of Cyclic Modulo Group",
"Definition:Multiplicative Group of Reduced Residues"
] | [
"Definition:Cyclic Group"
] |
proofwiki-22277 | Standard Topology on Locally Convex Space makes Seminorms Continuous | Let $\struct {X, \PP}$ be a locally convex space.
Let $\tau$ be the standard topology on $\struct {X, \PP}$.
Let $p \in \PP$.
Then $p$ is continuous as a function $\struct {X, \PP} \to \R$. | It is enough to show that for each $x \in X$ and $\epsilon > 0$, there exists $U_x \in \tau$ such that:
:$\size {\map p x - \map p y} < \epsilon$ for each $y \in U_x$.
From the definition of the standard topology, we have:
:$U_x = \set {y \in X : \map p {y - x} < \epsilon} \in \tau$
Then from Reverse Triangle Inequali... | Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]].
Let $\tau$ be the [[Definition:Locally Convex Space/Standard Topology|standard topology]] on $\struct {X, \PP}$.
Let $p \in \PP$.
Then $p$ is [[Definition:Continuous Function|continuous]] as a [[Definition:Function|function]] $\s... | It is enough to show that for each $x \in X$ and $\epsilon > 0$, there exists $U_x \in \tau$ such that:
:$\size {\map p x - \map p y} < \epsilon$ for each $y \in U_x$.
From the definition of the [[Definition:Locally Convex Space/Standard Topology|standard topology]], we have:
:$U_x = \set {y \in X : \map p {y - x} < ... | Standard Topology on Locally Convex Space makes Seminorms Continuous | https://proofwiki.org/wiki/Standard_Topology_on_Locally_Convex_Space_makes_Seminorms_Continuous | https://proofwiki.org/wiki/Standard_Topology_on_Locally_Convex_Space_makes_Seminorms_Continuous | [
"Locally Convex Spaces"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Continuous Function",
"Definition:Function"
] | [
"Definition:Locally Convex Space/Standard Topology",
"Reverse Triangle Inequality/Seminormed Vector Space",
"Category:Locally Convex Spaces"
] |
proofwiki-22278 | Lateral Face of Prismatoid is Triangle, Trapezium or Parallelogram | Let $\PP$ be a prismatoid.
Let $\FF$ be a lateral face of $\PP$.
Then $\FF$ is one of the following:
:a triangle
:a trapezium
:a parallelogram. | We have that an antiprism is {{apriori}} a prismatoid.
The lateral faces of an antiprism are triangles.
We have that an oblique prism is {{apriori}} a prismatoid.
The lateral faces of an oblique prism are parallelograms.
Suppose that $\FF$ were a polygon with more than $4$ sides.
Then at least $2$ of those adjacent sid... | Let $\PP$ be a [[Definition:Prismatoid|prismatoid]].
Let $\FF$ be a [[Definition:Lateral Face of Prismatoid|lateral face]] of $\PP$.
Then $\FF$ is one of the following:
:a [[Definition:Triangle (Geometry)|triangle]]
:a [[Definition:Trapezium|trapezium]]
:a [[Definition:Parallelogram|parallelogram]]. | We have that an [[Definition:Antiprism|antiprism]] is {{apriori}} a [[Definition:Prismatoid|prismatoid]].
The [[Definition:Lateral Face of Prismatoid|lateral faces]] of an [[Definition:Antiprism|antiprism]] are [[Definition:Triangle (Geometry)|triangles]].
We have that an [[Definition:Oblique Prism|oblique prism]] i... | Lateral Face of Prismatoid is Triangle, Trapezium or Parallelogram | https://proofwiki.org/wiki/Lateral_Face_of_Prismatoid_is_Triangle,_Trapezium_or_Parallelogram | https://proofwiki.org/wiki/Lateral_Face_of_Prismatoid_is_Triangle,_Trapezium_or_Parallelogram | [
"Lateral Faces of Prismatoids",
"Triangles",
"Trapezia",
"Parallelograms"
] | [
"Definition:Prismatoid",
"Definition:Prismatoid/Lateral Face",
"Definition:Triangle (Geometry)",
"Definition:Quadrilateral/Trapezium",
"Definition:Quadrilateral/Parallelogram"
] | [
"Definition:Antiprism",
"Definition:Prismatoid",
"Definition:Prismatoid/Lateral Face",
"Definition:Antiprism",
"Definition:Triangle (Geometry)",
"Definition:Oblique Prism",
"Definition:Prismatoid",
"Definition:Prismatoid/Lateral Face",
"Definition:Oblique Prism",
"Definition:Quadrilateral/Parallel... |
proofwiki-22279 | Homeomorphism Preserves System of Open Neighborhoods | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be topological spaces.
Let $f : T_1 \to T_2$ be a homeomorphism between $T_1$ and $T_2$.
Let $f^\to : \powerset{S_1}\to \powerset{S_2}$ denote the direct image mapping of $f : S_1 \to S_2$.
Then:
:$\forall x \in S_1 : f^\to \sqbrk {\map \UU x} = \map \UU... | Let $x \in S_1$. | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f : T_1 \to T_2$ be a [[Definition:Homeomorphism|homeomorphism]] between $T_1$ and $T_2$.
Let $f^\to : \powerset{S_1}\to \powerset{S_2}$ denote the [[Definition:Direct Image Mapping|direct im... | Let $x \in S_1$. | Homeomorphism Preserves System of Open Neighborhoods | https://proofwiki.org/wiki/Homeomorphism_Preserves_System_of_Open_Neighborhoods | https://proofwiki.org/wiki/Homeomorphism_Preserves_System_of_Open_Neighborhoods | [
"Homeomorphisms (Topological Spaces)",
"Systems of Open Neighborhoods"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism",
"Definition:Direct Image Mapping",
"Definition:System of Open Neighborhoods"
] | [] |
proofwiki-22280 | Frame Homomorphism Onto Two Induced by Completely Prime Filter | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the (Boolean Lattice) $\mathbf 2$.
