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-20500 | T3 Lindelöf Space is T4 | Let $T = \struct {S, \tau}$ be a $T_3$ Lindelöf topological space.
Then $T$ is a $T_4$ space. | From $T_3$ Lindelöf Space is Fully $T_4$:
:$T$ is a fully $T_4$ space.
From Fully $T_4$ Space is $T_4$:
:$T$ is a $T_4$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$]] [[Definition:Lindelöf Space|Lindelöf]] [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | From [[T3 Lindelöf Space is Fully T4|$T_3$ Lindelöf Space is Fully $T_4$]]:
:$T$ is a [[Definition:Fully T4 Space|fully $T_4$ space]].
From [[Fully T4 Space is T4|Fully $T_4$ Space is $T_4$]]:
:$T$ is a [[Definition:T4 Space|$T_4$ space]].
{{qed}} | T3 Lindelöf Space is T4/Proof 2 | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_T4 | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_T4/Proof_2 | [
"T3 Spaces",
"Lindelöf Spaces",
"T4 Spaces",
"T3 Lindelöf Space is T4"
] | [
"Definition:T3 Space",
"Definition:Lindelöf Space",
"Definition:Topological Space",
"Definition:T4 Space"
] | [
"T3 Lindelöf Space is Fully T4",
"Definition:Fully T4 Space",
"Fully T4 Space is T4",
"Definition:T4 Space"
] |
proofwiki-20501 | T3 Lindelöf Space is Fully T4 | Let $T = \struct {S, \tau}$ be a $T_3$ Lindelöf topological space.
Then $T$ is a fully $T_4$ space. | From $T_3$ Lindelöf Space is Paracompact:
:$T$ is a paracompact space
From $T_3$ Space is Fully $T_4$ iff Paracompact:
:$T$ is a fully $T_4$ space
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$]] [[Definition:Lindelöf Space|Lindelöf]] [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:Fully T4 Space|fully $T_4$ space]]. | From [[T3 Lindelöf Space is Paracompact|$T_3$ Lindelöf Space is Paracompact]]:
:$T$ is a [[Definition:Paracompact Space|paracompact space]]
From [[T3 Space is Fully T4 iff Paracompact|$T_3$ Space is Fully $T_4$ iff Paracompact]]:
:$T$ is a [[Definition:Fully T4 Space|fully $T_4$ space]]
{{qed}} | T3 Lindelöf Space is Fully T4 | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_Fully_T4 | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_Fully_T4 | [
"T3 Spaces",
"Lindelöf Spaces",
"Fully T4 Spaces"
] | [
"Definition:T3 Space",
"Definition:Lindelöf Space",
"Definition:Topological Space",
"Definition:Fully T4 Space"
] | [
"T3 Lindelöf Space is Paracompact",
"Definition:Paracompact Space",
"T3 Space is Fully T4 iff Paracompact",
"Definition:Fully T4 Space"
] |
proofwiki-20502 | Mazur's Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} } $ be a normed vector space over $\GF$ with weak topology $w$.
Let $C \subseteq X$ be a convex subset of $X$.
Then:
:$\map {\cl_w} C = \map \cl C$
where $\cl_w$ denotes the weak closure. | From Topological Closure in Coarser Topology is Larger:
:$\map \cl C \subseteq \map {\cl_w} C$
Now let $x \not \in \map \cl C$.
From Finite Topological Space is Compact, $\set x$ is compact.
Applying:
:Hahn-Banach Separation Theorem: Compact Convex Set and Closed Convex Set (Real Case) if $\GF = \R$
:Hahn-Banach Se... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} } $ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Let $C \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Then... | From [[Topological Closure in Coarser Topology is Larger]]:
:$\map \cl C \subseteq \map {\cl_w} C$
Now let $x \not \in \map \cl C$.
From [[Finite Topological Space is Compact]], $\set x$ is [[Definition:Compact Topological Space|compact]].
Applying:
:[[Hahn-Banach Separation Theorem/Normed Vector Space/Real Ca... | Mazur's Theorem | https://proofwiki.org/wiki/Mazur's_Theorem | https://proofwiki.org/wiki/Mazur's_Theorem | [
"Mazur's Theorem",
"Weak Topologies on Topological Vector Spaces",
"Convex Sets (Vector Spaces)",
"Weakly Closed Sets"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Weak Closure"
] | [
"Topological Closure in Coarser Topology is Larger",
"Finite Topological Space is Compact",
"Definition:Compact Topological Space",
"Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Compact Convex Set and Closed Convex Set",
"Hahn-Banach Separation Theorem/Normed Vector Space/Complex Case/Compac... |
proofwiki-20503 | Open Ball in Infinite-Dimensional Normed Vector Space is not Weakly Open | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$ with weak topology $w$.
Suppose that $\dim X = \infty$.
Let $\map {B_r} x$ be the open ball in $X$ with radius $r > 0$ and center $x \in X$.
Then $\map {B_r} x$ is not weakly open. | From Translation of Open Set in Normed Vector Space is Open and Dilation of Open Set in Normed Vector Space is Open, it suffices to show that $\map {B_1} 0$ is not weakly open.
{{AimForCont}} that $\map {B_1} 0$ is weakly open.
From Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex, $\... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Suppose that $\dim X = \infty$.
Let $\map {B_r} x$ be the [[Definition:Open Ball|open ball]] i... | From [[Translation of Open Set in Normed Vector Space is Open]] and [[Dilation of Open Set in Normed Vector Space is Open]], it suffices to show that $\map {B_1} 0$ is not [[Definition:Weakly Open Set|weakly open]].
{{AimForCont}} that $\map {B_1} 0$ is [[Definition:Weakly Open Set|weakly open]].
From [[Initial Topo... | Open Ball in Infinite-Dimensional Normed Vector Space is not Weakly Open | https://proofwiki.org/wiki/Open_Ball_in_Infinite-Dimensional_Normed_Vector_Space_is_not_Weakly_Open | https://proofwiki.org/wiki/Open_Ball_in_Infinite-Dimensional_Normed_Vector_Space_is_not_Weakly_Open | [
"Open Balls",
"Weakly Open Sets"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Weakly Open Set"
] | [
"Translation of Open Set in Normed Vector Space is Open",
"Dilation of Open Set in Normed Vector Space is Open",
"Definition:Weakly Open Set",
"Definition:Weakly Open Set",
"Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex",
"Definition:Locally Convex Space",
"Open Sets... |
proofwiki-20504 | Equation of Plane Wave is Particular Solution of Wave Equation/Direction Cosine Form | Let $\phi$ be a plane wave propagated with velocity $c$ in a Cartesian $3$-space.
Let $\phi$ be expressed as:
:$\map \phi {x, y, z, t} = \map f {l x + m y + n z - c t}$
where $l$, $m$ and $n$ are the direction cosines of the normal to $P$.
Then $\phi$ satisfies the wave equation. | The wave equation is expressible as:
:$\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$
Let $u = l x + m y + n z - c t$.
The second partial derivatives {{WRT|Differentiation}} $x$ are as ... | Let $\phi$ be a [[Definition:Plane Wave|plane wave]] [[Definition:Direction of Propagation of Wave|propagated]] with [[Definition:Velocity|velocity]] $c$ in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
Let $\phi$ be expressed as:
:$\map \phi {x, y, z, t} = \map f {l x + m y + n z - c t}$
where $l$, $m$ and ... | The [[Definition:Wave Equation|wave equation]] is expressible as:
:$\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$
Let $u = l x + m y + n z - c t$.
The second [[Definition:Partial D... | Equation of Plane Wave is Particular Solution of Wave Equation/Direction Cosine Form | https://proofwiki.org/wiki/Equation_of_Plane_Wave_is_Particular_Solution_of_Wave_Equation/Direction_Cosine_Form | https://proofwiki.org/wiki/Equation_of_Plane_Wave_is_Particular_Solution_of_Wave_Equation/Direction_Cosine_Form | [
"Equation of Plane Wave is Particular Solution of Wave Equation"
] | [
"Definition:Plane Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Velocity",
"Definition:Cartesian 3-Space",
"Definition:Direction Cosines",
"Definition:Normal Vector",
"Definition:Wave Equation"
] | [
"Definition:Wave Equation",
"Definition:Partial Derivative",
"Chain Rule for Partial Derivatives",
"Derivative of Constant Multiple",
"Chain Rule for Partial Derivatives",
"Definition:Partial Derivative",
"Definition:Wave Equation",
"Relation between Direction Cosines"
] |
proofwiki-20505 | Continuous Linear Transformation Space as Banach Algebra | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $*_X : X \times X \to X$ and $* : \map {CL} X \times \map {CL} X \to \map {CL} X$ be bilinear mappings.
Suppose $\struct {\struct {X, \norm {\, \cdot \,}_X}, *_X}$ is... | {{questionable|Please review the source. This cannot be true for any $*$, I believe this should be a composition. Mapping everything to zero is a counterexample.}}
{{ProofWanted}} | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $*_X : X \times X \to X$ and $* : \map {CL} X \times \map {CL} X \to \map {... | {{questionable|Please review the source. This cannot be true for any $*$, I believe this should be a composition. Mapping everything to zero is a counterexample.}}
{{ProofWanted}} | Continuous Linear Transformation Space as Banach Algebra | https://proofwiki.org/wiki/Continuous_Linear_Transformation_Space_as_Banach_Algebra | https://proofwiki.org/wiki/Continuous_Linear_Transformation_Space_as_Banach_Algebra | [] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Bilinear Mapping",
"Definition:Banach Algebra",
"Definition:Supremum Operator Norm",
"Definition:Banach Algebra"
] | [] |
proofwiki-20506 | Set of Points for which Seminorm is Zero is Vector Subspace | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $p$ be a seminorm on $X$.
Let:
:$U = \set {x \in X : \map p x = 0}$
Then $U$ is a vector subspace of $X$. | From Seminorm Maps Zero Vector to Zero, $\map p {\mathbf 0_X} = 0$.
So $\mathbf 0_X \in U$ and in particular $U \ne \O$.
So we look to apply One-Step Vector Subspace Test.
Let $x, y \in U$ and $\lambda, \mu \in \GF$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map p {\lambda x + \mu y}
| o = \le
| r = \map p {\lambda... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let:
:$U = \set {x \in X : \map p x = 0}$
Then $U$ is a [[Definition:Vector Subspace|vector subspace]] of $X$. | From [[Seminorm Maps Zero Vector to Zero]], $\map p {\mathbf 0_X} = 0$.
So $\mathbf 0_X \in U$ and in particular $U \ne \O$.
So we look to apply [[One-Step Vector Subspace Test]].
Let $x, y \in U$ and $\lambda, \mu \in \GF$.
Then we have:
{{begin-eqn}}
{{eqn | l = \map p {\lambda x + \mu y}
| o = \le
| r = \m... | Set of Points for which Seminorm is Zero is Vector Subspace | https://proofwiki.org/wiki/Set_of_Points_for_which_Seminorm_is_Zero_is_Vector_Subspace | https://proofwiki.org/wiki/Set_of_Points_for_which_Seminorm_is_Zero_is_Vector_Subspace | [
"Seminorms",
"Vector Subspaces"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Vector Subspace"
] | [
"Seminorm Maps Zero Vector to Zero",
"One-Step Vector Subspace Test",
"One-Step Vector Subspace Test",
"Definition:Vector Subspace"
] |
proofwiki-20507 | Open Ball with respect to Seminorm is Convex, Balanced and Absorbing | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $p$ be a seminorm on $X$.
Let $d_p$ be the pseudometric induced by $p$.
Let $B$ be the open unit ball in $\struct {X, d_p}$.
That is:
:$B = \set {x \in X : \map p x < 1}$
Then $B$ is convex, balanced and absorbing. | === Proof that $B$ is convex ===
Let $t \in \closedint 0 1$ and $x, y \in B$.
Then:
{{begin-eqn}}
{{eqn | l = \map p {t x + \paren {1 - t} y}
| o = \le
| r = t \map p x + \paren {1 - t} \map p y
| c = {{SeminormAxiom|2}}, {{SeminormAxiom|3}}
}}
{{eqn | o = <
| r = t + \paren {1 - t}
}}
{{eqn | r = 1
}}
{{end-e... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let $d_p$ be the [[Definition:Pseudometric Induced by Seminorm|pseudometric induced by $p$]].
Let $B$ be the [[Definition:Open Unit Ball|open unit ball]] in $\struct ... | === Proof that $B$ is convex ===
Let $t \in \closedint 0 1$ and $x, y \in B$.
Then:
{{begin-eqn}}
{{eqn | l = \map p {t x + \paren {1 - t} y}
| o = \le
| r = t \map p x + \paren {1 - t} \map p y
| c = {{SeminormAxiom|2}}, {{SeminormAxiom|3}}
}}
{{eqn | o = <
| r = t + \paren {1 - t}
}}
{{eqn | r = 1
}}
{{en... | Open Ball with respect to Seminorm is Convex, Balanced and Absorbing | https://proofwiki.org/wiki/Open_Ball_with_respect_to_Seminorm_is_Convex,_Balanced_and_Absorbing | https://proofwiki.org/wiki/Open_Ball_with_respect_to_Seminorm_is_Convex,_Balanced_and_Absorbing | [
"Convex Sets (Vector Spaces)",
"Absorbing Sets",
"Balanced Sets",
"Seminorms"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Pseudometric Induced by Seminorm",
"Definition:Open Unit Ball",
"Definition:Convex Set (Vector Space)",
"Definition:Balanced Set",
"Definition:Absorbing Set"
] | [] |
proofwiki-20508 | Closed Unit Ball in Normed Vector Space is Weakly Closed | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $B^-$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,} }$.
Then $B^-$ is weakly closed. | From Closed Unit Ball is Convex Set, $B^-$ is convex.
From Closed Ball is Closed, $B^-$ is $\norm {\, \cdot \,}$-closed.
From Mazur's Theorem: Corollary, we can conclude that $B^-$ is weakly closed.
{{qed}}
Category:Weakly Closed Sets
Category:Closed Balls
Category:Normed Vector Spaces
0z00xz69xza14gdzpcmaqakznsdyxz4 | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $B^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,} }$.
Then $B^-$ is [[Definition:Weakly Closed Set|weakly closed]]. | From [[Closed Unit Ball is Convex Set]], $B^-$ is [[Definition:Convex Set (Vector Space)|convex]].
From [[Closed Ball is Closed]], $B^-$ is [[Definition:Closed Set|$\norm {\, \cdot \,}$-closed]].
From [[Mazur's Theorem/Corollary|Mazur's Theorem: Corollary]], we can conclude that $B^-$ is [[Definition:Weakly Closed S... | Closed Unit Ball in Normed Vector Space is Weakly Closed | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Vector_Space_is_Weakly_Closed | https://proofwiki.org/wiki/Closed_Unit_Ball_in_Normed_Vector_Space_is_Weakly_Closed | [
"Normed Vector Spaces",
"Weakly Closed Sets",
"Closed Balls",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Unit Ball",
"Definition:Weakly Closed Set"
] | [
"Closed Unit Ball is Convex Set",
"Definition:Convex Set (Vector Space)",
"Closed Ball is Closed",
"Definition:Closed Set",
"Mazur's Theorem/Corollary",
"Definition:Weakly Closed Set",
"Category:Weakly Closed Sets",
"Category:Closed Balls",
"Category:Normed Vector Spaces"
] |
proofwiki-20509 | Upper Darboux Integral Never Smaller than Lower Darboux Integral | Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a bounded real function.
The lower Darboux integral of $f$ over $\closedint a b$ is less than or equal to the upper Darboux integral of $f$ over the same bounds.
That is:
:$\ds \underline {\int_a^b} \map f x \rd x \le \overline {\int_a^b}... | Let the value of the lower Darboux integral be $L$, and the value of the upper Darboux integral be $U$.
{{AimForCont}}, suppose $L > U$.
Then $\epsilon = \dfrac {L - U} 2$ is positive.
By the definitions of lower Darboux integral and upper Darboux integral:
:$\ds \sup_P \map L P > \inf_P \map U P$
where $P$ ranges over... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f: \closedint a b \to \R$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Real Function|real function]].
The [[Definition:Lower Darboux Integral|lower Darboux integral]] of $f$ over $\closedint a b$ is less ... | Let the value of the [[Definition:Lower Darboux Integral|lower Darboux integral]] be $L$, and the value of the [[Definition:Upper Darboux Integral|upper Darboux integral]] be $U$.
{{AimForCont}}, suppose $L > U$.
Then $\epsilon = \dfrac {L - U} 2$ is positive.