Let $p$ be a completely prime filter of $L$.
Let $\phi_p : L \to \mathbf 2$ be the mapping defined by:
::<nowiki>$\forall a \in L : \map {\phi_p} a = \begin{cases}
\top & : a \in... | === $\phi_p$ is Arbitrary Join Preserving ===
Let $\set{a_i : i \in I}$ be an indexed subset of elements of $L$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\phi_p} {\bigvee_{i \in I} a_i}
| r = \top
}}
{{eqn | ll = \leadstoandfrom
| l = \bigvee_{i \in I} a_i
| o = \in
| r = p
| c = Definitio... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the [[Definition:Two (Boolean Lattice)|(Boolean Lattice) $\mathbf 2$]].
Let $p$ be a [[Definition:Completely Prime Filter|completely prime filter]] of $L$.
Let $\phi_p : ... | === $\phi_p$ is Arbitrary Join Preserving ===
Let $\set{a_i : i \in I}$ be an [[Definition:Indexed Set|indexed subset]] of [[Definition:Element|elements]] of $L$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\phi_p} {\bigvee_{i \in I} a_i}
| r = \top
}}
{{eqn | ll = \leadstoandfrom
| l = \bigvee_{i \in I} a_i... | Frame Homomorphism Onto Two Induced by Completely Prime Filter | https://proofwiki.org/wiki/Frame_Homomorphism_Onto_Two_Induced_by_Completely_Prime_Filter | https://proofwiki.org/wiki/Frame_Homomorphism_Onto_Two_Induced_by_Completely_Prime_Filter | [
"Completely Prime Filters",
"Two (Boolean Lattice)",
"Frame Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Two (Boolean Lattice)",
"Definition:Completely Prime Filter",
"Definition:Mapping",
"Definition:Frame Homomorphism"
] | [
"Definition:Indexing Set/Indexed Set",
"Definition:Element",
"Characterization of Completely Prime Filter in Complete Lattice",
"Definition:Arbitrary Join Preserving Mapping",
"Definition:Indexing Set/Indexed Set",
"Definition:Element",
"Characterization of Completely Prime Filter in Complete Lattice"
] |
proofwiki-22281 | Completely Prime Filter Induced by Frame Homomorphism Onto Two | Let $\struct{L, \vee, \wedge, \preceq}$ be a frame.
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the (Boolean Lattice) $\mathbf 2$.
Let $\phi : L \to \mathbf 2$ be a frame homomorphism.
Then:
:$\map {\phi^{-1}} \top$ is a completely prime filter
where $\map {\phi^{-1}} \top$ denotes the preimage of $\top \in ... | From Singleton of Greatest Element in Two is Completely Prime Filter:
:$\set \top$ is a completely prime filter
From Inverse Image Mapping of Frame Homomorphism Preserves Completely Prime Filter:
:$\map {\phi^\gets}{\set \top}$ is a completely prime filter
We have:
{{begin-eqn}}
{{eqn | l = \map {\phi^\gets}{\set \top}... | Let $\struct{L, \vee, \wedge, \preceq}$ be a [[Definition:Frame (Lattice Theory)|frame]].
Let $\struct{\mathbf 2, \vee, \wedge, \preceq}$ denote the [[Definition:Two (Boolean Lattice)|(Boolean Lattice) $\mathbf 2$]].
Let $\phi : L \to \mathbf 2$ be a [[Definition:Frame Homomorphism|frame homomorphism]].
Then:
:$\... | From [[Singleton of Greatest Element in Two is Completely Prime Filter]]:
:$\set \top$ is a [[Definition:Completely Prime Filter|completely prime filter]]
From [[Inverse Image Mapping of Frame Homomorphism Preserves Completely Prime Filter]]:
:$\map {\phi^\gets}{\set \top}$ is a [[Definition:Completely Prime Filter|c... | Completely Prime Filter Induced by Frame Homomorphism Onto Two | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Frame_Homomorphism_Onto_Two | https://proofwiki.org/wiki/Completely_Prime_Filter_Induced_by_Frame_Homomorphism_Onto_Two | [
"Completely Prime Filters",
"Two (Boolean Lattice)",
"Frame Homomorphisms"
] | [
"Definition:Frame (Lattice Theory)",
"Definition:Two (Boolean Lattice)",
"Definition:Frame Homomorphism",
"Definition:Completely Prime Filter",
"Definition:Preimage/Mapping/Element",
"Definition:Mapping"
] | [
"Singleton of Greatest Element in Two is Completely Prime Filter",
"Definition:Completely Prime Filter",
"Inverse Image Mapping of Frame Homomorphism Preserves Completely Prime Filter",
"Definition:Completely Prime Filter",
"Category:Completely Prime Filters",
"Category:Two (Boolean Lattice)",
"Category... |
proofwiki-22282 | Total Variation is Smallest Measure exceeding Modulus of Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be either a signed measure or complex measure on $\struct {X, \Sigma}$.
Let $\nu$ be a measure such that:
:$\cmod {\map \mu A} \le \map \nu A$ for each $A \in \Sigma$.
Let $\cmod \mu$ be the variation of $\mu$.
Then:
:$\map {\cmod \mu} A \le \map \nu A$ for ea... | Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.
Let $\set {A_1, A_2, \ldots, A_n} \in \map P A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} }
| o = \le
| r = \sum_{j \mathop = 1}^n \map \nu {A_j}
}}
{{eqn | r = \map \nu {\sum_{j \mathop ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be either a [[Definition:Signed Measure|signed measure]] or [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be a [[Definition:Measure (Measure Theory)|measure]] such that:
:$\cmod {\map \mu A}... | Let $\map P A$ be the set of [[Definition:Finite Set|finite]] [[Definition:Set Partition|partitions]] of $A$ into [[Definition:Measurable Set|$\Sigma$-measurable sets]].