By the definitions of [[Definition:Lower Darboux Integra... | Upper Darboux Integral Never Smaller than Lower Darboux Integral | https://proofwiki.org/wiki/Upper_Darboux_Integral_Never_Smaller_than_Lower_Darboux_Integral | https://proofwiki.org/wiki/Upper_Darboux_Integral_Never_Smaller_than_Lower_Darboux_Integral | [
"Lower Darboux Integral",
"Upper Darboux Integral",
"Integral Calculus"
] | [
"Definition:Real Interval/Closed",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Real Function",
"Definition:Lower Darboux Integral",
"Definition:Upper Darboux Integral"
] | [
"Definition:Lower Darboux Integral",
"Definition:Upper Darboux Integral",
"Definition:Lower Darboux Integral",
"Definition:Upper Darboux Integral",
"Definition:Subdivision of Interval",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Characterizing Property of Supremum of Subset of Re... |
proofwiki-20510 | Minkowski Functional of Convex Absorbing Set is Finite | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $A \subseteq X$ be a convex absorbing set.
Let $\mu_A$ be the Minkowski functional of $A$.
Then for each $x \in X$, $\map {\mu_A} x$ is a finite extended real number.
That is:
:$\forall x \in X: \map {\mu_A} x < \infty$ | Let $x \in X$.
From Characterization of Convex Absorbing Set in Vector Space:
:$\exists t \in \R_{>0}: x \in t A$
where $t A$ denotes the dilation of $A$ by $t$.
Then:
:$x \in t^{-1} C$
so that:
:$t \in \set {t > 0 : t^{-1} x \in A}$
Then, we have:
:$\map {\mu_A} x \le t < \infty$
{{qed}}
Category:Minkowski Function... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $A \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Absorbing Set|absorbing set]].
Let $\mu_A$ be the [[Definition:Minkowski Functional|Minkowski functional]] of $A$.
Then for each $x... | Let $x \in X$.
From [[Characterization of Convex Absorbing Set in Vector Space]]:
:$\exists t \in \R_{>0}: x \in t A$
where $t A$ denotes the [[Definition:Dilation of Subset of Vector Space|dilation of $A$ by $t$]].
Then:
:$x \in t^{-1} C$
so that:
:$t \in \set {t > 0 : t^{-1} x \in A}$
Then, we have:
:$\map {\mu_... | Minkowski Functional of Convex Absorbing Set is Finite | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Finite | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Finite | [
"Minkowski Functionals"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional",
"Definition:Finite Extended Real Number"
] | [
"Characterization of Convex Absorbing Set in Vector Space",
"Definition:Linear Combination of Subsets of Vector Space/Dilation",
"Category:Minkowski Functionals"
] |
proofwiki-20511 | Minkowski Functional of Open Ball with respect to Seminorm is Seminorm | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $p$ be a seminorm on $X$.
Let:
:$B = \set {x \in X : \map p x < 1}$
be the open ball centred at ${\mathbf 0}_X$ with radius $1$ in the pseudometric induced by $p$.
Let $\mu_B$ the Minkowski functional of $B$.
Then $p = \mu_B$. | From Open Ball with respect to Seminorm is Convex, Balanced and Absorbing, $B$ is convex and absorbing.
Hence the definition is valid.
Let $x \in X$.
From {{SeminormAxiom|2}}:
:$\forall s > \map p x: \map p {\dfrac x s} = \dfrac 1 s \map p x < 1$
Then we have:
:$s \in \set {t > 0 : t^{-1} x \in B}$
so that:
:$\map {... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let:
:$B = \set {x \in X : \map p x < 1}$
be the [[Definition:Open Ball|open ball]] [[Definition:Center of Open Ball|centred]] at ${\mathbf 0}_X$ with [[Definition... | From [[Open Ball with respect to Seminorm is Convex, Balanced and Absorbing]], $B$ is [[Definition:Convex Set (Vector Space)|convex]] and [[Definition:Absorbing Set|absorbing]].
Hence the definition is valid.
Let $x \in X$.
From {{SeminormAxiom|2}}:
:$\forall s > \map p x: \map p {\dfrac x s} = \dfrac 1 s \map p x... | Minkowski Functional of Open Ball with respect to Seminorm is Seminorm | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Ball_with_respect_to_Seminorm_is_Seminorm | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Ball_with_respect_to_Seminorm_is_Seminorm | [
"Minkowski Functional",
"Minkowski Functionals",
"Seminorms",
"Minkowski Functionals"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Pseudometric Induced by Seminorm",
"Definition:Minkowski Functional"
] | [
"Open Ball with respect to Seminorm is Convex, Balanced and Absorbing",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Infimum of Set/Real Numbers"
] |
proofwiki-20512 | Continuous Linear Operator over Finite Dimensional Vector Space is Invertible | Let $\struct {X, \norm {\, \cdot\,}_X}$ be a normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $I \in \map {CL} X$ be the identity element.
Let $S, T \in \map {CL} X$.
Suppose the dimension of $X$ is finite:
:$d = \dim X < \infty$
Suppose $T \circ S = I$ where $... | Let $x \in X$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {T \circ S} } x
| r = \map I x
}}
{{eqn | ll = \leadsto
| l = \map T {\map {S} x}
| r = x
}}
{{end-eqn}}
Let $\mathbf 0 \in X$ be the zero vector of $X$.
Suppose $\map S x = \mathbf 0$.
Then:
{{begin-eqn}}
{{eqn | l = \map T {\map S x}
... | Let $\struct {X, \norm {\, \cdot\,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $I \in \map {CL} X$ be the [[Definition:Identity Element|identity element]].... | Let $x \in X$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren {T \circ S} } x
| r = \map I x
}}
{{eqn | ll = \leadsto
| l = \map T {\map {S} x}
| r = x
}}
{{end-eqn}}
Let $\mathbf 0 \in X$ be the [[Definition:Zero Vector|zero vector]] of $X$.
Suppose $\map S x = \mathbf 0$.
Then:
{{begin-eqn}}
{{... | Continuous Linear Operator over Finite Dimensional Vector Space is Invertible | https://proofwiki.org/wiki/Continuous_Linear_Operator_over_Finite_Dimensional_Vector_Space_is_Invertible | https://proofwiki.org/wiki/Continuous_Linear_Operator_over_Finite_Dimensional_Vector_Space_is_Invertible | [
"Continuous Linear Transformations",
"Inverse Mappings"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Dimension of Vector Space",
"Definition:Finite Cardinal",
"Definition:Composition of Mappings",
"Definition:Invertible Continuous Linear Operato... | [
"Definition:Zero Vector",
"Linear Transformation Maps Zero Vector to Zero Vector",
"Definition:Kernel of Linear Transformation/Vector Space",
"Definition:Basis of Vector Space",
"Definition:Scalar/Vector Space",
"Definition:Number Field",
"Definition:Linear Transformation/Vector Space",
"Definition:Li... |
proofwiki-20513 | Equivalence of Definitions of Riemann and Darboux Integrals | Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a real function.
Then:
:the Riemann integral of $f$ over $\closedint a b$ exists and is equal to $L$
{{iff}}:
:the Darboux integral of $f$ over $\closedint a b$ exists and is equal to $L$. | === Riemann Integral $\implies$ Darboux Integral ===
Let $L$ be the Riemann integral of $f$ over $\closedint a b$.
Then $\forall \epsilon > 0: \exists \delta > 0$ such that for every finite subdivision with norm $< \delta$
and every sequence of corresponding sample points, the Riemann sum of the subdivision is in $\ope... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f: \closedint a b \to \R$ be a [[Definition:Real Function|real function]].
Then:
:the [[Definition:Riemann Integral|Riemann integral]] of $f$ over $\closedint a b$ exists and is equal to $L$
{{iff}}:
:the [[Definition:Darboux Int... | === Riemann Integral $\implies$ Darboux Integral ===
Let $L$ be the [[Definition:Riemann Integral|Riemann integral]] of $f$ over $\closedint a b$.
Then $\forall \epsilon > 0: \exists \delta > 0$ such that for every [[Definition:Finite Subdivision|finite subdivision]] with [[Definition:Norm of Subdivision|norm]] $< \d... | Equivalence of Definitions of Riemann and Darboux Integrals | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Riemann_and_Darboux_Integrals | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Riemann_and_Darboux_Integrals | [
"Integral Calculus"
] | [
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Definite Integral/Riemann",
"Definition:Definite Integral/Darboux"
] | [
"Definition:Definite Integral/Riemann",
"Definition:Subdivision of Interval/Finite",
"Definition:Norm of Subdivision",
"Definition:Sequence",
"Definition:Riemann Sum",
"Definition:Lower Darboux Integral",
"Definition:Subdivision of Interval/Finite",
"Definition:Norm of Subdivision",
"Definition:Lowe... |
proofwiki-20514 | Wave Equation is Linear | Let $\phi_1$ and $\phi_2$ be particular solutions to the wave equation.
Then:
:$a_1 \phi_1 + a_2 \phi_2$ is also a particular solution to the wave equation. | {{begin-eqn}}
{{eqn | l = \map {\dfrac {\partial^2} {\partial t^2} } {a_1 \phi_1 + a_2 \phi_2}
| r = a_1 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_2}
| c = Linear Combination of Derivatives
}}
{{eqn | l = \map {\dfrac {\partial^2} {\partial x^... | Let $\phi_1$ and $\phi_2$ be [[Definition:Particular Solution to Differential Equation|particular solutions]] to the [[Definition:Wave Equation|wave equation]].
Then:
:$a_1 \phi_1 + a_2 \phi_2$ is also a [[Definition:Particular Solution to Differential Equation|particular solution]] to the [[Definition:Wave Equation|w... | {{begin-eqn}}
{{eqn | l = \map {\dfrac {\partial^2} {\partial t^2} } {a_1 \phi_1 + a_2 \phi_2}
| r = a_1 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_1} + a_2 \map {\dfrac {\partial^2} {\partial t^2} } {\phi_2}
| c = [[Linear Combination of Derivatives]]
}}
{{eqn | l = \map {\dfrac {\partial^2} {\partia... | Wave Equation is Linear | https://proofwiki.org/wiki/Wave_Equation_is_Linear | https://proofwiki.org/wiki/Wave_Equation_is_Linear | [
"Wave Equation"
] | [
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Wave Equation",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Wave Equation"
] | [
"Linear Combination of Derivatives",
"Linear Combination of Derivatives",
"Linear Combination of Derivatives",
"Linear Combination of Derivatives",
"Definition:Wave Equation",
"Definition:Wave Equation"
] |
proofwiki-20515 | Well-Defined Jordan Content Equals Content | Let $M$ be a bounded subspace of Euclidean space.
Let the Jordan content of $M$ be $\map m M$.
Then the content $\map V M = \map m M$. | Let $C$ be a finite covering of $M$.
By {{EuclidCommonNotionLink|5}}, $\map V C \ge \map V M$.
Therefore, $\map V M$ is a lower bound of all $\map V C$.
So by the definition of greatest lower bound:
:$\map V M \le \map {m^*} M$
Let $D$ be a finite covering of $S \setminus M$.
By the same reasoning:
:$\map V {S \setminu... | Let $M$ be a [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspace]] of [[Definition:Euclidean Space|Euclidean space]].
Let the [[Definition:Jordan Content|Jordan content]] of $M$ be $\map m M$.
Then the [[Definition:Content (Measure Theory)|content]] $\map V M = \map m M$. | Let $C$ be a [[Definition:Finite Cover|finite covering]] of $M$.
By {{EuclidCommonNotionLink|5}}, $\map V C \ge \map V M$.
Therefore, $\map V M$ is a [[Definition:Lower Bound|lower bound]] of all $\map V C$.
So by the definition of [[Definition:Greatest Lower Bound|greatest lower bound]]:
:$\map V M \le \map {m^*} M... | Well-Defined Jordan Content Equals Content | https://proofwiki.org/wiki/Well-Defined_Jordan_Content_Equals_Content | https://proofwiki.org/wiki/Well-Defined_Jordan_Content_Equals_Content | [] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Jordan Content",
"Definition:Content (Measure Theory)"
] | [
"Definition:Cover of Set/Finite",
"Definition:Lower Bound",
"Definition:Infimum of Set/Real Numbers",
"Definition:Cover of Set/Finite"
] |
proofwiki-20516 | Outer Jordan Content Never Smaller than Inner Jordan Content | Let $M$ be a bounded subspace of Euclidean space.
Then:
:$\map {m^*} M \ge \map {m_*} M$
where:
:$m^*$ denotes the outer Jordan content
:$m_*$ denotes the inner Jordan content | Let $\RR$ be a closed rectangle that contains $M$.
By definition of inner Jordan content:
:$\map {m_*} M = \map V \RR - \map {m^*} {\RR \setminus M}$
But since:
:$\RR = M \cup \paren {\RR \setminus M}$
we have by Outer Jordan Content is Subadditive that:
:$\map {m^*} \RR \le \map {m^*} M + \map {m^*} {\RR \setminus M}$... | Let $M$ be a [[Definition:Bounded (Metric Space)|bounded]] [[Definition:Subspace|subspace]] of [[Definition:Euclidean Space|Euclidean space]].
Then:
:$\map {m^*} M \ge \map {m_*} M$
where:
:$m^*$ denotes the [[Definition:Outer Jordan Content|outer Jordan content]]
:$m_*$ denotes the [[Definition:Inner Jordan Content|i... | Let $\RR$ be a [[Definition:Closed Rectangle|closed rectangle]] that [[Definition:Set Containment|contains]] $M$.
By definition of [[Definition:Inner Jordan Content|inner Jordan content]]:
:$\map {m_*} M = \map V \RR - \map {m^*} {\RR \setminus M}$
But since:
:$\RR = M \cup \paren {\RR \setminus M}$
we have by [[Oute... | Outer Jordan Content Never Smaller than Inner Jordan Content | https://proofwiki.org/wiki/Outer_Jordan_Content_Never_Smaller_than_Inner_Jordan_Content | https://proofwiki.org/wiki/Outer_Jordan_Content_Never_Smaller_than_Inner_Jordan_Content | [
"Outer Jordan Content"
] | [
"Definition:Bounded Metric Space",
"Definition:Subspace",
"Definition:Euclidean Space",
"Definition:Outer Jordan Content",
"Definition:Inner Jordan Content"
] | [
"Definition:Closed Rectangle",
"Definition:Subset",
"Definition:Inner Jordan Content",
"Outer Jordan Content is Subadditive",
"Outer Jordan Content of Closed Rectangle",
"Category:Outer Jordan Content"
] |
proofwiki-20517 | Sum of Equal and Opposite Harmonic Waves form Stationary Wave | Let $\phi_1$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.
Let $\phi_2$ be a harmonic wave travelling with constant velocity $-c$, that is, at the same speed as $\phi_1$ but in the opposite direction.
Then their sum $\phi_1 + \phi_2$ describes a stationa... | From Equation of Harmonic Wave: Wave Number and Frequency:
:$(1): \quad \map {\phi_1} {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$
where:
:$k$ denotes the wave number of $\phi_1$
:$\nu$ denotes the frequency of $\phi_1$.
From Equation of Wave with Constant Velocity: Corollary, the equation of $\phi_2$ is:
:$(2):... | Let $\phi_1$ be a [[Definition:Harmonic Wave|harmonic wave]] which is [[Definition:Direction of Propagation of Wave|propagated]] along the [[Definition:X-Axis|$x$-axis]] in the [[Definition:Positive Direction|positive direction]] with [[Definition:Constant|constant]] [[Definition:Velocity|velocity]] $c$.
Let $\phi_2$... | From [[Equation of Harmonic Wave/Wave Number and Frequency|Equation of Harmonic Wave: Wave Number and Frequency]]:
:$(1): \quad \map {\phi_1} {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$
where:
:$k$ denotes the [[Definition:Wave Number of Periodic Wave|wave number]] of $\phi_1$
:$\nu$ denotes the [[Definition:Fr... | Sum of Equal and Opposite Harmonic Waves form Stationary Wave | https://proofwiki.org/wiki/Sum_of_Equal_and_Opposite_Harmonic_Waves_form_Stationary_Wave | https://proofwiki.org/wiki/Sum_of_Equal_and_Opposite_Harmonic_Waves_form_Stationary_Wave | [
"Harmonic Waves",
"Stationary Waves"
] | [
"Definition:Harmonic Wave",
"Definition:Wave/Direction of Propagation",
"Definition:Axis/X-Axis",
"Definition:Axis/Positive Direction",
"Definition:Constant",
"Definition:Velocity",
"Definition:Harmonic Wave",
"Definition:Constant",
"Definition:Velocity",
"Definition:Speed",
"Definition:Addition... | [
"Equation of Harmonic Wave/Wave Number and Frequency",
"Definition:Periodic Wave/Wave Number",
"Definition:Periodic Wave/Frequency",
"Equation of Wave with Constant Velocity/Corollary",
"Prosthaphaeresis Formulas/Cosine plus Cosine",
"Cosine Function is Even",
"Definition:Stationary Wave"
] |
proofwiki-20518 | Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 2 | :the family of continuous mappings $\family{f_{U,V}}_{\tuple{U,V} \in \AA}$ separates points from closed sets | Let $F$ be a closed subset of $T$.
Let $x \in S \setminus F$.
By definition of closed subset:
:$S \setminus F$ is open in $S$
Let $V = S \setminus F$.
By definition of regular space:
:$\exists U \in \tau: x \subseteq U, U^- \subseteq V$
Hence $\tuple{U, V} \in \AA$.
Consider the Urysohn function $f_{U, V}$ for $U^-$ an... | :the [[Definition:Indexed Family|family]] of [[Definition:Continuous Mapping|continuous mappings]] $\family{f_{U,V}}_{\tuple{U,V} \in \AA}$ [[Definition:Mappings Separating Points from Closed Sets|separates points from closed sets]] | Let $F$ be a [[Definition:Closed Set (Topology)|closed subset]] of $T$.
Let $x \in S \setminus F$.
By definition of [[Definition:Closed Set (Topology)|closed subset]]:
:$S \setminus F$ is [[Definition:Open Set (Topology)|open]] in $S$
Let $V = S \setminus F$.
By definition of [[Definition:Regular Space|regular spa... | Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 2 | https://proofwiki.org/wiki/Regular_Second-Countable_Space_is_Homeomorphic_to_Subspace_of_Hilbert_Cube/Lemma_2 | https://proofwiki.org/wiki/Regular_Second-Countable_Space_is_Homeomorphic_to_Subspace_of_Hilbert_Cube/Lemma_2 | [
"Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube"
] | [
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping",
"Definition:Mappings Separating Points from Closed Sets"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Regular Space",
"Definition:Urysohn Function",
"Definition:Urysohn Function",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping",
"Definition:Mappings Separating Points from... |
proofwiki-20519 | Minkowski Functional of Convex Absorbing Set is Sublinear | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $p$ be a seminorm on $X$.
Let $A$ be a convex absorbing set.
Let $\mu_A$ be the Minkowski functional of $A$.
Then for each $x, y \in X$ we have:
:$\map {\mu_A} {x + y} \le \map {\mu_A} x + \map {\mu_A} y$ | Let $\epsilon > 0$.
By the definition of infimum, we can pick $t > 0$ such that:
{{explain|domain of $t$ -- this is not completely obvious, as the underlying set of $X$ includes the case where it's $\C$.}}
:$\map {\mu_A} x \le t \le \map {\mu_A} x + \epsilon$
and $t^{-1} x \in A$.
We can also pick $s > 0$ such that:
... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let $A$ be a [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Absorbing Set|absorbing set]].
Let $\mu_A$ be the [[Definition:Minkowski Functional|Minkows... | Let $\epsilon > 0$.
By the definition of [[Definition:Infimum of Subset of Real Numbers|infimum]], we can pick $t > 0$ such that:
{{explain|domain of $t$ -- this is not completely obvious, as the underlying set of $X$ includes the case where it's $\C$.}}
:$\map {\mu_A} x \le t \le \map {\mu_A} x + \epsilon$
and $t... | Minkowski Functional of Convex Absorbing Set is Sublinear | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Sublinear | https://proofwiki.org/wiki/Minkowski_Functional_of_Convex_Absorbing_Set_is_Sublinear | [
"Minkowski Functionals"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Convex Set (Vector Space)",
"Definition:Absorbing Set",
"Definition:Minkowski Functional"
] | [
"Definition:Infimum of Set/Real Numbers",
"Definition:Convex Set (Vector Space)"
] |
proofwiki-20520 | Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\struct {\map D T, T}$ be a symmetric densely-defined linear operator.
Let $\struct {\map D {T^\ast}, T^\ast}$ be the adjoint of $T$.
Then $\map D T \subseteq \map D {T^\ast}$ and:
:$T x = T^\ast x$ for each $x \in \map D T$. | For each $y \in \HH$, define the linear functional $f_x : \map D T \to \Bbb F$ by:
:$\map {f_y} x = \innerprod {T x} y$ for each $x \in \map D T$.
We show that for $y \in \map D T$, $f_y$ is bounded.
Let $y \in \map D T$, then:
:$\innerprod {T x} y = \innerprod x {T y}$ for each $x \in \map D T$.
Then we have:
{{begi... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\struct {\map D T, T}$ be a [[Definition:Symmetric Densely-Defined Linear Operator|symmetric]] [[Definition:Densely-Defined Linear Operator|densely-defined linear operator]].
Let $\struct {\map D {T^\ast}, T^\ast}$ be th... | For each $y \in \HH$, define the [[Definition:Linear Functional|linear functional]] $f_x : \map D T \to \Bbb F$ by:
:$\map {f_y} x = \innerprod {T x} y$ for each $x \in \map D T$.
We show that for $y \in \map D T$, $f_y$ is [[Definition:Bounded Linear Functional|bounded]].
Let $y \in \map D T$, then:
:$\innerprod ... | Adjoint of Symmetric Densely-Defined Linear Operator Extends Operator | https://proofwiki.org/wiki/Adjoint_of_Symmetric_Densely-Defined_Linear_Operator_Extends_Operator | https://proofwiki.org/wiki/Adjoint_of_Symmetric_Densely-Defined_Linear_Operator_Extends_Operator | [
"Symmetric Densely-Defined Linear Operators",
"Adjoints (Densely-Defined Linear Operators)"
] | [
"Definition:Hilbert Space",
"Definition:Symmetric Densely-Defined Linear Operator",
"Definition:Densely-Defined Linear Operator",
"Definition:Adjoint of Densely-Defined Linear Operator"
] | [
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces",
"Definition:Bounded Linear Functional",
"Definition:Symmetric Densely-Defined Linear Operator",
"Definition:Adjoint of Densely-Defined Linear Operator",
"Inner Product is... |
proofwiki-20521 | Spectrum of Bounded Linear Operator is Compact | Let $X$ be a Banach space over $\C$.
Let $T : X \to X$ be a bounded linear operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then $\map \sigma T$ is compact, and:
:$\map \sigma T \subseteq \set {\lambda \in \C : \cmod \lambda \le \norm T_{\map \BB X} }$ | From Spectrum of Bounded Linear Operator is Closed, $\map \sigma T$ is closed in $\C$.
It therefore suffices to show that $\map \sigma T$ is bounded.
Hence it will suffice to show that:
:$\map \sigma T \subseteq \set {\lambda \in \C : \cmod \lambda \le \norm T_{\map \BB X} }$
Let $\lambda \in \C$ be such that $\cmod \... | Let $X$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map \sigma T$ be the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$.
Then $\map \sigma T$ is [[Definition:Compact Subset of Complex Pla... | From [[Spectrum of Bounded Linear Operator is Closed]], $\map \sigma T$ is [[Definition:Closed Set/Complex Analysis|closed]] in $\C$.
It therefore suffices to show that $\map \sigma T$ is [[Definition:Bounded Subset of Complex Plane|bounded]].
Hence it will suffice to show that:
:$\map \sigma T \subseteq \set {\lam... | Spectrum of Bounded Linear Operator is Compact | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_is_Compact | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_is_Compact | [
"Spectra (Bounded Linear Operators)",
"Compact Spaces (Complex Analysis)"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Compact Space/Metric Space/Complex"
] | [
"Spectrum of Bounded Linear Operator is Closed",
"Definition:Closed Set/Complex Analysis",
"Definition:Bounded Metric Space/Complex",
"Invertibility of Identity Minus Operator",
"Definition:Invertible Bounded Linear Operator",
"Definition:Vector Space",
"Definition:Invertible Bounded Linear Operator",
... |
proofwiki-20522 | Resolvent Set of Bounded Linear Operator is Open | Let $X$ be a Banach space over $\C$.
Let $T : X \to X$ be a bounded linear operator.
Let $\map \rho T$ be the resolvent set of $T$.
Then $\map \rho T$ is open. | Let $\lambda \in \map \rho T$.
Then $T - \lambda I$ is invertible as a bounded linear operator.
Let $\delta > 0$ be such that:
:$\cmod \delta < \norm {\paren {T - \lambda I}^{-1} }_{\map \BB X}^{-1}$
Then, we have:
:$\norm {\delta I}_{\map \BB X} \norm {\paren {T - \lambda I}^{-1} } < 1$
From Invertibility of Iden... | Let $X$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map \rho T$ be the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent set]] of $T$.
Then $\map \rho T$ is [[Definition:Open Set (Complex Analys... | Let $\lambda \in \map \rho T$.
Then $T - \lambda I$ is [[Definition:Invertible Bounded Linear Operator|invertible as a bounded linear operator]].
Let $\delta > 0$ be such that:
:$\cmod \delta < \norm {\paren {T - \lambda I}^{-1} }_{\map \BB X}^{-1}$
Then, we have:
:$\norm {\delta I}_{\map \BB X} \norm {\paren ... | Resolvent Set of Bounded Linear Operator is Open | https://proofwiki.org/wiki/Resolvent_Set_of_Bounded_Linear_Operator_is_Open | https://proofwiki.org/wiki/Resolvent_Set_of_Bounded_Linear_Operator_is_Open | [
"Resolvent Sets (Bounded Linear Operators)"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Open Set/Complex Analysis"
] | [
"Definition:Invertible Bounded Linear Operator",
"Invertibility of Identity Minus Operator/Corollary",
"Definition:Invertible Bounded Linear Operator",
"Definition:Invertible Bounded Linear Operator",
"Definition:Open Neighborhood",
"Definition:Open Set/Complex Analysis"
] |
proofwiki-20523 | Spectrum of Adjoint of Bounded Linear Operator | Let $X$ be a Banach space over $\C$.
Let $T : X \to X$ be a bounded linear operator.
Let $T^\ast : X \to X$ be the adjoint of $T$.
Let $\map \sigma T$ and $\map \sigma {T^\ast}$ be the spectrum of $T$ and $T^\ast$ respectively.
Then:
:$\map \sigma {T^\ast} = \set {\overline \lambda : \lambda \in \map \sigma T}$
wh... | We show that for $\lambda \in \C$, we have $\lambda \not \in \map \sigma T$ {{iff}} $\overline \lambda \not \in \map \sigma {T^\ast}$.
Let $\lambda \in \C$ have $\lambda \not \in \map \sigma T$.
Then $T - \lambda I$ is invertible as a bounded linear operator.
Note that $I^\ast = I$ from Adjoint of Identity Transformati... | Let $X$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $T^\ast : X \to X$ be the [[Definition:Adjoint Linear Transformation|adjoint]] of $T$.
Let $\map \sigma T$ and $\map \sigma {T^\ast}$ be the [[Definition:Spect... | We show that for $\lambda \in \C$, we have $\lambda \not \in \map \sigma T$ {{iff}} $\overline \lambda \not \in \map \sigma {T^\ast}$.
Let $\lambda \in \C$ have $\lambda \not \in \map \sigma T$.
Then $T - \lambda I$ is [[Definition:Invertible Bounded Linear Operator|invertible as a bounded linear operator]].
Note th... | Spectrum of Adjoint of Bounded Linear Operator | https://proofwiki.org/wiki/Spectrum_of_Adjoint_of_Bounded_Linear_Operator | https://proofwiki.org/wiki/Spectrum_of_Adjoint_of_Bounded_Linear_Operator | [
"Spectra (Bounded Linear Operators)",
"Adjoints"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator",
"Definition:Adjoint Linear Transformation",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Complex Conjugate"
] | [
"Definition:Invertible Bounded Linear Operator",
"Adjoint of Identity Transformation",
"Adjoint is Conjugate Linear",
"Adjoining Commutes with Inverting",
"Definition:Invertible Bounded Linear Operator",
"Adjoint is Involutive",
"Complex Conjugation is Involution"
] |
proofwiki-20524 | Weak-* Topology is Hausdorff | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $X^\ast$ be the topological dual space of $X$.
Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.
Then $\struct {X^\ast, w^\ast}$ is Hausdorff space. | From Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex, $w^\ast$ can be induced by:
:$\PP = \set {p_{x^\wedge} : x \in X}$
where we define $p_{x^\wedge} : X^\ast \to \R_{\ge 0}$ by:
:$\map {p_{x^\wedge} } f = \cmod {\map {x^\wedge} f} = \cmod {\map f x}$
From Locally Convex Space is H... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
Let $w^\ast$ be the [[Definition:Weak-* Topology|weak-$\ast$ topology]] on $X^\ast$.
Then $\struct {X^\... | From [[Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex]], $w^\ast$ can be [[Definition:Locally Convex Space/Standard Topology|induced by]]:
:$\PP = \set {p_{x^\wedge} : x \in X}$
where we define $p_{x^\wedge} : X^\ast \to \R_{\ge 0}$ by:
:$\map {p_{x^\wedge} } f = \cmod {\map {x^... | Weak-* Topology is Hausdorff | https://proofwiki.org/wiki/Weak-*_Topology_is_Hausdorff | https://proofwiki.org/wiki/Weak-*_Topology_is_Hausdorff | [
"Weak-* Topologies",
"Hausdorff Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Topological Dual Space",
"Definition:Weak-* Topology",
"Definition:T2 Space"
] | [
"Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex",
"Definition:Locally Convex Space/Standard Topology",
"Locally Convex Space is Hausdorff iff induces Hausdorff Topology",
"Definition:Locally Convex Space/Hausdorff",
"Definition:Separating Family of Seminorms on Vector Spa... |
proofwiki-20525 | Vector Addition is Continuous in Weak Topology | Let $K$ be a topological field.
Let $X$ be a topological vector space over $K$ with weak topology $w$.
Define $s : \struct {X, w} \times \struct {X, w} \to \struct {X, w}$ by:
:$\map s {x, y} = x + y$
for each $x, y \in X$.
Then $s$ is continuous.
That is, vector addition remains continuous when restricting to the ... | Let $X^\ast$ be the topological dual space of $X$.
From Continuity in Initial Topology, it suffices to show that for each $f \in X^\ast$ we have:
:$f \circ s : \struct {X, w} \times \struct {X, w} \to K$ is continuous.
Define the projections $\pr_1 : \struct {X, w} \times {X, w} \to \struct {X, w}$ and $\pr_2 : \struc... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Define $s : \struct {X, w} \times \struct {X, w} \to \struct {X, w}$ by:
:$\map s ... | Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
From [[Continuity in Initial Topology]], it suffices to show that for each $f \in X^\ast$ we have:
:$f \circ s : \struct {X, w} \times \struct {X, w} \to K$ is [[Definition:Continuous Mapping (Topology)|continuous]].
Define the... | Vector Addition is Continuous in Weak Topology | https://proofwiki.org/wiki/Vector_Addition_is_Continuous_in_Weak_Topology | https://proofwiki.org/wiki/Vector_Addition_is_Continuous_in_Weak_Topology | [
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Vector Addition",
"Definition:Weak Topology on Topological Vector Space"
] | [
"Definition:Topological Dual Space",
"Continuity in Initial Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Topology",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Product Topology/Factor Space",
"Definition:Product Topology",
"Definition:Continuous M... |
proofwiki-20526 | Scalar Multiplication is Continuous in Weak Topology | Let $K$ be a topological field.
Let $X$ be a topological vector space over $K$ with weak topology $w$.
Define $m : K \times \struct {X, w} \to \struct {X, w}$ by:
:$\map m {\lambda, x} = \lambda x$
for each $\lambda \in K$, $x \in X$.
Then $m$ is continuous.
That is, scalar multiplication remains continuous when re... | Let $X^\ast$ be the topological dual space of $X$.
From Continuity in Initial Topology, it suffices to show that for each $f \in X^\ast$ we have:
:$f \circ m : K \times \struct {X, w} \to K$ is continuous.
Define the projections $\pr_1 : K \times {X, w} \to \struct {X, w}$ and $\pr_2 : K \times {X, w} \to \struct {X, ... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Define $m : K \times \struct {X, w} \to \struct {X, w}$ by:
:$\map m {\lambda, x} ... | Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
From [[Continuity in Initial Topology]], it suffices to show that for each $f \in X^\ast$ we have:
:$f \circ m : K \times \struct {X, w} \to K$ is [[Definition:Continuous Mapping (Topology)|continuous]].
Define the [[Definition... | Scalar Multiplication is Continuous in Weak Topology | https://proofwiki.org/wiki/Scalar_Multiplication_is_Continuous_in_Weak_Topology | https://proofwiki.org/wiki/Scalar_Multiplication_is_Continuous_in_Weak_Topology | [
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Scalar Multiplication",
"Definition:Weak Topology on Topological Vector Space"
] | [
"Definition:Topological Dual Space",
"Continuity in Initial Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Topology",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Projection (Mapping Theory)/Se... |
proofwiki-20527 | Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology | Let $K$ be a topological field.