Let $\set {A_1, A_2, \ldots, A_n} \in \map P A$.
Then we have:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} }
| o = \le
... | Total Variation is Smallest Measure exceeding Modulus of Measure | https://proofwiki.org/wiki/Total_Variation_is_Smallest_Measure_exceeding_Modulus_of_Measure | https://proofwiki.org/wiki/Total_Variation_is_Smallest_Measure_exceeding_Modulus_of_Measure | [
"Variation of Complex Measure",
"Variation of Signed Measure"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Complex Measure",
"Definition:Measure (Measure Theory)",
"Definition:Variation"
] | [
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Measurable Set",
"Definition:Supremum of Set/Real Numbers",
"Category:Variation of Complex Measure",
"Category:Variation of Signed Measure"
] |
proofwiki-22283 | Positive and Negative Parts of Signed Measure are Mutually Singular | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then $\mu^+$ and $\mu^-$ are mutually singular. | From the Hahn Decomposition Theorem, there exists $\mu$-positive set and a $\mu$-negative set such that:
:$X = P \cup N$
and:
:$P \cap N = \O$
From the Jordan Decomposition Theorem, we have:
:$\map {\mu^+} A = \map \mu {A \cap P}$
and:
:$\map {\mu^-} A = -\map \mu {A \cap N}$
for each $A \in \Sigma$.
From the definiti... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then $\mu^+$ and $\mu^-$ are [[Definition:Mutu... | From the [[Hahn Decomposition Theorem]], there exists [[Definition:Positive Set|$\mu$-positive set]] and a [[Definition:Negative Set|$\mu$-negative set]] such that:
:$X = P \cup N$
and:
:$P \cap N = \O$
From the [[Jordan Decomposition Theorem]], we have:
:$\map {\mu^+} A = \map \mu {A \cap P}$
and:
:$\map {\mu^-} A =... | Positive and Negative Parts of Signed Measure are Mutually Singular | https://proofwiki.org/wiki/Positive_and_Negative_Parts_of_Signed_Measure_are_Mutually_Singular | https://proofwiki.org/wiki/Positive_and_Negative_Parts_of_Signed_Measure_are_Mutually_Singular | [
"Signed Measures",
"Mutually Singular Measures",
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Jordan Decomposition",
"Definition:Mutually Singular Measures"
] | [
"Hahn Decomposition Theorem",
"Definition:Positive Set",
"Definition:Negative Set",
"Jordan Decomposition Theorem",
"Definition:Signed Measure",
"Definition:Concentration on Measurable Set",
"Definition:Concentration on Measurable Set",
"Definition:Mutually Singular Measures",
"Category:Mutually Sin... |
proofwiki-22284 | Lebesgue Decomposition Theorem/Uniqueness | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be either:
:a $\sigma$-finite measure
:a complex measure.
Then the Lebesgue decomposition of $\nu$ with respect to $\mu$ is unique. | Suppose that:
:$\nu = \nu_a + \nu_s = \nu_a' + \nu_s'$
where:
:$\nu_a$ and $\nu_a'$ are absolutely continuous with respect to $\mu$
:$\nu_s$ and $\nu_s'$ are mutually singular with respect to $\mu$. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure|measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be either:
:a [[Definition:Sigma-Finite Measure|$\sigma$-finite measure]]
:a [[Definition:Complex Measure|complex measure]].
Then the [[Definition:Lebes... | Suppose that:
:$\nu = \nu_a + \nu_s = \nu_a' + \nu_s'$
where:
:$\nu_a$ and $\nu_a'$ are [[Definition:Absolute Continuity/Complex Measure|absolutely continuous]] with respect to $\mu$
:$\nu_s$ and $\nu_s'$ are [[Definition:Mutually Singular Measures|mutually singular]] with respect to $\mu$. | Lebesgue Decomposition Theorem/Uniqueness | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Uniqueness | https://proofwiki.org/wiki/Lebesgue_Decomposition_Theorem/Uniqueness | [
"Lebesgue Decomposition Theorem"
] | [
"Definition:Measurable Space",
"Definition:Measure",
"Definition:Sigma-Finite Measure",
"Definition:Complex Measure",
"Definition:Lebesgue Decomposition"
] | [
"Definition:Absolute Continuity/Complex Measure",
"Definition:Mutually Singular Measures",
"Definition:Mutually Singular Measures",
"Definition:Absolute Continuity/Complex Measure",
"Definition:Absolute Continuity/Complex Measure",
"Definition:Absolute Continuity/Complex Measure",
"Definition:Mutually S... |
proofwiki-22285 | Triangle Inequality for Variation of Complex Measure | Let $\struct {X, \Sigma}$ be measurable space.
Let $\mu$ and $\nu$ be two complex measures.
Let $\cmod {\mu + \nu}$, $\cmod \mu$ and $\cmod \nu$ be the variation of $\mu + \nu$, $\mu$ and $\nu$ respectively.
Then:
:$\cmod {\mu + \nu} \le \cmod \mu + \cmod \nu$ | Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.
Let $\set {A_1, A_2, \ldots, A_n} \in \map P A$.
Then from Triangle Inequality for Complex Numbers we have:
:$\ds \sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} + \map \nu {A_j} } \le \sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } + \su... | Let $\struct {X, \Sigma}$ be [[Definition:Measurable Space|measurable space]].
Let $\mu$ and $\nu$ be two [[Definition:Complex Measure|complex measures]].
Let $\cmod {\mu + \nu}$, $\cmod \mu$ and $\cmod \nu$ be the [[Definition:Variation of Complex Measure|variation]] of $\mu + \nu$, $\mu$ and $\nu$ respectively.