Let $X$ be a topological vector space over $K$ admitting a weak topology $w$.
Then $\struct {X, w}$ is a topological vector space. | Follows from combining:
:Vector Addition is Continuous in Weak Topology
:Scalar Multiplication is Continuous in Weak Topology
{{qed}}
Category:Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology
Category:Weak Topologies on Topological Vector Spaces
624hwlo3fjhh236wdxrued... | Let $K$ be a [[Definition:Topological Field|topological field]].
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $K$ admitting a [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Then $\struct {X, w}$ is a [[Definition:Topological Vector Space|topological... | Follows from combining:
:[[Vector Addition is Continuous in Weak Topology]]
:[[Scalar Multiplication is Continuous in Weak Topology]]
{{qed}}
[[Category:Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology]]
[[Category:Weak Topologies on Topological Vector Spaces]]
624h... | Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology | https://proofwiki.org/wiki/Topological_Vector_Space_over_Topological_Field_remains_Topological_Vector_Space_with_Weak_Topology | https://proofwiki.org/wiki/Topological_Vector_Space_over_Topological_Field_remains_Topological_Vector_Space_with_Weak_Topology | [
"Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology",
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Topological Field",
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Topological Vector Space"
] | [
"Vector Addition is Continuous in Weak Topology",
"Scalar Multiplication is Continuous in Weak Topology",
"Category:Topological Vector Space over Topological Field remains Topological Vector Space with Weak Topology",
"Category:Weak Topologies on Topological Vector Spaces"
] |
proofwiki-20528 | Spectrum of Compact Linear Operator on Infinite-Dimensional Banach Space contains Zero | Let $X$ be an infinite-dimensional Banach space over $\C$.
Let $T : X \to X$ be a compact linear operator.
Let $\map \sigma T$ be the spectrum of $T$.
Then $0 \in \map \sigma T$.
That is, $T$ is not invertible as a bounded linear operator. | Suppose $0 \not \in \map \sigma T$.
Then $T$ is invertible as a bounded linear operator with bounded inverse $T^{-1}$ so that:
:$T T^{-1} = I$
From Left Composition of Compact Linear Transformation with Bounded Linear Transformation is Compact, this implies that $I$ is compact.
From Identity Operator is Compact iff F... | Let $X$ be an [[Definition:Finite Dimensional Vector Space|infinite-dimensional]] [[Definition:Banach Space|Banach space]] over $\C$.
Let $T : X \to X$ be a [[Definition:Compact Linear Operator|compact linear operator]].
Let $\map \sigma T$ be the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$.
... | Suppose $0 \not \in \map \sigma T$.
Then $T$ is [[Definition:Invertible Bounded Linear Operator|invertible as a bounded linear operator]] with [[Definition:Bounded Linear Operator|bounded]] [[Definition:Inverse Linear Operator|inverse]] $T^{-1}$ so that:
:$T T^{-1} = I$
From [[Left Composition of Compact Linear Tr... | Spectrum of Compact Linear Operator on Infinite-Dimensional Banach Space contains Zero | https://proofwiki.org/wiki/Spectrum_of_Compact_Linear_Operator_on_Infinite-Dimensional_Banach_Space_contains_Zero | https://proofwiki.org/wiki/Spectrum_of_Compact_Linear_Operator_on_Infinite-Dimensional_Banach_Space_contains_Zero | [
"Compact Linear Transformations",
"Spectra (Bounded Linear Operators)"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Banach Space",
"Definition:Compact Linear Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Invertible Bounded Linear Operator"
] | [
"Definition:Invertible Bounded Linear Operator",
"Definition:Bounded Linear Operator",
"Definition:Inverse Linear Operator",
"Compact Linear Transformations Composed with Bounded Linear Operator",
"Definition:Compact Linear Operator",
"Identity Operator is Compact iff Finite-Dimensional Normed Vector Spac... |
proofwiki-20529 | Identity Operator is Compact iff Finite-Dimensional Normed Vector Space | Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $I : X \to X$ be the identity operator.
Then $I$ is compact {{iff}} $X$ is finite-dimensional. | Let $\operatorname {ball} X$ be the closed unit ball in $\struct {X, \norm {\, \cdot \,} }$.
From the definition of a compact operator, we have that $I$ is compact {{iff}}:
:$\overline {\operatorname {ball} X}$ is compact in $\struct {X, \norm \cdot_X}$.
From Closed Ball is Closed and Set is Closed iff Equals Topologic... | Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $I : X \to X$ be the [[Definition:Identity Operator|identity operator]].
Then $I$ is [[Definition:Compact Linear Operator|compact]] {{iff}} $X$ is [[Definition:Finite Dimensional Vector Space|finite-dimensional... | Let $\operatorname {ball} X$ be the [[Definition:Closed Unit Ball|closed unit ball]] in $\struct {X, \norm {\, \cdot \,} }$.
From the definition of a [[Definition:Compact Linear Operator|compact operator]], we have that $I$ is [[Definition:Compact Linear Operator|compact]] {{iff}}:
:$\overline {\operatorname {ball} X... | Identity Operator is Compact iff Finite-Dimensional Normed Vector Space | https://proofwiki.org/wiki/Identity_Operator_is_Compact_iff_Finite-Dimensional_Normed_Vector_Space | https://proofwiki.org/wiki/Identity_Operator_is_Compact_iff_Finite-Dimensional_Normed_Vector_Space | [
"Finite-Dimensional Vector Spaces",
"Finite Dimensional Vector Spaces",
"Finite Dimensional Vector Spaces",
"Compact Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Identity Mapping",
"Definition:Compact Linear Operator",
"Definition:Dimension of Vector Space/Finite"
] | [
"Definition:Closed Unit Ball",
"Definition:Compact Linear Operator",
"Definition:Compact Linear Operator",
"Definition:Compact Topological Space/Subspace",
"Closed Ball is Closed",
"Set is Closed iff Equals Topological Closure",
"Definition:Compact Topological Space/Subspace",
"Normed Vector Space is ... |
proofwiki-20530 | Continuous Linear Operator over Infinite Dimensional Vector Space is not necessarily Invertible | Let $\struct {X, \norm {\, \cdot\,}_X}$ be a normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $I \in \map {CL} X$ be the identity element.
Let $S, T \in \map {CL} X$.
Suppose the dimension of $X$ is finite:
:$d = \dim X = \infty$
Suppose $T \circ S = I$ where $... | Let $X, Y$ be $2$-sequence spaces.
Let $T = L : X \to Y$ be the left shift operator.
Let $S = R : X \to Y$ be the right shift operator.
Let $x := \tuple {a_1, a_2, \ldots} \in X$.
We have that:
{{begin-eqn}}
{{eqn | l = L \circ R \circ x
| r = \map L {\map R {\tuple{a_1, a_2, \ldots} } }
}}
{{eqn | r = \map L {\t... | Let $\struct {X, \norm {\, \cdot\,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $I \in \map {CL} X$ be the [[Definition:Identity Element|identity element]].... | Let $X, Y$ be [[Definition:P-Sequence Space|$2$-sequence spaces]].
Let $T = L : X \to Y$ be the [[Definition:Left Shift Operator|left shift operator]].
Let $S = R : X \to Y$ be the [[Definition:Right Shift Operator|right shift operator]].
Let $x := \tuple {a_1, a_2, \ldots} \in X$.
We have that:
{{begin-eqn}}
{{eq... | Continuous Linear Operator over Infinite Dimensional Vector Space is not necessarily Invertible | https://proofwiki.org/wiki/Continuous_Linear_Operator_over_Infinite_Dimensional_Vector_Space_is_not_necessarily_Invertible | https://proofwiki.org/wiki/Continuous_Linear_Operator_over_Infinite_Dimensional_Vector_Space_is_not_necessarily_Invertible | [
"Continuous Linear Transformations",
"Inverse Mappings"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Dimension of Vector Space",
"Definition:Finite Cardinal",
"Definition:Composition of Mappings",
"Definition:Invertible Continuous Linear Operato... | [
"Definition:P-Sequence Space",
"Definition:Left Shift Operator",
"Definition:Right Shift Operator"
] |
proofwiki-20531 | Invertible Continuous Linear Operator has Unique Inverse | Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $I \in \map {CL} X$ be the identity element.
Suppose $A \in \map {CL} X$ is invertible.
Then there is a unique $B \in \map {CL} X$ such that $A \circ B = B \circ A =... | Let $B_1, B_2 \in \map {CL} X$.
Suppose:
:$A \circ B_1 = I = B_1 \circ A$
:$A \circ B_2 = I = B_2 \circ A$
Then:
{{begin-eqn}}
{{eqn | l = B_1
| r = I \circ B_1
| c = {{Defof|Identity Element}}
}}
{{eqn | r = B_2 \circ A \circ B_1
}}
{{eqn | r = B_2 \circ I
}}
{{eqn | r = B_2
| c = {{Defof|Identity El... | Let $\struct {X, \norm {\, \cdot \,} }$ be the [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $I \in \map {CL} X$ be the [[Definition:Identity Element|identity element]... | Let $B_1, B_2 \in \map {CL} X$.
Suppose:
:$A \circ B_1 = I = B_1 \circ A$
:$A \circ B_2 = I = B_2 \circ A$
Then:
{{begin-eqn}}
{{eqn | l = B_1
| r = I \circ B_1
| c = {{Defof|Identity Element}}
}}
{{eqn | r = B_2 \circ A \circ B_1
}}
{{eqn | r = B_2 \circ I
}}
{{eqn | r = B_2
| c = {{Defof|Identi... | Invertible Continuous Linear Operator has Unique Inverse | https://proofwiki.org/wiki/Invertible_Continuous_Linear_Operator_has_Unique_Inverse | https://proofwiki.org/wiki/Invertible_Continuous_Linear_Operator_has_Unique_Inverse | [
"Continuous Linear Transformations",
"Inverse Mappings"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Invertible Continuous Linear Operator",
"Definition:Unique"
] | [] |
proofwiki-20532 | Positive Part of Darboux Integrable Function is Integrable | Let $f$ be a real function that is Darboux integrable over $\closedint a b$.
Let $f^+$ be the positive part of $f$.
Then $f^+$ is Darboux integrable over $\closedint a b$. | Let $\epsilon > 0$ be a strictly positive real number.
By Condition for Darboux Integrability, there is a finite subdivision $P = \sequence {x_i}_{0 \mathop \le i \mathop \le n}$ such that:
:$\map {U_f} P - \map {L_f} P < \epsilon$
where $\map {U_f} P$ and $\map {L_f} P$ are the upper Darboux sum and lower Darboux sum,... | Let $f$ be a [[Definition:Real Function|real function]] that is [[Definition:Darboux Integrable Function|Darboux integrable]] over $\closedint a b$.
Let $f^+$ be the [[Definition:Positive Part|positive part]] of $f$.
Then $f^+$ is [[Definition:Darboux Integrable Function|Darboux integrable]] over $\closedint a b$. | Let $\epsilon > 0$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
By [[Condition for Darboux Integrability]], there is a [[Definition:Finite Subdivision|finite subdivision]] $P = \sequence {x_i}_{0 \mathop \le i \mathop \le n}$ such that:
:$\map {U_f} P - \map {L_f} P < \epsilon$
wher... | Positive Part of Darboux Integrable Function is Integrable | https://proofwiki.org/wiki/Positive_Part_of_Darboux_Integrable_Function_is_Integrable | https://proofwiki.org/wiki/Positive_Part_of_Darboux_Integrable_Function_is_Integrable | [
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Positive Part",
"Definition:Darboux Integrable Function"
] | [
"Definition:Strictly Positive/Real Number",
"Condition for Darboux Integrability",
"Definition:Subdivision of Interval/Finite",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Supremum does not Precede Infimum",
"Definition:Infimum of Mapping/Real-Valued Function",
"Definition:Supremu... |
proofwiki-20533 | Positive Part of Darboux Integrable Function is Integrable/Negative Part | Let $f$ be a real function that is Darboux integrable over $\closedint a b$.
Let $f^-$ be the negative part of $f$.
Then $f^-$ is Darboux integrable over $\closedint a b$. | $f^-$ is the positive part of $\map g x = -\map f x$.
From Linear Combination of Definite Integrals, it follows that:
:$\ds \int_a^b \map g x \rd x = -\int_a^b \map f x \rd x$
Therefore, by Positive Part of Darboux Integrable Function is Integrable, $f^-$ is Darboux integrable over $\closedint a b$.
{{qed}}
Category:De... | Let $f$ be a [[Definition:Real Function|real function]] that is [[Definition:Darboux Integrable Function|Darboux integrable]] over $\closedint a b$.
Let $f^-$ be the [[Definition:Negative Part|negative part]] of $f$.
Then $f^-$ is [[Definition:Darboux Integrable Function|Darboux integrable]] over $\closedint a b$. | $f^-$ is the [[Definition:Positive Part|positive part]] of $\map g x = -\map f x$.
From [[Linear Combination of Integrals/Definite|Linear Combination of Definite Integrals]], it follows that:
:$\ds \int_a^b \map g x \rd x = -\int_a^b \map f x \rd x$
Therefore, by [[Positive Part of Darboux Integrable Function is Inte... | Positive Part of Darboux Integrable Function is Integrable/Negative Part | https://proofwiki.org/wiki/Positive_Part_of_Darboux_Integrable_Function_is_Integrable/Negative_Part | https://proofwiki.org/wiki/Positive_Part_of_Darboux_Integrable_Function_is_Integrable/Negative_Part | [
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Darboux Integrable Function",
"Definition:Negative Part",
"Definition:Darboux Integrable Function"
] | [
"Definition:Positive Part",
"Linear Combination of Integrals/Definite",
"Positive Part of Darboux Integrable Function is Integrable",
"Definition:Darboux Integrable Function",
"Category:Definite Integrals"
] |
proofwiki-20534 | Topological Evaluation Mapping is Continuous | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \... | For each $i \in I$, let:
:$\pr_i : Y \to Y_i$ denote the $i$th projection on $Y$
From Composite of Evaluation Mapping and Projection:
:$\forall i \in I : \pr_i \circ f = f_i$
By assumption:
:$\forall i \in I : \pr_i \circ f$ is continuous
From Continuous Mapping to Product Space:
:$f$ is continuous
{{qed}}
Category:Eva... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | For each $i \in I$, let:
:$\pr_i : Y \to Y_i$ denote the [[Definition:Projection (Mapping Theory)|$i$th projection]] on $Y$
From [[Composite of Evaluation Mapping and Projection]]:
:$\forall i \in I : \pr_i \circ f = f_i$
By assumption:
:$\forall i \in I : \pr_i \circ f$ is [[Definition:Continuous Mapping|continuous]... | Topological Evaluation Mapping is Continuous | https://proofwiki.org/wiki/Topological_Evaluation_Mapping_is_Continuous | https://proofwiki.org/wiki/Topological_Evaluation_Mapping_is_Continuous | [
"Evaluation Mappings (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)",
"Definition:Evaluation Mapping (Topology)",
"Definition:Co... | [
"Definition:Projection (Mapping Theory)",
"Composite of Evaluation Mapping and Projection",
"Definition:Continuous Mapping",
"Continuous Mapping to Product Space",
"Definition:Continuous Mapping",
"Category:Evaluation Mappings (Topological Spaces)"
] |
proofwiki-20535 | Characterization for Topological Evaluation Mapping to be Embedding | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \... | === Necessary Condition ===
Let $f$ be an embedding.
{{:Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition}}{{qed|lemma}} | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | === [[Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition|Necessary Condition]] ===
Let $f$ be an [[Definition:Embedding (Topology)|embedding]].
{{:Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition}}{{qed|lemma}} | Characterization for Topological Evaluation Mapping to be Embedding | https://proofwiki.org/wiki/Characterization_for_Topological_Evaluation_Mapping_to_be_Embedding | https://proofwiki.org/wiki/Characterization_for_Topological_Evaluation_Mapping_to_be_Embedding | [
"Evaluation Mappings (Topological Spaces)",
"Embeddings (Topology)",
"Characterization for Topological Evaluation Mapping to be Embedding"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)",
"Definition:Evaluation Mapping (Topology)",
"Definition:Em... | [
"Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition",
"Definition:Embedding (Topology)"
] |
proofwiki-20536 | Eigenvectors Corresponding to Distinct Eigenvalues of Linear Operator are Linearly Independent | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $T : X \to X$ be a linear operator.
Let $\lambda_1, \lambda_2, \ldots, \lambda_n \in K$ be distinct eigenvalues of $T$.
Let $x_1, x_2, \ldots, x_n$ be eigenvectors corresponding to $\lambda_1, \lambda_2, \ldots, \lambda_n$.
Then $\set {x_1, \ldots, x_n}$ is... | Proof by induction:
Let $\map P n$ be the proposition:
:for any $n$ eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n \in K$ and eigenvectors $x_1, x_2, \ldots, x_n$ corresponding to $\lambda_1, \lambda_2, \ldots, \lambda_n$, we have that:
::$\set {x_1, \ldots, x_n}$ is a linearly independent set. | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $T : X \to X$ be a [[Definition:Linear Operator|linear operator]].
Let $\lambda_1, \lambda_2, \ldots, \lambda_n \in K$ be distinct [[Definition:Eigenvalue of Linear Operator|eigenvalues]]... | Proof by [[Principle of Mathematical Induction|induction]]:
Let $\map P n$ be the proposition:
:for any $n$ [[Definition:Eigenvalue of Linear Operator|eigenvalues]] $\lambda_1, \lambda_2, \ldots, \lambda_n \in K$ and [[Definition:Eigenvector of Linear Operator|eigenvectors]] $x_1, x_2, \ldots, x_n$ corresponding to $... | Eigenvectors Corresponding to Distinct Eigenvalues of Linear Operator are Linearly Independent | https://proofwiki.org/wiki/Eigenvectors_Corresponding_to_Distinct_Eigenvalues_of_Linear_Operator_are_Linearly_Independent | https://proofwiki.org/wiki/Eigenvectors_Corresponding_to_Distinct_Eigenvalues_of_Linear_Operator_are_Linearly_Independent | [
"Eigenvalues of Linear Operators",
"Linear Operators"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Operator",
"Definition:Eigenvalue/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Linearly Independent/Set"
] | [
"Principle of Mathematical Induction",
"Definition:Eigenvalue/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Linearly Independent/Set",
"Definition:Linearly Independent/Set",
"Definition:Eigenvalue/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Linearl... |
proofwiki-20537 | Spectrum of Bounded Linear Operator contains Point Spectrum | Let $X$ be a Banach space over $\C$.
Let $T : X \to X$ be a bounded linear operator.
Let $\map {\sigma_p} T$ be the point spectrum of $T$.
Let $\map \sigma T$ be the spectrum of $T$.
Then $\map {\sigma_p} T \subseteq \map \sigma T$. | Let $\lambda \in \map {\sigma_p} T$.
Then there exists $x \ne \mathbf 0_X$ such that $T x = \lambda x$.
So $\paren {T - \lambda I} x = 0$ for some $x \ne \mathbf 0_X$.
So $\ker T \ne \set {\mathbf 0_X}$.
So from Linear Transformation is Injective iff Kernel Contains Only Zero, $T$ is not injective.
So $T$ cannot be ... | Let $X$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map {\sigma_p} T$ be the [[Definition:Point Spectrum of Linear Operator|point spectrum]] of $T$.
Let $\map \sigma T$ be the [[Definition:Spectrum of Bounded ... | Let $\lambda \in \map {\sigma_p} T$.
Then there exists $x \ne \mathbf 0_X$ such that $T x = \lambda x$.
So $\paren {T - \lambda I} x = 0$ for some $x \ne \mathbf 0_X$.
So $\ker T \ne \set {\mathbf 0_X}$.
So from [[Linear Transformation is Injective iff Kernel Contains Only Zero]], $T$ is not [[Definition:Injectiv... | Spectrum of Bounded Linear Operator contains Point Spectrum | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_contains_Point_Spectrum | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_contains_Point_Spectrum | [
"Point Spectra (Linear Operators)",
"Spectra (Bounded Linear Operators)"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator",
"Definition:Point Spectrum of Linear Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator"
] | [
"Linear Transformation is Injective iff Kernel Contains Only Zero",
"Definition:Injective",
"Definition:Invertible Bounded Linear Operator",
"Category:Point Spectra (Linear Operators)",
"Category:Spectra (Bounded Linear Operators)"
] |
proofwiki-20538 | Spectrum of Bounded Linear Operator on Finite-Dimensional Banach Space is equal to Point Spectrum | Let $X$ be a finite-dimensional Banach space over $\C$.
Let $T : X \to X$ be a bounded linear operator.
Let $\map {\sigma_p} T$ be the point spectrum of $T$.
Let $\map \sigma T$ be the spectrum of $T$.
Then $\map \sigma T = \map {\sigma_p} T$. | We have that $\lambda \in \map \sigma T$ {{iff}} $T - \lambda I$ is not invertible as a bounded linear transformation.
So $T - \lambda I$ is not bijective or its inverse $\paren {T - \lambda I}^{-1}$ is not bounded.
From Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous, every linear... | Let $X$ be a [[Definition:Finite Dimensional Vector Space|finite-dimensional]] [[Definition:Banach Space|Banach space]] over $\C$.
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\map {\sigma_p} T$ be the [[Definition:Point Spectrum of Linear Operator|point spectrum]] of $... | We have that $\lambda \in \map \sigma T$ {{iff}} $T - \lambda I$ is not [[Definition:Invertible Bounded Linear Transformation|invertible as a bounded linear transformation]].
So $T - \lambda I$ is not [[Definition:Bijective|bijective]] or its [[Definition:Inverse Mapping|inverse]] $\paren {T - \lambda I}^{-1}$ is not ... | Spectrum of Bounded Linear Operator on Finite-Dimensional Banach Space is equal to Point Spectrum | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_on_Finite-Dimensional_Banach_Space_is_equal_to_Point_Spectrum | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_on_Finite-Dimensional_Banach_Space_is_equal_to_Point_Spectrum | [
"Spectra (Bounded Linear Operators)",
"Point Spectra (Linear Operators)",
"Finite Dimensional Vector Spaces"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Banach Space",
"Definition:Bounded Linear Operator",
"Definition:Point Spectrum of Linear Operator",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator"
] | [
"Definition:Invertible Bounded Linear Transformation",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Definition:Bounded Linear Operator",
"Linear Transformations between Finite-Dimensional Normed Vector Spaces are Continuous",
"Definition:Linear Operator",
"Definition:Bounded Linear Operator",
... |
proofwiki-20539 | Evaluation Mapping on T1 Space is Embedding if Mappings Separate Points from Closed Sets | Let $X$ be a $T_1$ topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\family {f_i}_{i \mathop \in I}$ separate points from closed sets.
Let $\d... | Let $\BB = \set{f_i^{-1} \sqbrk V : i \in I, V \text{ is open in } Y_i}$.
From Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets:
:$\BB$ is a basis for $X$
From Analytic Basis is Analytic Sub-Basis:
:$\BB$ is a sub-basis for $X$
By definition of a $T_1$ space:
:all points of $X$ ... | Let $X$ be a [[Definition:T1 Space|$T_1$ topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an ... | Let $\BB = \set{f_i^{-1} \sqbrk V : i \in I, V \text{ is open in } Y_i}$.
From [[Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets]]:
:$\BB$ is a [[Definition:Analytic Basis|basis]] for $X$
From [[Analytic Basis is Analytic Sub-Basis]]:
:$\BB$ is a [[Definition:Analytic Sub-Ba... | Evaluation Mapping on T1 Space is Embedding if Mappings Separate Points from Closed Sets | https://proofwiki.org/wiki/Evaluation_Mapping_on_T1_Space_is_Embedding_if_Mappings_Separate_Points_from_Closed_Sets | https://proofwiki.org/wiki/Evaluation_Mapping_on_T1_Space_is_Embedding_if_Mappings_Separate_Points_from_Closed_Sets | [
"Continuous Mappings",
"Embeddings (Topology)"
] | [
"Definition:T1 Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Mappings Separating Points from Closed Sets",
"Definition:Product Space (Topology)",
"Definiti... | [
"Preimage of Open Sets forms Basis if Continuous Mappings Separate Points from Closed Sets",
"Definition:Basis (Topology)/Analytic Basis",
"Analytic Basis is Analytic Sub-Basis",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:T1 Space",
"Definition:Closed Point",
"Definition:Mappings Separating ... |
proofwiki-20540 | Subspace of Product Space has Initial Topology with respect to Restricted Projections | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $\XX$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:
:$\ds \XX := \prod_{i \mathop \in I} X_i$
For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the projection o... | By definition of product topology:
:$\tau_\XX$ is the initial topology with respect to the mappings $\family {\pr_i : \XX \to X_i}_{i \mathop \in I}$
From Subspace Topology on Initial Topology is Initial Topology on Restrictions:
:$\tau_Y$ is the initial topology on $Y$ with respect to the mappings $\family {\pr_i \res... | Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\XX$ be the [[Definition:Cartesian Product of Family|cartesian product]] of $\family {... | By definition of [[Definition:Product Topology|product topology]]:
:$\tau_\XX$ is the [[Definition:Initial Topology|initial topology]] with respect to the [[Definition:Mapping|mappings]] $\family {\pr_i : \XX \to X_i}_{i \mathop \in I}$
From [[Subspace Topology on Initial Topology is Initial Topology on Restrictions]]... | Subspace of Product Space has Initial Topology with respect to Restricted Projections | https://proofwiki.org/wiki/Subspace_of_Product_Space_has_Initial_Topology_with_respect_to_Restricted_Projections | https://proofwiki.org/wiki/Subspace_of_Product_Space_has_Initial_Topology_with_respect_to_Restricted_Projections | [
"Product Topology",
"Topological Subspaces",
"Initial Topology"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Cartesian Product/Family of Sets",
"Definition:Projection (Mapping Theory)",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Product Topology",
"D... | [
"Definition:Product Topology",
"Definition:Initial Topology",
"Definition:Mapping",
"Subspace Topology on Initial Topology is Initial Topology on Restrictions",
"Definition:Initial Topology",
"Definition:Mapping",
"Category:Product Topology",
"Category:Topological Subspaces",
"Category:Initial Topol... |
proofwiki-20541 | Invertible Continuous Linear Operator is Bijective | Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $I \in \map {CL} X$ be the identity element.
Suppose $A \in \map {CL} X$ is invertible.
Then $A$ is bijective. | === $A$ is injective ===
Let $x, y \in X$ be such that $\map A x = \map A y$.
Then:
:$A^{-1} \circ \map A x = A^{-1} \circ \map A y$
where $A^{-1}$ is the inverse of $A$.
By definition:
:$A^{-1} \circ A = I$
Hence:
:$x = y$
By definition, $A$ is injective.
{{qed|lemma}} | Let $\struct {X, \norm {\, \cdot \,} }$ be the [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $I \in \map {CL} X$ be the [[Definition:Identity Element|identity element]... | === $A$ is [[Definition:Injection/Definition 1|injective]] ===
Let $x, y \in X$ be such that $\map A x = \map A y$.
Then:
:$A^{-1} \circ \map A x = A^{-1} \circ \map A y$
where $A^{-1}$ is the [[Definition:Inverse of Continuous Linear Operator|inverse of $A$]].
By [[Definition:Inverse of Continuous Linear Operator... | Invertible Continuous Linear Operator is Bijective | https://proofwiki.org/wiki/Invertible_Continuous_Linear_Operator_is_Bijective | https://proofwiki.org/wiki/Invertible_Continuous_Linear_Operator_is_Bijective | [
"Continuous Linear Transformations",
"Bijections"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Invertible Continuous Linear Operator",
"Definition:Bijection"
] | [
"Definition:Injection/Definition 1",
"Definition:Inverse of Continuous Linear Operator",
"Definition:Inverse of Continuous Linear Operator",
"Definition:Injection/Definition 1"
] |
proofwiki-20542 | Power Series Expansion for Reciprocal of Square Root of 1 - x | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\sqrt {1 - x} }
| r = \sum_{k \mathop = 0}^\infty \frac {\paren {2 k}!} {\paren {2^k k!}^2} x^k
| c =
}}
{{eqn | r = 1 + \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x... | {{begin-eqn}}
{{eqn | l = \frac 1 {\sqrt {1 - x} }
| r = \paren {1 - x}^{-\frac 1 2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 2}^{\underline k} } {k!} \paren {-x}^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = ... | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\sqrt {1 - x} }
| r = \sum_{k \mathop = 0}^\infty \frac {\paren {2 k}!} {\paren {2^k k!}^2} x^k
| c =
}}
{{eqn | r = 1 + \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6}... | {{begin-eqn}}
{{eqn | l = \frac 1 {\sqrt {1 - x} }
| r = \paren {1 - x}^{-\frac 1 2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 2}^{\underline k} } {k!} \paren {-x}^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \matho... | Power Series Expansion for Reciprocal of Square Root of 1 - x/Proof 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_Root_of_1_-_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_Root_of_1_-_x/Proof_2 | [
"Power Series Expansion for Reciprocal of Square Root of 1 - x",
"Examples of Power Series"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Translation of Index Variable of Product",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-20543 | Bijective Continuous Linear Operator is not necessarily Invertible | Let $\struct {X, \norm {\, \cdot \,} }$ be the normed vector space.
Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.
Let $I \in \map {CL} X$ be the identity element.
Suppose $A \in \map {CL} X$ is bijective.
Then $A$ is not necessarily invertible. | Let $\mathbb F \in \set {\R, \C}$.
Let $\map {c_{00} } {\mathbb F}$ be the space of almost-zero sequences on $\mathbb F$.
Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_N, 0, \ldots} \in c_{00}$.
Let $A : c_{00} \to c_{00}$ be a mapping such that:
:$\map A {\tuple {x_1, x_2, x_3, \ldots} } = \tuple {x_1, \dfrac {x_2} 2,... | Let $\struct {X, \norm {\, \cdot \,} }$ be the [[Definition:Normed Vector Space|normed vector space]].
Let $\map {CL} X := \map {CL} {X, X}$ be a [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $I \in \map {CL} X$ be the [[Definition:Identity Element|identity element]... | Let $\mathbb F \in \set {\R, \C}$.
Let $\map {c_{00} } {\mathbb F}$ be the [[Definition:Space of Almost-Zero Sequences|space of almost-zero sequences]] on $\mathbb F$.
Let $\mathbf x = \tuple {x_1, x_2, \ldots, x_N, 0, \ldots} \in c_{00}$.
Let $A : c_{00} \to c_{00}$ be a [[Definition:Mapping|mapping]] such that:
... | Bijective Continuous Linear Operator is not necessarily Invertible | https://proofwiki.org/wiki/Bijective_Continuous_Linear_Operator_is_not_necessarily_Invertible | https://proofwiki.org/wiki/Bijective_Continuous_Linear_Operator_is_not_necessarily_Invertible | [
"Continuous Linear Transformations",
"Bijections",
"Inverse Mappings"
] | [
"Definition:Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Bijection",
"Definition:Logical Not",
"Definition:Conditional/Necessary Condition",
"Definition:Invertible Continuous Linear Operator"
] | [
"Definition:Space of Almost-Zero Sequences",
"Definition:Mapping",
"Definition:Space of Bounded Sequences/Normed Vector Space",
"Space of Almost-Zero Sequences is Subspace of Space of Bounded Sequences",
"Space of Almost-Zero Sequences with Supremum Norm is Normed Vector Space"
] |
proofwiki-20544 | Linear Transformation has Finite Rank iff Domain Quotiented by Kernel is Finite Dimensional | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be a linear transformation.
Let $\ker T$ be the kernel of $T$.
Let $X/\ker T$ be the quotient vector space of $X$ modulo $\ker T$.
Then $T$ has finite rank {{iff}} $X/\ker T$ is finite dimensional. | Let $q : X \to X/\ker T$ be the quotient mapping. | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $\ker T$ be the [[Definition:Kernel of Linear Transformation|kernel]] of $T$.