T... | Let $\map P A$ be the set of [[Definition:Finite Set|finite]] [[Definition:Set Partition|partitions]] of $A$ into [[Definition:Measurable Set|$\Sigma$-measurable sets]].
Let $\set {A_1, A_2, \ldots, A_n} \in \map P A$.
Then from [[Triangle Inequality for Complex Numbers]] we have:
:$\ds \sum_{j \mathop = 1}^n \cmod ... | Triangle Inequality for Variation of Complex Measure | https://proofwiki.org/wiki/Triangle_Inequality_for_Variation_of_Complex_Measure | https://proofwiki.org/wiki/Triangle_Inequality_for_Variation_of_Complex_Measure | [
"Variation of Complex Measure",
"Triangle Inequality"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Variation/Complex Measure"
] | [
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Measurable Set",
"Triangle Inequality/Complex Numbers",
"Definition:Variation/Complex Measure",
"Definition:Supremum of Set/Real Numbers",
"Category:Variation of Complex Measure",
"Category:Triangle Inequality"
] |
proofwiki-22286 | Measure is Discrete iff Concentrated on Countable Set | Let $\struct {X, \Sigma}$ be a measurable space such that:
:$\set x \in \Sigma$ for all $x \in X$.
Let $\mu$ be a measure.
Then $\mu$ is discrete {{iff}}:
:there exists a countable set $C \subseteq X$ such that $\map \mu {X \setminus C} = 0$. | === Necessary Condition ===
Suppose that $\mu$ is discrete.
Then there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $X$ and a sequence of non-negative real numbers $\sequence {\lambda_n}_{n \mathop \in \N}$ such that:
:$\ds \mu = \sum_{n \mathop = 1}^\infty \lambda_n \delta_{x_n}$
where $\delta_{x_n}$ is t... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]] such that:
:$\set x \in \Sigma$ for all $x \in X$.
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]].
Then $\mu$ is [[Definition:Discrete Measure|discrete]] {{iff}}:
:there exists a [[Definition:Countable Set|countable set]... | === Necessary Condition ===
Suppose that $\mu$ is [[Definition:Discrete Measure|discrete]].
Then there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ in $X$ and a [[Definition:Sequence|sequence]] of [[Definition:Non-Negative Real Number|non-negative real numbers]] $\sequence {\lambda_n... | Measure is Discrete iff Concentrated on Countable Set | https://proofwiki.org/wiki/Measure_is_Discrete_iff_Concentrated_on_Countable_Set | https://proofwiki.org/wiki/Measure_is_Discrete_iff_Concentrated_on_Countable_Set | [
"Discrete Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Discrete Measure",
"Definition:Countable Set"
] | [
"Definition:Discrete Measure",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Positive/Real Number",
"Definition:Dirac Measure",
"Definition:Countable Set",
"Definition:Countable Set"
] |
proofwiki-22287 | Intersection Measures preserve Absolute Continuity | Let $\struct {X, \Sigma}$ be a measurable space.
Let $A \in \Sigma$.
Let $\mu$ and $\nu$ be measures such that:
:$\mu$ is absolutely continuous {{WRT}} $\nu$.
Let $\mu_A$ be the intersection measure of $\mu$ {{WRT}} $A$.
Then $\mu_A$ is absolutely continuous {{WRT}} $\nu$. | Let $B \in \Sigma$ be such that:
:$\map \nu B = 0$
From Null Sets Closed under Subset, we have that $\map \nu {A \cap B} = 0$.
Hence since $\mu$ is absolutely continuous {{WRT}} $\nu$, we have $\map \mu {A \cap B} = 0$.
That is, $\map {\mu_A} B = 0$.
Hence whenever $B \in \Sigma$ is a $\nu$-null set, it is a $\mu_A$-n... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $A \in \Sigma$.
Let $\mu$ and $\nu$ be [[Definition:Measure (Measure Theory)|measures]] such that:
:$\mu$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] {{WRT}} $\nu$.
Let $\mu_A$ be the [[Definition:Intersect... | Let $B \in \Sigma$ be such that:
:$\map \nu B = 0$
From [[Null Sets Closed under Subset]], we have that $\map \nu {A \cap B} = 0$.
Hence since $\mu$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] {{WRT}} $\nu$, we have $\map \mu {A \cap B} = 0$.
That is, $\map {\mu_A} B = 0$.
Hence whenever ... | Intersection Measures preserve Absolute Continuity | https://proofwiki.org/wiki/Intersection_Measures_preserve_Absolute_Continuity | https://proofwiki.org/wiki/Intersection_Measures_preserve_Absolute_Continuity | [
"Absolutely Continuous Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Absolute Continuity/Measure",
"Definition:Intersection Measure",
"Definition:Absolute Continuity/Measure"
] | [
"Null Sets Closed under Subset",
"Definition:Absolute Continuity/Measure",
"Definition:Null Set",
"Definition:Null Set",
"Definition:Absolute Continuity/Measure",
"Category:Absolutely Continuous Measures"
] |
proofwiki-22288 | Intersection Measures preserve Mutual Singularity | Let $\struct {X, \Sigma}$ be a measurable space.
Let $A \in \Sigma$.
Let $\mu$ and $\nu$ be measures such that:
:$\mu$ is mutually singular {{WRT}} $\nu$.
Let $\mu_A$ be the intersection measure of $\mu$ {{WRT}} $A$.
Then $\mu_A$ is mutually singular {{WRT}} $\nu$. | Since $\mu$ is mutually singular with respect to $\nu$, there exists a $\nu$-null set $N$ such that $\map \mu {N^c} = 0$.
From Null Sets Closed under Subset, we have $\map \mu {A \cap N^c} = 0$.
Hence $\map {\mu_A} {N^c} = 0$.
Hence $N$ is a $\nu$-null set such that $\map {\mu_A} {N^c} = 0$.
Hence $\mu_A$ is mutually ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $A \in \Sigma$.
Let $\mu$ and $\nu$ be [[Definition:Measure (Measure Theory)|measures]] such that:
:$\mu$ is [[Definition:Mutually Singular Measures|mutually singular]] {{WRT}} $\nu$.
Let $\mu_A$ be the [[Definition:Intersection Mea... | Since $\mu$ is [[Definition:Mutually Singular Measures|mutually singular]] with respect to $\nu$, there exists a [[Definition:Null Set|$\nu$-null set]] $N$ such that $\map \mu {N^c} = 0$.
From [[Null Sets Closed under Subset]], we have $\map \mu {A \cap N^c} = 0$.
Hence $\map {\mu_A} {N^c} = 0$.
Hence $N$ is a [[De... | Intersection Measures preserve Mutual Singularity | https://proofwiki.org/wiki/Intersection_Measures_preserve_Mutual_Singularity | https://proofwiki.org/wiki/Intersection_Measures_preserve_Mutual_Singularity | [
"Mutually Singular Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Mutually Singular Measures",
"Definition:Intersection Measure",
"Definition:Mutually Singular Measures"
] | [
"Definition:Mutually Singular Measures",
"Definition:Null Set",
"Null Sets Closed under Subset",
"Definition:Null Set",
"Definition:Mutually Singular Measures",
"Category:Mutually Singular Measures"
] |
proofwiki-22289 | Lebesgue Decomposition of Finite Borel Measure | Let $\map \BB \R$ be the Borel $\sigma$-algebra of the real number line $\R$.
Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$.
Let $\mu$ be a finite Borel measure on $\struct {\R, \map \BB \R}$.
Then there exists:
:a discrete measure $\mu_{pp}$
:a measure $\mu_{sc}$ that is continuous and mutually... | Let:
:$C = \set {x \in \R : \map \mu {\set x} > 0}$
We argue that $C$ is countable.
Note that:
:$\ds C = \bigcup_{n \mathop = 1}^\infty \set {x \in \R : \map \mu {\set x} > \frac 1 n}$
We show that each set:
:$\ds C_n = \set {x \in \R : \map \mu {\set x} > \frac 1 n}$
is finite.
{{AimForCont}} that $C_n$ is infinite f... | Let $\map \BB \R$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] of the [[Definition:Real Number Line|real number line]] $\R$.
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $\struct {\R, \map \BB \R}$.
Let $\mu$ be a [[Definition:Finite Measure|finite]] [[Definition:Borel... | Let:
:$C = \set {x \in \R : \map \mu {\set x} > 0}$
We argue that $C$ is [[Definition:Countable Set|countable]].
Note that:
:$\ds C = \bigcup_{n \mathop = 1}^\infty \set {x \in \R : \map \mu {\set x} > \frac 1 n}$
We show that each set:
:$\ds C_n = \set {x \in \R : \map \mu {\set x} > \frac 1 n}$
is [[Definition:Fi... | Lebesgue Decomposition of Finite Borel Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_of_Finite_Borel_Measure | https://proofwiki.org/wiki/Lebesgue_Decomposition_of_Finite_Borel_Measure | [
"Lebesgue Decomposition Theorem",
"Discrete Measures",
"Mutually Singular Measures",
"Absolutely Continuous Measures"
] | [
"Definition:Borel Sigma-Algebra",
"Definition:Real Number/Real Number Line",
"Definition:Lebesgue Measure",
"Definition:Finite Measure",
"Definition:Borel Measure",
"Definition:Discrete Measure",
"Definition:Measure",
"Definition:Continuous Measure",
"Definition:Mutually Singular Measures",
"Defin... | [
"Definition:Countable Set",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Countably Infinite/Set",
"Definition:Finite Measure",
"Definition:Finite Set",
"Countable Union of Finite Sets is Countable",
"Definition:Countable Set",
"Measure is Discrete iff Concentrated on Countable Set... |
proofwiki-22290 | Graph of Linear Transformation is Vector Subspace | Let $K$ be a field.
Let $X$ and $Y$ be vector spaces over $K$.
Let $X \times Y$ be the direct product of $X$ and $Y$.
Let $T : X \to Y$ be a linear transformation.
Define:
:$\map \GG T = \set {\tuple {x, T x} : x \in X} \subseteq X \times Y$
Then $\map \GG T$ is a vector subspace of $X \times Y$. | Since $\map T { {\mathbf 0}_X} = {\mathbf 0}_Y$, we have:
:$\tuple { {\mathbf 0}_X, {\mathbf 0}_Y} \in \map \GG T$
So $\map \GG T \ne \O$.
Hence from One-Step Vector Subspace Test, it is enough to show that $u + \lambda v \in \map \GG T$ for $u, v \in \map \GG T$ and $\lambda \in K$.
Let $\alpha \in K$ and $\tuple {x... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $K$.
Let $X \times Y$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $X$ and $Y$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
... | Since $\map T { {\mathbf 0}_X} = {\mathbf 0}_Y$, we have:
:$\tuple { {\mathbf 0}_X, {\mathbf 0}_Y} \in \map \GG T$
So $\map \GG T \ne \O$.
Hence from [[One-Step Vector Subspace Test]], it is enough to show that $u + \lambda v \in \map \GG T$ for $u, v \in \map \GG T$ and $\lambda \in K$.
Let $\alpha \in K$ and $\t... | Graph of Linear Transformation is Vector Subspace | https://proofwiki.org/wiki/Graph_of_Linear_Transformation_is_Vector_Subspace | https://proofwiki.org/wiki/Graph_of_Linear_Transformation_is_Vector_Subspace | [
"Direct Product of Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Linear Transformation",
"Definition:Vector Subspace"
] | [
"One-Step Vector Subspace Test",
"Definition:Linear Transformation",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Category:Direct Product of Vector Spaces"
] |
proofwiki-22291 | Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Consider $\HH \times \HH$ with the direct product norm $\norm {\, \cdot \,}_{\HH \times \HH}$.