Let $X/\ker T$ be the [[Definiti... | Let $q : X \to X/\ker T$ be the [[Definition:Quotient Mapping|quotient mapping]]. | Linear Transformation has Finite Rank iff Domain Quotiented by Kernel is Finite Dimensional | https://proofwiki.org/wiki/Linear_Transformation_has_Finite_Rank_iff_Domain_Quotiented_by_Kernel_is_Finite_Dimensional | https://proofwiki.org/wiki/Linear_Transformation_has_Finite_Rank_iff_Domain_Quotiented_by_Kernel_is_Finite_Dimensional | [
"Finite Rank Operators",
"Quotient Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Kernel of Linear Transformation",
"Definition:Quotient Vector Space",
"Definition:Finite Rank Operator",
"Definition:Dimension of Vector Space/Finite"
] | [
"Definition:Quotient Mapping"
] |
proofwiki-20545 | Power Series Expansion for Square Root of 1 - x | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sqrt {1 - x}
| r = 1 - \sum_{k \mathop = 1}^\infty \frac {\paren {2 \paren {k - 1} }!} {2^{2 k - 1} k! \paren {k - 1}!} x^k
| c =
}}
{{eqn | r = 1 - \frac 1 2 x - \frac 1 {2 \times 4} x^2 - \frac {1 \times 3} {2 \times 4 \times 6} x^... | {{begin-eqn}}
{{eqn | l = \sqrt {1 + x}
| r = 1 + \frac 1 2 x - \frac 1 {2 \times 4} x^2 + \frac {1 \times 3} {2 \times 4 \times 6} x^3 - \cdots
| c = Power Series Expansion for $\sqrt {1 + x}$
}}
{{eqn | ll= \leadsto
| l = \sqrt {1 - x}
| r = 1 + \frac 1 2 \paren {-x} - \frac 1 {2 \times 4} \pa... | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sqrt {1 - x}
| r = 1 - \sum_{k \mathop = 1}^\infty \frac {\paren {2 \paren {k - 1} }!} {2^{2 k - 1} k! \paren {k - 1}!} x^k
| c =
}}
{{eqn | r = 1 - \frac 1 2 x - \frac 1 {2 \times 4} x^2 - \frac {1 \times 3} {2 \times 4 \times 6} ... | {{begin-eqn}}
{{eqn | l = \sqrt {1 + x}
| r = 1 + \frac 1 2 x - \frac 1 {2 \times 4} x^2 + \frac {1 \times 3} {2 \times 4 \times 6} x^3 - \cdots
| c = [[Power Series Expansion for Square Root of 1 + x|Power Series Expansion for $\sqrt {1 + x}$]]
}}
{{eqn | ll= \leadsto
| l = \sqrt {1 - x}
| r = ... | Power Series Expansion for Square Root of 1 - x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Square_Root_of_1_-_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Square_Root_of_1_-_x | [
"Examples of Power Series"
] | [] | [
"Power Series Expansion for Square Root of 1 + x"
] |
proofwiki-20546 | Non-Zero Subspace of Topological Vector Space is not von Neumann-Bounded | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $Y \ne \set {\mathbf 0_X}$ be a non-trivial subspace of $X$.
Then $Y$ is not von Neumann-bounded. | Let $x \in Y \setminus \set {\mathbf 0_X}$.
Then $t x \in Y$ for $t > 0$.
Since $X$ is Hausdorff, there exists an open neighborhood $V$ of $x$ that does not contain $\mathbf 0_X$.
Then $t x \not \in t V$ for each $t > 0$.
So we do not have $Y \subseteq t V$ for any $t > 0$.
So $Y$ is not von Neumann-bounded.
{{qed}} | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $Y \ne \set {\mathbf 0_X}$ be a [[Definition:Zero Subspace|non-trivial subspace]] of $X$.
Then $Y$ is not [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-boun... | Let $x \in Y \setminus \set {\mathbf 0_X}$.
Then $t x \in Y$ for $t > 0$.
Since $X$ is [[Definition:Hausdorff Space|Hausdorff]], there exists an [[Definition:Open Neighborhood|open neighborhood]] $V$ of $x$ that does not contain $\mathbf 0_X$.
Then $t x \not \in t V$ for each $t > 0$.
So we do not have $Y \subset... | Non-Zero Subspace of Topological Vector Space is not von Neumann-Bounded | https://proofwiki.org/wiki/Non-Zero_Subspace_of_Topological_Vector_Space_is_not_von_Neumann-Bounded | https://proofwiki.org/wiki/Non-Zero_Subspace_of_Topological_Vector_Space_is_not_von_Neumann-Bounded | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Zero Subspace",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:T2 Space",
"Definition:Open Neighborhood",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] |
proofwiki-20547 | Linear Transformation between Normed Vector Spaces is Bounded iff Bounded as Linear Transformation between Topological Vector Spaces | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $T : X \to Y$ be linear transformations.
Then $T$ is bounded as a linear transformation between normed vector spaces {{iff}} it is bounded as a linear transformation b... | === Sufficient Condition ===
Suppose that $T$ is bounded as a linear transformation between topological vector spaces
Then:
:for each von Neumann-bounded subset $E$ of $X$, $T \sqbrk E$ is von Neumann-bounded.
From Characterization of von Neumann-Boundedness in Normed Vector Space, this is equivalent to:
:if $E \subse... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $T : X \to Y$ be [[Definition:Linear Transformation|linear transformations]].
Then $T$ is [[Definition:Bounded Linear Transforma... | === Sufficient Condition ===
Suppose that $T$ is [[Definition:Bounded Linear Transformation/Topological Vector Space|bounded as a linear transformation between topological vector spaces]]
Then:
:for each [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded subset]] $E$ of $X$, $T \... | Linear Transformation between Normed Vector Spaces is Bounded iff Bounded as Linear Transformation between Topological Vector Spaces | https://proofwiki.org/wiki/Linear_Transformation_between_Normed_Vector_Spaces_is_Bounded_iff_Bounded_as_Linear_Transformation_between_Topological_Vector_Spaces | https://proofwiki.org/wiki/Linear_Transformation_between_Normed_Vector_Spaces_is_Bounded_iff_Bounded_as_Linear_Transformation_between_Topological_Vector_Spaces | [
"Bounded Linear Transformations (Topological Vector Spaces)",
"Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation/Normed Vector Space",
"Definition:Bounded Linear Transformation/Topological Vector Space"
] | [
"Definition:Bounded Linear Transformation/Topological Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Characterization of von Neumann-Boundedness in Normed Vector Space",
"Definition:Bounded Linear Tran... |
proofwiki-20548 | Homeomorphic Topology of Initial Topology is Initial Topology | Let $\struct {X_\alpha, \tau_\alpha}, \struct {X_\beta, \tau_\beta}$ be topological spaces.
Let $\ds \family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $\ds \family {f_i: X_\beta \to Y_i}_{i \mathop \in I}$ be an indexed family of mapp... | Let $\SS_\beta = \set {f_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$.
By definition of initial topology:
:$\SS_\beta$ is a sub-basis for $\tau_\beta$
From Inverse of Homeomorphism is Homeomorphism:
:$\phi^{-1}$ is a homeomorphism
From Homeomorphic Image of Sub-Basis is Sub-Basis:
:$\SS_\alpha = \set {\phi^{-1} \sqbrk {f_... | Let $\struct {X_\alpha, \tau_\alpha}, \struct {X_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $\ds \family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary ... | Let $\SS_\beta = \set {f_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$.
By definition of [[Definition:Initial Topology|initial topology]]:
:$\SS_\beta$ is a [[Definition:Analytic Sub-Basis|sub-basis]] for $\tau_\beta$
From [[Inverse of Homeomorphism is Homeomorphism]]:
:$\phi^{-1}$ is a [[Definition:Homeomorphism (Topol... | Homeomorphic Topology of Initial Topology is Initial Topology | https://proofwiki.org/wiki/Homeomorphic_Topology_of_Initial_Topology_is_Initial_Topology | https://proofwiki.org/wiki/Homeomorphic_Topology_of_Initial_Topology_is_Initial_Topology | [
"Homeomorphisms (Topological Spaces)",
"Initial Topology"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Mapping",
"Definition:Indexing Set",
"Definition:Initial Topology",
"Definition:Homeomorphism/Topological Spaces",
"Definition:... | [
"Definition:Initial Topology",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Inverse of Homeomorphism is Homeomorphism",
"Definition:Homeomorphism/Topological Spaces",
"Homeomorphic Image of Sub-Basis is Sub-Basis",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Preimage of Subset under Composite Mapping",
... |
proofwiki-20549 | Net Characterization of von Neumann-Boundedness in Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$.
Let $E \subseteq X$.
Let $\struct {\Lambda, \preceq}$ be a directed set.
Let $E$ be von Neumann-bounded.
Then:
:for each net $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda}$ in $E$ and each net $\sequence {\alpha_\lambda}_{\lambda \mat... | === Necessary Condition ===
Let $E \subseteq X$ be von Neumann-bounded.
Let $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a net in $E$.
Let $\sequence {\alpha_\lambda}_{\lambda \mathop \in \Lambda}$ be a net in $\GF$ such that $\alpha_\lambda \to 0$.
Let $U$ be an open neighborhood of ${\mathbf 0}_X$.
From... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $E \subseteq X$.
Let $\struct {\Lambda, \preceq}$ be a [[Definition:Directed Set|directed set]].
Let $E$ be [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-boun... | === Necessary Condition ===
Let $E \subseteq X$ be [[Definition:Von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded]].
Let $\sequence {x_\lambda}_{\lambda \mathop \in \Lambda}$ be a [[Definition:Net (Set Theory)|net]] in $E$.
Let $\sequence {\alpha_\lambda}_{\lambda \mathop \in \Lambda}$ be a... | Net Characterization of von Neumann-Boundedness in Topological Vector Space | https://proofwiki.org/wiki/Net_Characterization_of_von_Neumann-Boundedness_in_Topological_Vector_Space | https://proofwiki.org/wiki/Net_Characterization_of_von_Neumann-Boundedness_in_Topological_Vector_Space | [
"Von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Directed Preordering",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Net",
"Definition:Net",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Von Neumann-Bounded Subset of Topological Vector Space"
] | [
"Definition:Von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Net (Set Theory)",
"Definition:Net (Set Theory)",
"Definition:Open Neighborhood",
"Open Neighborhood of Origin in Topological Vector Space contains Balanced Open Neighborhood",
"Definition:Balanced Set",
"Definition:Open Ne... |
proofwiki-20550 | Primitive of Reciprocal of 1 plus Sine of a x/Corollary | :$\ds \int \frac {\d x} {1 + \sin x} = \map \tan {\frac x 2 - \frac \pi 4} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 + \sin a x}
| r = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C
| c = Primitive of $\dfrac 1 {1 + \sin a x}$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \sin x}
| r = -\map \tan {\frac \pi 4 - \frac x 2} + C
| c = setting $a ... | :$\ds \int \frac {\d x} {1 + \sin x} = \map \tan {\frac x 2 - \frac \pi 4} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 + \sin a x}
| r = -\frac 1 a \map \tan {\frac \pi 4 - \frac {a x} 2} + C
| c = [[Primitive of Reciprocal of 1 plus Sine of a x|Primitive of $\dfrac 1 {1 + \sin a x}$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 + \sin x}
| r = -\map \tan {\f... | Primitive of Reciprocal of 1 plus Sine of a x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Sine_of_a_x/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_plus_Sine_of_a_x/Corollary | [
"Primitive of Reciprocal of 1 plus Sine of a x"
] | [] | [
"Primitive of Reciprocal of 1 plus Sine of a x",
"Tangent Function is Odd"
] |
proofwiki-20551 | Newton's Method/Sequence of Approximations Converges Quadratically | Let $\map f x$ be a real function.
Let $\alpha$ be a root of $\map f x$.
Let $\epsilon > 0$ be a positive real number, and $I = \closedint {\alpha - \epsilon} {\alpha + \epsilon}$.
Let $f$ have a continuous second derivative on $I$.
Let $\ds M = \map \sup {\size {\frac {\map {f' '} s} {\map {f'} t} } }$ over all $s, t ... | Suppose that the sequence is produced up to $x_n$.
Suppose also that $x_n \in I$.
{{begin-eqn}}
{{eqn | l = \map f {x_n} + \paren {\alpha - x_n} \map {f'} {x_n} + \frac {\map {f' '} {\zeta_n} } 2 \paren {\alpha - x_n}^2
| r = \map f \alpha
| c = Taylor's Theorem, for some $\zeta_n \in \openint {x_n} \alpha$... | Let $\map f x$ be a [[Definition:Real Function|real function]].
Let $\alpha$ be a [[Definition:Root of Function|root]] of $\map f x$.
Let $\epsilon > 0$ be a [[Definition:Strictly Positive Real Number|positive real number]], and $I = \closedint {\alpha - \epsilon} {\alpha + \epsilon}$.
Let $f$ have a [[Definition:Co... | Suppose that the sequence is produced up to $x_n$.
Suppose also that $x_n \in I$.
{{begin-eqn}}
{{eqn | l = \map f {x_n} + \paren {\alpha - x_n} \map {f'} {x_n} + \frac {\map {f' '} {\zeta_n} } 2 \paren {\alpha - x_n}^2
| r = \map f \alpha
| c = [[Taylor's Theorem]], for some $\zeta_n \in \openint {x_n} \... | Newton's Method/Sequence of Approximations Converges Quadratically | https://proofwiki.org/wiki/Newton's_Method/Sequence_of_Approximations_Converges_Quadratically | https://proofwiki.org/wiki/Newton's_Method/Sequence_of_Approximations_Converges_Quadratically | [
"Newton's Method"
] | [
"Definition:Real Function",
"Definition:Root of Mapping",
"Definition:Strictly Positive/Real Number",
"Definition:Continuous Real Function/Subset",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Non-Vanishing",
"Newton's Method",
"Definition:Convergent Sequence/Real Numbers"... | [
"Taylor's Theorem",
"Newton's Method",
"Principle of Mathematical Induction/Zero-Based",
"Principle of Recursive Definition",
"Principle of Mathematical Induction/Zero-Based",
"Definition:Order of Convergence"
] |
proofwiki-20552 | Primitive of Reciprocal of 1 minus Sine of a x/Corollary 2 | :$\ds \int \frac {\d x} {1 - \sin x} = -\map \cot {\frac x 2 - \frac \pi 4} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \sin a x}
| r = \map \tan {\frac x 2 + \frac \pi 4} + C
| c = Primitive of $\dfrac 1 {1 - \sin x}$: Tangent form
}}
{{eqn | r = \dfrac \pi 2 - \map \cot {\frac \pi 4 + \frac x 2} + C
| c = setting $a \gets 1$
}}
{{eqn | r =
| c =
}}
{{end-eqn}}
... | :$\ds \int \frac {\d x} {1 - \sin x} = -\map \cot {\frac x 2 - \frac \pi 4} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \sin a x}
| r = \map \tan {\frac x 2 + \frac \pi 4} + C
| c = [[Primitive of Reciprocal of 1 minus Sine of a x/Corollary 1|Primitive of $\dfrac 1 {1 - \sin x}$: Tangent form]]
}}
{{eqn | r = \dfrac \pi 2 - \map \cot {\frac \pi 4 + \frac x 2} + C
| c = s... | Primitive of Reciprocal of 1 minus Sine of a x/Corollary 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Sine_of_a_x/Corollary_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Sine_of_a_x/Corollary_2 | [
"Primitive of Reciprocal of 1 minus Sine of a x"
] | [] | [
"Primitive of Reciprocal of 1 minus Sine of a x/Corollary 1"
] |
proofwiki-20553 | Primitive of Reciprocal of 1 minus Sine of a x/Corollary 1 | :$\ds \int \frac {\d x} {1 - \sin x} = \map \tan {\frac x 2 + \frac \pi 4} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \sin a x}
| r = \frac 1 a \map \tan {\frac \pi 4 + \frac {a x} 2} + C
| c = Primitive of $\dfrac 1 {1 - \sin a x}$
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 - \sin x}
| r = \map \tan {\frac \pi 4 + \frac x 2} + C
| c = setting $a \g... | :$\ds \int \frac {\d x} {1 - \sin x} = \map \tan {\frac x 2 + \frac \pi 4} + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {1 - \sin a x}
| r = \frac 1 a \map \tan {\frac \pi 4 + \frac {a x} 2} + C
| c = [[Primitive of Reciprocal of 1 minus Sine of a x|Primitive of $\dfrac 1 {1 - \sin a x}$]]
}}
{{eqn | ll= \leadsto
| l = \int \frac {\d x} {1 - \sin x}
| r = \map \tan {\fr... | Primitive of Reciprocal of 1 minus Sine of a x/Corollary 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Sine_of_a_x/Corollary_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_1_minus_Sine_of_a_x/Corollary_1 | [
"Primitive of Reciprocal of 1 minus Sine of a x"
] | [] | [
"Primitive of Reciprocal of 1 minus Sine of a x"
] |
proofwiki-20554 | Homeomorphic Image of Sub-Basis is Sub-Basis | Let $T_\alpha = \struct{X_\alpha, \tau_\alpha}, T_\beta = \struct{X_\beta, \tau_\beta}$ be topological spaces.
Let $\SS_\alpha \subseteq \tau_\alpha$ be a sub-basis for $\tau_\alpha$.
Let $\phi: T_\alpha \to T_\beta$ be a homeomorphism.
Let $\SS_\beta = \set{\phi \sqbrk S : S \in \SS_\alpha}$.
Then:
:$\SS_\beta$ is a s... | By definition of homeomorphism:
:$\forall U \subseteq X_\alpha : U \in \tau_\alpha \iff \phi \sqbrk U \in \tau_\beta$
By definition of sub-basis:
:$\SS_\alpha \subseteq \tau_\alpha$
Hence:
:$\SS_\beta \subseteq \tau_\beta$
Let $\ds \BB_\alpha = \set {\bigcap \FF: \FF \subseteq \SS_\alpha, \FF \text{ is finite} }$.