Let $\tuple {\map D T, T}$ be a densely-defined linear operator.
Let $\map \GG T$ be the graph of $T$.
Then $T$ is closable {{iff}}:
:whenever $\tuple ... | === Necessary Condition ===
Suppose that $T$ is closed.
Then $\map \cl {\map \GG T} = \map \GG T$ from Set is Closed iff Equals Topological Closure.
Then if $\tuple { {\mathbf 0}_X, y} \in \map \GG T$, we have $y = \map T { {\mathbf 0}_X}$ from the definition of the graph.
So $y = {\mathbf 0}_Y$ since $T$ is linear.
{... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Consider $\HH \times \HH$ with the [[Definition:Direct Product Norm|direct product norm]] $\norm {\, \cdot \,}_{\HH \times \HH}$.
Let $\tuple {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-... | === Necessary Condition ===
Suppose that $T$ is [[Definition:Closed Linear Transformation|closed]].
Then $\map \cl {\map \GG T} = \map \GG T$ from [[Set is Closed iff Equals Topological Closure]].
Then if $\tuple { {\mathbf 0}_X, y} \in \map \GG T$, we have $y = \map T { {\mathbf 0}_X}$ from the definition of the [[... | Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph | https://proofwiki.org/wiki/Characterization_of_Closable_Densely-Defined_Linear_Operators_in_terms_of_Closure_of_Graph | https://proofwiki.org/wiki/Characterization_of_Closable_Densely-Defined_Linear_Operators_in_terms_of_Closure_of_Graph | [
"Closable Densely-Defined Linear Operators",
"Closed Linear Transformations",
"Closable Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Direct Product Norm",
"Definition:Densely-Defined Linear Operator",
"Definition:Graph of Mapping",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Closure (Topology)"
] | [
"Definition:Closed Linear Transformation",
"Set is Closed iff Equals Topological Closure",
"Definition:Graph of Mapping",
"Definition:Linear Transformation",
"Definition:Closed Linear Transformation"
] |
proofwiki-22292 | Linear Transformation defined from Graph | Let $K$ be a field.
Let $X$ and $Y$ be vector spaces over $K$.
Let $X \times Y$ be the direct product of $X$ and $Y$.
Let $U$ be a vector subspace of $X \times Y$ such that:
:whenever $\tuple { {\mathbf 0}_X, y} \in U$, we have $y = {\mathbf 0}_Y$
Define:
:$\map D T = \set {x \in X : \tuple {x, y} \in U \text { for so... | We first show that $\map D T$ is a vector subspace of $X$.
We have $\tuple { {\mathbf 0}_X, {\mathbf 0}_Y} \in U$, so ${\mathbf 0}_X \in \map D T$.
Hence $\map D T \ne \O$.
Let $x, y \in \map D T$ and $\alpha \in \C$.
Then there exists $u, v \in Y$ such that $\tuple {x, u} \in U$ and $\tuple {y, v} \in U$.
Since $U$... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ and $Y$ be [[Definition:Vector Space|vector spaces]] over $K$.
Let $X \times Y$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $X$ and $Y$.
Let $U$ be a [[Definition:Vector Subspace|vector subspace]] of $X \times Y$ such th... | We first show that $\map D T$ is a [[Definition:Vector Subspace|vector subspace]] of $X$.
We have $\tuple { {\mathbf 0}_X, {\mathbf 0}_Y} \in U$, so ${\mathbf 0}_X \in \map D T$.
Hence $\map D T \ne \O$.
Let $x, y \in \map D T$ and $\alpha \in \C$.
Then there exists $u, v \in Y$ such that $\tuple {x, u} \in U$ an... | Linear Transformation defined from Graph | https://proofwiki.org/wiki/Linear_Transformation_defined_from_Graph | https://proofwiki.org/wiki/Linear_Transformation_defined_from_Graph | [
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Vector Subspace",
"Definition:Linear Transformation",
"Definition:Graph of Mapping"
] | [
"Definition:Vector Subspace",
"Definition:Vector Subspace",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace",
"Definition:Vector Subspace",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Category:Linear Transformations"
] |
proofwiki-22293 | Closable Densely-Defined Linear Operator has Smallest Closed Extension | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a closable densely-defined linear operator.
Then there exists a closed densely-defined linear extension $\struct {\map D S, S}$ of $T$ such that:
:the graph $\map \GG S$ is $\subseteq$-minimal among the graphs of clo... | From Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph:
:whenever $\tuple { {\mathbf 0}_\HH, y} \in \map \cl {\map \GG T}$, we have $y = {\mathbf 0}_\HH$.
Define:
:$\map D S = \set {x \in \HH : \tuple {x, y} \in \map \cl {\map \GG T} \text { for some } y \in \HH}$
From Linear Tr... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Closable Densely-Defined Linear Operator|closable densely-defined linear operator]].
Then there exists a [[Definition:Closed Linear Transformation|closed densely-defined... | From [[Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph]]:
:whenever $\tuple { {\mathbf 0}_\HH, y} \in \map \cl {\map \GG T}$, we have $y = {\mathbf 0}_\HH$.
Define:
:$\map D S = \set {x \in \HH : \tuple {x, y} \in \map \cl {\map \GG T} \text { for some } y \in \HH}$
From [[L... | Closable Densely-Defined Linear Operator has Smallest Closed Extension | https://proofwiki.org/wiki/Closable_Densely-Defined_Linear_Operator_has_Smallest_Closed_Extension | https://proofwiki.org/wiki/Closable_Densely-Defined_Linear_Operator_has_Smallest_Closed_Extension | [
"Closable Densely-Defined Linear Operators",
"Closed Linear Transformations"
] | [
"Definition:Hilbert Space",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Closed Linear Transformation",
"Definition:Extension of Mapping",
"Definition:Graph of Mapping",
"Definition:Minimal/Element",
"Definition:Graph of Mapping",
"Definition:Closed Linear Transformation",
"Def... | [
"Characterization of Closable Densely-Defined Linear Operators in terms of Closure of Graph",
"Linear Transformation defined from Graph",
"Definition:Linear Transformation",
"Definition:Closed Linear Transformation",
"Definition:Extension of Mapping",
"Definition:Closed Set",
"Set Closure Preserves Set ... |
proofwiki-22294 | Closable Densely-Defined Operator is Closed iff Equal to Closure | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a closable densely-defined linear operator.
Let $\tuple {\map D {\overline T}, \overline T}$ be the closure of $\tuple {\map D T, T}$.
Then $\tuple {\map D T, T}$ is closed {{iff}} $T = \overline T$. | Let $\struct {\HH \times \HH, \norm {\, \cdot \,}_{\HH \times \HH} }$ be the direct product of $\HH$ with itself equipped with the direct product norm.
Suppose that $\tuple {\map D T, T}$ is closed.
From Set is Closed iff Equals Topological Closure, we have that $\map \cl {\map \GG T} = \map \GG T$.
From Closable Dens... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Closable Densely-Defined Linear Operator|closable densely-defined linear operator]].
Let $\tuple {\map D {\overline T}, \overline T}$ be the [[Definition:Closure of Closa... | Let $\struct {\HH \times \HH, \norm {\, \cdot \,}_{\HH \times \HH} }$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $\HH$ with itself equipped with the [[Definition:Direct Product Norm|direct product norm]].
Suppose that $\tuple {\map D T, T}$ is [[Definition:Closed Linear Transformation|clo... | Closable Densely-Defined Operator is Closed iff Equal to Closure | https://proofwiki.org/wiki/Closable_Densely-Defined_Operator_is_Closed_iff_Equal_to_Closure | https://proofwiki.org/wiki/Closable_Densely-Defined_Operator_is_Closed_iff_Equal_to_Closure | [
"Closable Densely-Defined Linear Operators",
"Closed Linear Transformations",
"Closable Densely-Defined Linear Operators"
] | [
"Definition:Hilbert Space",
"Definition:Closable Densely-Defined Linear Operator",
"Definition:Closure of Closable Densely-Defined Linear Operator",
"Definition:Closed Linear Transformation"
] | [
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Definition:Closed Linear Transformation",
"Set is Closed iff Equals Topological Closure",
"Closable Densely-Defined Linear Operator has Smallest Closed Extension",
"Definition:Closed Set",
"Definition:Closed Linear Transfor... |
proofwiki-22295 | Square of V Operator on Hilbert Space | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the Hilbert space direct sum of $\HH$ with itself.
Define $V : \HH \times \HH \to \HH \times \HH$ by:
:$\map V {x, y} = \tuple {-y, x}$
for each $\tuple {x, y} \in \HH \t... | For each $x, y \in \HH$ we have:
{{begin-eqn}}
{{eqn | l = \map {V^2} {x, y}
| r = \map V {-y, x}
}}
{{eqn | r = \tuple {-x, -y}
}}
{{eqn | r = -\tuple {x, y}
}}
{{end-eqn}}
{{qed}} | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the [[Definition:Hilbert Space Direct Sum|Hilbert space direct sum]] of $\HH$ with itself.
Define $V : \HH \times \HH \to \HH \times \HH$ b... | For each $x, y \in \HH$ we have:
{{begin-eqn}}
{{eqn | l = \map {V^2} {x, y}
| r = \map V {-y, x}
}}
{{eqn | r = \tuple {-x, -y}
}}
{{eqn | r = -\tuple {x, y}
}}
{{end-eqn}}
{{qed}} | Square of V Operator on Hilbert Space | https://proofwiki.org/wiki/Square_of_V_Operator_on_Hilbert_Space | https://proofwiki.org/wiki/Square_of_V_Operator_on_Hilbert_Space | [
"V Operators on Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Hilbert Space Direct Sum"
] | [] |
proofwiki-22296 | Adjoint of Densely-Defined Linear Operator is Closed | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.
Let $\tuple {\map D T, T}$ be a densely-defined linear operator.
Let $\tuple {\map D {T^\ast}, T^\ast}$ be the adjoint of $\tuple {\map D T, T}$.
Then $\tuple {\map D {T^\ast}, T^\ast}$ is closed. | Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the Hilbert space direct sum of $\HH$ with itself.
From Equivalent Norms on Direct Product of Normed Vector Spaces, the inner product norm on $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ is equivalent to the direct produ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]] over $\C$.
Let $\tuple {\map D T, T}$ be a [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\tuple {\map D {T^\ast}, T^\ast}$ be the [[Definition:Adjoint of Densely-Defined Linear Operator|a... | Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the [[Definition:Hilbert Space Direct Sum|Hilbert space direct sum]] of $\HH$ with itself.
From [[Equivalent Norms on Direct Product of Normed Vector Spaces]], the [[Definition:Inner Product Norm|inner product norm]] on $\struct {\HH \times \H... | Adjoint of Densely-Defined Linear Operator is Closed | https://proofwiki.org/wiki/Adjoint_of_Densely-Defined_Linear_Operator_is_Closed | https://proofwiki.org/wiki/Adjoint_of_Densely-Defined_Linear_Operator_is_Closed | [
"Adjoints (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Densely-Defined Linear Operator",
"Definition:Adjoint of Densely-Defined Linear Operator",
"Definition:Closed Linear Transformation"
] | [
"Definition:Hilbert Space Direct Sum",
"Equivalent Norms on Direct Product of Normed Vector Spaces",
"Definition:Inner Product Norm",
"Definition:Equivalence of Norms",
"Definition:Direct Product Norm",
"Open Sets in Vector Spaces with Equivalent Norms Coincide",
"Definition:Closed Set",
"Graph of Adj... |
proofwiki-22297 | Hilbert Space Isomorphism preserves Orthocomplements | Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be Hilbert spaces.