Let ... | Let $T_\alpha = \struct{X_\alpha, \tau_\alpha}, T_\beta = \struct{X_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $\SS_\alpha \subseteq \tau_\alpha$ be a [[Definition:Analytic Sub-Basis|sub-basis]] for $\tau_\alpha$.
Let $\phi: T_\alpha \to T_\beta$ be a [[Definition:Homeomorphism (... | By definition of [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]]:
:$\forall U \subseteq X_\alpha : U \in \tau_\alpha \iff \phi \sqbrk U \in \tau_\beta$
By definition of [[Definition:Analytic Sub-Basis|sub-basis]]:
:$\SS_\alpha \subseteq \tau_\alpha$
Hence:
:$\SS_\beta \subseteq \tau_\beta$
Let $\ds ... | Homeomorphic Image of Sub-Basis is Sub-Basis | https://proofwiki.org/wiki/Homeomorphic_Image_of_Sub-Basis_is_Sub-Basis | https://proofwiki.org/wiki/Homeomorphic_Image_of_Sub-Basis_is_Sub-Basis | [
"Homeomorphisms (Topological Spaces)",
"Topological Bases"
] | [
"Definition:Topological Space",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Sub-Basis/Analytic Sub-Basis"
] | [
"Definition:Homeomorphism/Topological Spaces",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Image of Intersection under Injection",
"Inverse of Homeomorphism is Homeomorphism",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Sub-Basis/Analytic Sub... |
proofwiki-20555 | Image of von Neumann-Bounded Set under Equicontinuous Family of Linear Transformations is Contained in von Neumann-Bounded Set | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be a topological vector space over $\GF$.
Let $\family {T_\alpha}_{\alpha \in I}$ be an equicontinuous family of linear transformations.
Let $E \subseteq X$ be von Neumann-bounded.
Then there exists a von Neumann-bounded set $F \subseteq Y$ such that:
:$T_\alpha \sqbrk ... | Let:
:$\ds F = \bigcup_{\alpha \mathop \in I} T_\alpha \sqbrk E$
Then $T_\alpha \sqbrk E \subseteq F$ for each $\alpha \in I$.
It is enough to show that $F$ is von Neumann-bounded.
Let $W$ be an open neighborhood of $\mathbf 0_Y$.
Since $\family {T_\alpha}_{\alpha \in I}$ is equicontinuous, there exists an open neighbo... | Let $\GF \in \set {\R, \C}$.
Let $X$ and $Y$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$.
Let $\family {T_\alpha}_{\alpha \in I}$ be an [[Definition:Equicontinuous Family of Linear Transformations between Topological Vector Spaces|equicontinuous family]] of [[Definition:Linear T... | Let:
:$\ds F = \bigcup_{\alpha \mathop \in I} T_\alpha \sqbrk E$
Then $T_\alpha \sqbrk E \subseteq F$ for each $\alpha \in I$.
It is enough to show that $F$ is [[Definition:von Neumann-Bounded Subset of Topological Vector Space|von Neumann-bounded]].
Let $W$ be an [[Definition:Open Neighborhood|open neighborhood]] ... | Image of von Neumann-Bounded Set under Equicontinuous Family of Linear Transformations is Contained in von Neumann-Bounded Set | https://proofwiki.org/wiki/Image_of_von_Neumann-Bounded_Set_under_Equicontinuous_Family_of_Linear_Transformations_is_Contained_in_von_Neumann-Bounded_Set | https://proofwiki.org/wiki/Image_of_von_Neumann-Bounded_Set_under_Equicontinuous_Family_of_Linear_Transformations_is_Contained_in_von_Neumann-Bounded_Set | [
"Equicontinuous Families of Linear Transformations between Topological Vector Spaces",
"von Neumann-Bounded Subsets of Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Equicontinuous Family of Linear Transformations between Topological Vector Spaces",
"Definition:Linear Transformation",
"Definition:von Neumann-Bounded Subset of Topological Vector Space",
"Definition:von Neumann-Bounded Subset of Topological Vector Space"
... | [
"Definition:von Neumann-Bounded Subset of Topological Vector Space",
"Definition:Open Neighborhood",
"Definition:Equicontinuous Family of Linear Transformations between Topological Vector Spaces",
"Definition:Open Neighborhood",
"Definition:von Neumann-Bounded Subset of Topological Vector Space",
"Image o... |
proofwiki-20556 | Integration by Partial Fractions | Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a rational function over $\R$ such that the degree of the polynomial $P$ is strictly smaller than the degree of the polynomial $Q$.
Consider the primitive:
:$\ds \int \map R x \rd x$
Let $\map R x$ be expressible by the partial fractions expansion:
:$\map R x = \ds \sum_... | {{begin-eqn}}
{{eqn | l = \int \map R x \rd x
| r = \int \paren {\sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x} } \rd x
| c = Definition of $\map R x$: {{hypothesis}}
}}
{{eqn | r = \sum_{k \mathop = 0}^n \int \dfrac {\map {p_k} x} {\map {q_k} x} \rd x
| c = Linear Combination of Integral... | Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a [[Definition:Real Rational Function|rational function over $\R$]] such that the [[Definition:Degree of Polynomial|degree]] of the [[Definition:Real Polynomial Function|polynomial $P$]] is strictly smaller than the [[Definition:Degree of Polynomial|degree]] of the [[Def... | {{begin-eqn}}
{{eqn | l = \int \map R x \rd x
| r = \int \paren {\sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x} } \rd x
| c = Definition of $\map R x$: {{hypothesis}}
}}
{{eqn | r = \sum_{k \mathop = 0}^n \int \dfrac {\map {p_k} x} {\map {q_k} x} \rd x
| c = [[Linear Combination of Integr... | Integration by Partial Fractions | https://proofwiki.org/wiki/Integration_by_Partial_Fractions | https://proofwiki.org/wiki/Integration_by_Partial_Fractions | [
"Integration by Partial Fractions",
"Integral Calculus",
"Partial Fractions Expansions",
"Named Theorems",
"Proof Techniques"
] | [
"Definition:Rational Function/Real",
"Definition:Degree of Polynomial",
"Definition:Polynomial Function/Real",
"Definition:Degree of Polynomial",
"Definition:Polynomial Function/Real",
"Definition:Primitive (Calculus)",
"Definition:Partial Fractions Expansion",
"Definition:Polynomial Function",
"Def... | [
"Linear Combination of Integrals"
] |
proofwiki-20557 | Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \... | === $(1)$ The Topology on $X$ is the Initial Topology ===
Let $f \sqbrk X$ denote the image of $f$.
Let $\tau_{f \sqbrk X}$ be the subspace topology on $f \sqbrk X$.
By definition of embedding:
:$f \restriction_{X \times f \sqbrk X}$ is a homeomorphism between $X$ and $f \sqbrk X$
From Subspace of Product Space has Ini... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | === $(1)$ The Topology on $X$ is the Initial Topology ===
Let $f \sqbrk X$ denote the [[Definition:Image of Mapping|image]] of $f$.
Let $\tau_{f \sqbrk X}$ be the [[Definition:Subspace Topology|subspace topology]] on $f \sqbrk X$.
By definition of [[Definition:Embedding (Topology)|embedding]]:
:$f \restriction_{X \t... | Characterization for Topological Evaluation Mapping to be Embedding/Necessary Condition | https://proofwiki.org/wiki/Characterization_for_Topological_Evaluation_Mapping_to_be_Embedding/Necessary_Condition | https://proofwiki.org/wiki/Characterization_for_Topological_Evaluation_Mapping_to_be_Embedding/Necessary_Condition | [
"Characterization for Topological Evaluation Mapping to be Embedding"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)",
"Definition:Evaluation Mapping (Topology)",
"Definition:Em... | [
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Topological Subspace",
"Definition:Embedding (Topology)",
"Definition:Homeomorphism",
"Subspace of Product Space has Initial Topology with respect to Restricted Projections",
"Definition:Initial Topology",
"Definition:Mapping",
"Definition:T... |
proofwiki-20558 | Characterization for Topological Evaluation Mapping to be Embedding/Sufficient Condition | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \... | From Evaluation Mapping is Injective iff Mappings Separate Points:
:$f$ is an injection
From Injection to Image is Bijection:
:$f \restriction_{X \times f \sqbrk X} \mathop : X \to f \sqbrk X$ is a bijection
From Topological Evaluation Mapping is Continuous:
:$f$ is continuous
From Continuity of Composite of Inclusion ... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | From [[Evaluation Mapping is Injective iff Mappings Separate Points]]:
:$f$ is an [[Definition:Injection|injection]]
From [[Injection to Image is Bijection]]:
:$f \restriction_{X \times f \sqbrk X} \mathop : X \to f \sqbrk X$ is a [[Definition:Bijection|bijection]]
From [[Topological Evaluation Mapping is Continuous... | Characterization for Topological Evaluation Mapping to be Embedding/Sufficient Condition | https://proofwiki.org/wiki/Characterization_for_Topological_Evaluation_Mapping_to_be_Embedding/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_for_Topological_Evaluation_Mapping_to_be_Embedding/Sufficient_Condition | [
"Characterization for Topological Evaluation Mapping to be Embedding"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)",
"Definition:Evaluation Mapping (Topology)",
"Definition:To... | [
"Evaluation Mapping is Injective iff Mappings Separate Points",
"Definition:Injection",
"Injection to Image is Bijection",
"Definition:Bijection",
"Topological Evaluation Mapping is Continuous",
"Definition:Continuous Mapping (Topology)",
"Continuity of Composite with Inclusion/Inclusion on Mapping",
... |
proofwiki-20559 | There are 83 Right-Truncatable Primes in Base 10 | In base $10$, there are $83$ right-truncatable primes:
:$2$, $3$, $5$, $7$,
:$23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$,
:$233$, $239$, $293$, $311$, $313$, $317$, $373$, $379$, $593$, $599$, $719$, $733$, $739$, $797$,
:$2333$, $2339$, $2393$, $2399$, $2939$, $3119$, $3137$, $3733$, $3739$, $3793$, $3797$, $... | Of the $1$-digit numbers, only $2$, $3$, $5$, $7$ are primes.
Of the $2$-digit numbers starting with $2$, only $23$ and $29$ are primes.
Of the $2$-digit numbers starting with $3$, only $31$ and $37$ are primes.
Of the $2$-digit numbers starting with $5$, only $53$ and $59$ are primes.
Of the $2$-digit numbers starting... | In [[Definition:Decimal Notation|base $10$]], there are $83$ [[Definition:Right-Truncatable Prime|right-truncatable primes]]:
:$2$, $3$, $5$, $7$,
:$23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$,
:$233$, $239$, $293$, $311$, $313$, $317$, $373$, $379$, $593$, $599$, $719$, $733$, $739$, $797$,
:$2333$, $2339$, $... | Of the $1$-[[Definition:Digit|digit]] [[Definition:Natural Number|numbers]], only $2$, $3$, $5$, $7$ are [[Definition:Prime Number|primes]].
Of the $2$-[[Definition:Digit|digit]] [[Definition:Natural Number|numbers]] starting with $2$, only $23$ and $29$ are [[Definition:Prime Number|primes]].
Of the $2$-[[Definitio... | There are 83 Right-Truncatable Primes in Base 10 | https://proofwiki.org/wiki/There_are_83_Right-Truncatable_Primes_in_Base_10 | https://proofwiki.org/wiki/There_are_83_Right-Truncatable_Primes_in_Base_10 | [
"Right-Truncatable Primes"
] | [
"Definition:Decimal Notation",
"Definition:Right-Truncatable Prime"
] | [
"Definition:Digit",
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Digit",
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Digit",
"Definition:Natural Numbers",
"Definition:Prime Number",
"Definition:Digit",
"Definition:Natural Numbers",
"Definition:P... |
proofwiki-20560 | Stone's Representation Theorem for Boolean Algebras | Let $B$ be a Boolean algebra.
Let $S$ be the Stone space of $B$.
Then:
:The set of clopen sets in $S$ is a Boolean algebra under union, intersection, and complementation in $S$.
:That Boolean algebra is isomorphic to $B$. | {{Proofread}}
{{Explain| Explain how the Ultrafilter Lemma/Boolean prime ideal theorem or Axiom of choice/Zorn's Lemma is explicitly used in the proof.}}
First, the statement of the proof has to be shown to be equivalent to the form:
Let $B$ be a Boolean algebra.
This proof will assume the Ultrafilter Lemma holds.
Then... | Let $B$ be a [[Definition:Boolean Algebra|Boolean algebra]].
Let $S$ be the [[Definition:Stone Space|Stone space]] of $B$.
Then:
:The [[Definition:Set|set]] of [[Definition:Clopen Set|clopen sets]] in $S$ is a [[Definition:Boolean Algebra|Boolean algebra]] under [[Definition:Set Union|union]], [[Definition:Set Inters... | {{Proofread}}
{{Explain| Explain how the Ultrafilter Lemma/Boolean prime ideal theorem or Axiom of choice/Zorn's Lemma is explicitly used in the proof.}}
First, the statement of the proof has to be shown to be equivalent to the form:
Let $B$ be a [[Definition:Boolean Algebra|Boolean algebra]].
This proof will assume... | Stone's Representation Theorem for Boolean Algebras | https://proofwiki.org/wiki/Stone's_Representation_Theorem_for_Boolean_Algebras | https://proofwiki.org/wiki/Stone's_Representation_Theorem_for_Boolean_Algebras | [
"Boolean Algebras",
"Stone Spaces"
] | [
"Definition:Boolean Algebra",
"Definition:Stone Space",
"Definition:Set",
"Definition:Clopen Set",
"Definition:Boolean Algebra",
"Definition:Set Union",
"Definition:Set Intersection",
"Definition:Set Complement",
"Definition:Boolean Algebra",
"Definition:Isomorphism (Abstract Algebra)"
] | [
"Definition:Boolean Algebra",
"Ultrafilter Lemma"
] |
proofwiki-20561 | There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers | Let $n \in \N$ be a natural number.
Let $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ be the $n$th, $n + 1$th, $n + 2$th and $n + 3$th triangular numbers respectively.
Then it is not the case that all of $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are sphenic numbers. | Let $\map \Omega n$ denote the number of prime factors of $n$ counted with multiplicity.
{{AimForCont}} there exists an $n$ such that $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are all sphenic numbers.
Thus from Closed Form for Triangular Numbers:
{{begin-eqn}}
{{eqn | l = T_n
| r = \dfrac {n \paren {n + 1}... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ be the $n$th, $n + 1$th, $n + 2$th and $n + 3$th [[Definition:Triangular Number|triangular numbers]] respectively.
Then it is not the case that all of $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$... | Let $\map \Omega n$ denote the number of [[Definition:Prime Factor|prime factors]] of $n$ counted with [[Definition:Multiplicity of Prime Factor|multiplicity]].
{{AimForCont}} there exists an $n$ such that $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are all [[Definition:Sphenic Number|sphenic numbers]].
Thus fr... | There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers/Proof 1 | https://proofwiki.org/wiki/There_exist_no_4_Consecutive_Triangular_Numbers_which_are_all_Sphenic_Numbers | https://proofwiki.org/wiki/There_exist_no_4_Consecutive_Triangular_Numbers_which_are_all_Sphenic_Numbers/Proof_1 | [
"Triangular Numbers",
"Sphenic Numbers",
"There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Triangular Number",
"Definition:Sphenic Number"
] | [
"Definition:Prime Factor",
"Definition:Prime Decomposition/Multiplicity",
"Definition:Sphenic Number",
"Closed Form for Triangular Numbers",
"Definition:Sphenic Number",
"Definition:Prime Number",
"Closed Form for Triangular Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Alge... |
proofwiki-20562 | There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers | Let $n \in \N$ be a natural number.
Let $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ be the $n$th, $n + 1$th, $n + 2$th and $n + 3$th triangular numbers respectively.
Then it is not the case that all of $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are sphenic numbers. | {{AimForCont}} there exists an $n$ such that $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are all sphenic numbers.
Observe from Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic that there are no such $n$ for $n < 12$.
Thus from Closed Form for Triangular Numbers:
{{begin-eqn}}
{{eqn | l = T_n... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ be the $n$th, $n + 1$th, $n + 2$th and $n + 3$th [[Definition:Triangular Number|triangular numbers]] respectively.
Then it is not the case that all of $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$... | {{AimForCont}} there exists an $n$ such that $T_n$, $T_{n + 1}$, $T_{n + 2}$, and $T_{n + 3}$ are all [[Definition:Sphenic Number|sphenic numbers]].
Observe from [[Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic]] that there are no such $n$ for $n < 12$.