Let $T : \HH \to \KK$ be a Hilbert space isomorphism.
Let $U \subseteq \HH$.
Then:
:$T \sqbrk {U^\bot} = \paren {T \sqbrk U}^\bot$ | Let $x \in \KK$.
Since $T$ is a Hilbert space isomorphism, there exists $v \in \HH$ such that $x = T v$.
We then have $x \in \paren {T \sqbrk U}^\bot$ {{iff}}:
:$\innerprod x {T u}_\KK = 0$ for each $u \in U$.
That is:
:$\innerprod {T v} {T u}_\KK = 0$
Since $T$ is a Hilbert space isomorphism, this is equivalent to $\i... | Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ and $\struct {\KK, \innerprod \cdot \cdot_\KK}$ be [[Definition:Hilbert Space|Hilbert spaces]].
Let $T : \HH \to \KK$ be a [[Definition:Isomorphism (Hilbert Spaces)|Hilbert space isomorphism]].
Let $U \subseteq \HH$.
Then:
:$T \sqbrk {U^\bot} = \paren {T \sqbrk U}^\bo... | Let $x \in \KK$.
Since $T$ is a [[Definition:Isomorphism (Hilbert Spaces)|Hilbert space isomorphism]], there exists $v \in \HH$ such that $x = T v$.
We then have $x \in \paren {T \sqbrk U}^\bot$ {{iff}}:
:$\innerprod x {T u}_\KK = 0$ for each $u \in U$.
That is:
:$\innerprod {T v} {T u}_\KK = 0$
Since $T$ is a [[De... | Hilbert Space Isomorphism preserves Orthocomplements | https://proofwiki.org/wiki/Hilbert_Space_Isomorphism_preserves_Orthocomplements | https://proofwiki.org/wiki/Hilbert_Space_Isomorphism_preserves_Orthocomplements | [
"Orthocomplements",
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Isomorphism (Hilbert Spaces)"
] | [
"Definition:Isomorphism (Hilbert Spaces)",
"Definition:Isomorphism (Hilbert Spaces)",
"Category:Orthocomplements",
"Category:Hilbert Spaces"
] |
proofwiki-22298 | Solution to Quadratic Congruence | The quadratic congruence:
:$a x^2 + b x + c \equiv 0 \pmod n$
can be solved by solving the congruence:
:$y^2 \equiv \paren {b^2 - 4 a c} \pmod n$
and the linear congruence:
:$2 a x + b \equiv y \pmod n$ | {{ProofWanted|use the method of completing the square}} | The [[Definition:Quadratic Congruence|quadratic congruence]]:
:$a x^2 + b x + c \equiv 0 \pmod n$
can be solved by solving the [[Definition:Congruence Modulo Integer|congruence]]:
:$y^2 \equiv \paren {b^2 - 4 a c} \pmod n$
and the [[Definition:Linear Congruence|linear congruence]]:
:$2 a x + b \equiv y \pmod n$ | {{ProofWanted|use the method of completing the square}} | Solution to Quadratic Congruence | https://proofwiki.org/wiki/Solution_to_Quadratic_Congruence | https://proofwiki.org/wiki/Solution_to_Quadratic_Congruence | [
"Quadratic Congruences"
] | [
"Definition:Quadratic Congruence",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Linear Congruence"
] | [] |
proofwiki-22299 | Tannery's Theorem | Let $\sequence {p_n}$ be an increasing, unbounded above sequence of natural numbers.
For every $r, n \in \N$, let $v_r$ be a mapping:
:$\map {v_r} n \in \C$
Let $\sequence {w_r}$ be a sequence of complex numbers such that, for every $r \in \N$:
:$\ds w_r = \lim_{n \mathop \to \infty} \map {v_r} n$
Let $\sequence {M_r}$... | First, observe that for all $r \in \N$:
{{begin-eqn}}
{{eqn | l = \size {w_r}
| r = \size {\lim_{n \mathop \to \infty} \map {v_r} n}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \size {\map {v_r} n}
| c = Complex Modulus Function is Continuous
}}
{{eqn | o = \le
| r = M_r
| c = Lower and Upper Bou... | Let $\sequence {p_n}$ be an [[Definition:Increasing Sequence|increasing]], [[Definition:Unbounded Above Sequence|unbounded above sequence]] of [[Definition:Natural Number|natural numbers]].
For every $r, n \in \N$, let $v_r$ be a [[Definition:Mapping|mapping]]:
:$\map {v_r} n \in \C$
Let $\sequence {w_r}$ be a [[Defi... | First, observe that for all $r \in \N$:
{{begin-eqn}}
{{eqn | l = \size {w_r}
| r = \size {\lim_{n \mathop \to \infty} \map {v_r} n}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \size {\map {v_r} n}
| c = [[Complex Modulus Function is Continuous]]
}}
{{eqn | o = \le
| r = M_r
| c = [[Lower and Upp... | Tannery's Theorem | https://proofwiki.org/wiki/Tannery's_Theorem | https://proofwiki.org/wiki/Tannery's_Theorem | [
"Analysis",
"Series"
] | [
"Definition:Increasing/Sequence",
"Definition:Bounded Above Sequence/Unbounded",
"Definition:Natural Numbers",
"Definition:Mapping",
"Definition:Sequence/Infinite Sequence",
"Definition:Complex Number",
"Definition:Sequence/Infinite Sequence",
"Definition:Positive/Real Number",
"Definition:Convergen... | [
"Complex Modulus Function is Continuous",
"Lower and Upper Bounds for Sequences",
"Tail of Convergent Series tends to Zero",
"Definition:Bounded Above Sequence/Unbounded",
"Definition:Increasing/Sequence",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Triangle Inequality/Complex Numbers/General... |
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