Thus from [[Closed Form for Triangular ... | There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers/Proof 2 | https://proofwiki.org/wiki/There_exist_no_4_Consecutive_Triangular_Numbers_which_are_all_Sphenic_Numbers | https://proofwiki.org/wiki/There_exist_no_4_Consecutive_Triangular_Numbers_which_are_all_Sphenic_Numbers/Proof_2 | [
"Triangular Numbers",
"Sphenic Numbers",
"There exist no 4 Consecutive Triangular Numbers which are all Sphenic Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Triangular Number",
"Definition:Sphenic Number"
] | [
"Definition:Sphenic Number",
"Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic",
"Closed Form for Triangular Numbers",
"Definition:Sphenic Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Sphenic Number",
"Definition:Sphenic Nu... |
proofwiki-20563 | Number of form 4666...6669 is Divisible by 7 | Let $x$ be a natural number in the form:
:$\sqbrk {4 \underbrace {666 \cdots 6}_n 9}_{10}$
Then $x$ is divisible by $7$. | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {4 \underbrace {666 \cdots 6}_n 9}_{10}
| r = 4 \times 10^{n + 1} + 6 \times 10^n + \cdots + 6 \times 10^1 + 9
| c =
}}
{{eqn | r = 4 \times 10^{n + 1} + \dfrac {6 \times \paren {10^n - 1} } {\paren {10 - 1} } \times 10 + 9
| c = Sum of Geometric Sequence
}}
... | Let $x$ be a [[Definition:Natural Number|natural number]] in the form:
:$\sqbrk {4 \underbrace {666 \cdots 6}_n 9}_{10}$
Then $x$ is [[Definition:Divisor of Integer|divisible]] by $7$. | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {4 \underbrace {666 \cdots 6}_n 9}_{10}
| r = 4 \times 10^{n + 1} + 6 \times 10^n + \cdots + 6 \times 10^1 + 9
| c =
}}
{{eqn | r = 4 \times 10^{n + 1} + \dfrac {6 \times \paren {10^n - 1} } {\paren {10 - 1} } \times 10 + 9
| c = [[Sum of Geometric Sequence]... | Number of form 4666...6669 is Divisible by 7 | https://proofwiki.org/wiki/Number_of_form_4666...6669_is_Divisible_by_7 | https://proofwiki.org/wiki/Number_of_form_4666...6669_is_Divisible_by_7 | [
"Divisibility by 7"
] | [
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Sum of Geometric Sequence",
"Definition:Common Denominator",
"Definition:Addition",
"Definition:Digit",
"Definition:Multiple/Integer",
"Divisibility by 9/Corollary",
"Definition:Integer",
"Congruence of Powers",
"Congruence of Product",
"Congruence of Quotient",
"Category:Divisibility by 7"
] |
proofwiki-20564 | Number of form 28000...0007 is Divisible by 7 | Let $x$ be a natural number in the form:
:$\sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}$
Then $x$ is divisible by $7$. | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}
| r = 2 \times 10^{n + 2} + 8 \times 10^{n + 1} + 0 \times 10^n + \cdots + 0 \times 10^1 + 7
| c =
}}
{{eqn | r = 28 \times 10^{n + 1} + 7
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = 28 \times 10^{n + 1} + 7
... | Let $x$ be a [[Definition:Natural Number|natural number]] in the form:
:$\sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}$
Then $x$ is [[Definition:Divisor of Integer|divisible]] by $7$. | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}
| r = 2 \times 10^{n + 2} + 8 \times 10^{n + 1} + 0 \times 10^n + \cdots + 0 \times 10^1 + 7
| c =
}}
{{eqn | r = 28 \times 10^{n + 1} + 7
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = 28 \times 10^{n + 1} ... | Number of form 28000...0007 is Divisible by 7/Proof 1 | https://proofwiki.org/wiki/Number_of_form_28000...0007_is_Divisible_by_7 | https://proofwiki.org/wiki/Number_of_form_28000...0007_is_Divisible_by_7/Proof_1 | [
"Number of form 28000...0007 is Divisible by 7",
"Divisibility by 7"
] | [
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Fermat's Little Theorem",
"Congruence of Powers",
"Congruence of Product"
] |
proofwiki-20565 | Number of form 28000...0007 is Divisible by 7 | Let $x$ be a natural number in the form:
:$\sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}$
Then $x$ is divisible by $7$. | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}
| r = 7 \times \sqbrk {4 \underbrace {000 \cdots 0}_n 1}_{10}
| c =
}}
{{end-eqn}}
and the result follows by definition of divisibility.
{{qed}} | Let $x$ be a [[Definition:Natural Number|natural number]] in the form:
:$\sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}$
Then $x$ is [[Definition:Divisor of Integer|divisible]] by $7$. | We have:
{{begin-eqn}}
{{eqn | l = \sqbrk {28 \underbrace {000 \cdots 0}_n 7}_{10}
| r = 7 \times \sqbrk {4 \underbrace {000 \cdots 0}_n 1}_{10}
| c =
}}
{{end-eqn}}
and the result follows by definition of [[Definition:Divisor of Integer|divisibility]].
{{qed}} | Number of form 28000...0007 is Divisible by 7/Proof 2 | https://proofwiki.org/wiki/Number_of_form_28000...0007_is_Divisible_by_7 | https://proofwiki.org/wiki/Number_of_form_28000...0007_is_Divisible_by_7/Proof_2 | [
"Number of form 28000...0007 is Divisible by 7",
"Divisibility by 7"
] | [
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-20566 | Derivative of Hyperbolic Tangent/Corollary | :$\map {\dfrac \d {\d x} } {\tanh x} = 1 - \tanh^2 x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
| r = \sech^2 x
| c = Derivative of Hyperbolic Tangent
}}
{{eqn | r = 1 - \tanh^2 x
| c = Sum of Squares of Hyperbolic Secant and Tangent
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\tanh x} = 1 - \tanh^2 x$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\tanh x}
| r = \sech^2 x
| c = [[Derivative of Hyperbolic Tangent]]
}}
{{eqn | r = 1 - \tanh^2 x
| c = [[Sum of Squares of Hyperbolic Secant and Tangent]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Tangent/Corollary | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent/Corollary | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Tangent/Corollary | [
"Hyperbolic Tangent Function"
] | [] | [
"Derivative of Hyperbolic Tangent",
"Sum of Squares of Hyperbolic Secant and Tangent"
] |
proofwiki-20567 | Derivative of Hyperbolic Cotangent/Corollary | :$\map {\dfrac \d {\d x} } {\coth x} = 1 - \coth^2 x$ | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \coth x
| r = -\csch^2 x
| c = Derivative of Hyperbolic Cotangent
}}
{{eqn | r = 1 - \coth^2 x
| c = Difference of Squares of Hyperbolic Cotangent and Cosecant
}}
{{end-eqn}}
{{qed}} | :$\map {\dfrac \d {\d x} } {\coth x} = 1 - \coth^2 x$ | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \coth x
| r = -\csch^2 x
| c = [[Derivative of Hyperbolic Cotangent]]
}}
{{eqn | r = 1 - \coth^2 x
| c = [[Difference of Squares of Hyperbolic Cotangent and Cosecant]]
}}
{{end-eqn}}
{{qed}} | Derivative of Hyperbolic Cotangent/Corollary | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent/Corollary | https://proofwiki.org/wiki/Derivative_of_Hyperbolic_Cotangent/Corollary | [
"Derivative of Hyperbolic Cotangent"
] | [] | [
"Derivative of Hyperbolic Cotangent",
"Difference of Squares of Hyperbolic Cotangent and Cosecant"
] |
proofwiki-20568 | Interior of Proper Subspace of Normed Vector Space is Empty | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $U$ be a proper vector subspace of $X$.
Let $U^\circ$ be the interior of $U$.
Then $U^\circ = \O$. | {{AimForCont}} $U^\circ \ne \O$.
Take $x \in U^\circ$.
Then there exists $\epsilon > 0$ such that $\map {B_\epsilon} x \subseteq U$, where $\map {B_\epsilon} x$ is the open ball centered at $x$ with radius $\epsilon$.
Since $U$ is a vector subspace, we have:
:$\map {B_\epsilon} 0 = \map {B_\epsilon} x - x \subseteq U$
... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $U$ be a [[Definition:Proper Subset|proper]] [[Definition:Vector Subspace|vector subspace]] of $X$.
Let $U^\circ$ be the [[Definition:Interior (Topology)|interior]] of $... | {{AimForCont}} $U^\circ \ne \O$.
Take $x \in U^\circ$.
Then there exists $\epsilon > 0$ such that $\map {B_\epsilon} x \subseteq U$, where $\map {B_\epsilon} x$ is the [[Definition:Open Ball|open ball]] [[Definition:Center of Open Ball|centered]] at $x$ with [[Definition:Radius of Open Ball|radius]] $\epsilon$.
Sinc... | Interior of Proper Subspace of Normed Vector Space is Empty | https://proofwiki.org/wiki/Interior_of_Proper_Subspace_of_Normed_Vector_Space_is_Empty | https://proofwiki.org/wiki/Interior_of_Proper_Subspace_of_Normed_Vector_Space_is_Empty | [
"Normed Vector Spaces",
"Set Interiors"
] | [
"Definition:Normed Vector Space",
"Definition:Proper Subset",
"Definition:Vector Subspace",
"Definition:Interior (Topology)"
] | [
"Definition:Open Ball",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Definition:Vector Subspace",
"Definition:Vector Subspace",
"Definition:Contradiction",
"Proof by Contradition",
"Category:Normed Vector Spaces",
"Category:Set Interiors"
] |
proofwiki-20569 | Weak Topology on Infinite Dimensional Normed Vector Space is not Metrizable | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an infinite dimensional normed vector space over $\GF$.
Let $w$ be the weak topology on $X$.
Then $w$ is not metrizable. | Let $X^\ast$ be the normed dual space of $X$.
From Metric Space is First-Countable, it suffices to show that $\struct {X, w}$ is not first-countable.
{{AimForCont}} that $\struct {X, w}$ is first-countable.
Let $\sequence {U_n}_{n \mathop \in \N}$ be a local basis for $\mathbf 0_X$.
From Open Sets in Weak Topology of ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be an [[Definition:Infinite Dimensional Vector Space|infinite dimensional]] [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $w$ be the [[Definition:Weak Topology on Topological Vector Space|weak topology]] on $X$.
Then $w$ ... | Let $X^\ast$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
From [[Metric Space is First-Countable]], it suffices to show that $\struct {X, w}$ is not [[Definition:First-Countable Space|first-countable]].
{{AimForCont}} that $\struct {X, w}$ is [[Definition:First-Countable Space|first-countable]].... | Weak Topology on Infinite Dimensional Normed Vector Space is not Metrizable | https://proofwiki.org/wiki/Weak_Topology_on_Infinite_Dimensional_Normed_Vector_Space_is_not_Metrizable | https://proofwiki.org/wiki/Weak_Topology_on_Infinite_Dimensional_Normed_Vector_Space_is_not_Metrizable | [
"Weak Topologies on Topological Vector Spaces",
"Metrizable Spaces"
] | [
"Definition:Infinite Dimensional Vector Space",
"Definition:Normed Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Metrizable Space"
] | [
"Definition:Normed Dual Space",
"Metric Space is First-Countable",
"Definition:First-Countable Space",
"Definition:First-Countable Space",
"Definition:Local Basis",
"Open Sets in Weak Topology of Topological Vector Space",
"Definition:Finite Set",
"Definition:Local Basis",
"Condition for Linear Depe... |
proofwiki-20570 | Open Sets in Weak Topology of Topological Vector Space | Let $\GF \in \set {\R, \C}$.
Let $X$ be a topological vector space over $\GF$ with weak topology $w$.
Let $U \subseteq X$.
Then $U$ is open in $\struct {X, w}$ {{iff}} for each $x \in U$ there exists $f_1, f_2, \ldots, f_n \in X^\ast$ and $\epsilon > 0$ such that:
:$\set {y \in X : \cmod {\map {f_i} {y - x} } < \ep... | Let $X^\ast$ be the topological dual space of $X$.
By the definition of the weak topology, $w$ is the initial topology on $X$ with respect to $X^\ast$.
For each $f \in X^\ast$, define $p_f : X \to \hointr 0 \infty$ by:
:$\map {p_f} x = \cmod {\map f x}$
for each $x \in X$, and set:
:$\PP = \set {p_f : f \in X^\ast}$... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Topological Vector Space|topological vector space]] over $\GF$ with [[Definition:Weak Topology on Topological Vector Space|weak topology]] $w$.
Let $U \subseteq X$.
Then $U$ is [[Definition:Open Set (Topology)|open]] in $\struct {X, w}$ {{iff}} for each $x \... | Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
By the definition of the [[Definition:Weak Topology on Topological Vector Space|weak topology]], $w$ is the [[Definition:Initial Topology|initial topology]] on $X$ with respect to $X^\ast$.
For each $f \in X^\ast$, define $p_f :... | Open Sets in Weak Topology of Topological Vector Space | https://proofwiki.org/wiki/Open_Sets_in_Weak_Topology_of_Topological_Vector_Space | https://proofwiki.org/wiki/Open_Sets_in_Weak_Topology_of_Topological_Vector_Space | [
"Weak Topologies on Topological Vector Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Open Set/Topology"
] | [
"Definition:Topological Dual Space",
"Definition:Weak Topology on Topological Vector Space",
"Definition:Initial Topology",
"Initial Topology on Vector Space Generated by Linear Functionals is Locally Convex",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Locally Convex Space",
"Open ... |
proofwiki-20571 | 41 is Smallest Number whose Period of Reciprocal is 5 | $41$ is the first positive integer the decimal expansion of whose reciprocal has a period of $5$:
:$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$ | From Reciprocal of $41$:
{{:Reciprocal of 41}}
Counting the digits, it is seen that this has a period of recurrence of $5$.
It remains to be shown that $41$ is the smallest positive integer which has this property.
{{ProofWanted}}
Category:41
Category:Examples of Reciprocals
thl87a0mi1vnburqpl26wstyuqqgoy2 | $41$ is the first [[Definition:Positive Integer|positive integer]] the [[Definition:Decimal Expansion|decimal expansion]] of whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $5$:
:$\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$ | From [[Reciprocal of 41|Reciprocal of $41$]]:
{{:Reciprocal of 41}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $5$.
It remains to be shown that $41$ is the smallest [[Definition:Positive Integer|positive integer]] which has this property.
{{ProofWante... | 41 is Smallest Number whose Period of Reciprocal is 5 | https://proofwiki.org/wiki/41_is_Smallest_Number_whose_Period_of_Reciprocal_is_5 | https://proofwiki.org/wiki/41_is_Smallest_Number_whose_Period_of_Reciprocal_is_5 | [
"41",
"Examples of Reciprocals"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 41",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Positive/Integer",
"Category:41",
"Category:Examples of Reciprocals"
] |
proofwiki-20572 | Evaluation Mapping is Injective iff Mappings Separate Points | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \... | We have:
:$f$ is an injection
{{begin-eqn}}
{{eqn | lll = \iff
| ll = \forall x, y \in X : x \ne y :
| l = \map f x
| o = \ne
| r = \map f y
| c = {{Defof|Injection}}
}}
{{eqn | lll = \iff
| ll = \forall x, y \in X : x \ne y :
| l = \family{ \map {f_i} x }_{i \in I}
| o =... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | We have:
:$f$ is an injection
{{begin-eqn}}
{{eqn | lll = \iff
| ll = \forall x, y \in X : x \ne y :
| l = \map f x
| o = \ne
| r = \map f y
| c = {{Defof|Injection}}
}}
{{eqn | lll = \iff
| ll = \forall x, y \in X : x \ne y :
| l = \family{ \map {f_i} x }_{i \in I}
| o =... | Evaluation Mapping is Injective iff Mappings Separate Points | https://proofwiki.org/wiki/Evaluation_Mapping_is_Injective_iff_Mappings_Separate_Points | https://proofwiki.org/wiki/Evaluation_Mapping_is_Injective_iff_Mappings_Separate_Points | [
"Evaluation Mappings (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)",
"Definition:Evaluation Mapping (Topology)",
"Definition:In... | [
"Category:Evaluation Mappings (Topological Spaces)"
] |
proofwiki-20573 | Injection is Open Mapping iff Image of Sub-Basis Set is Open | Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be topological spaces.
Let $\SS \subseteq \powerset {X_1}$ be a sub-basis of $\tau_1$.
Let $f: X_1 \to X_2$ be an injection.
Then:
:$f$ is an open mapping
{{iff}}:
:$\forall U \in \SS: f \sqbrk U \in \tau_2$ | === Necessary Condition ===
Let $f$ be an open mapping.
By definition of open mapping:
:$\forall U \in \tau_1 : f \sqbrk U \in \tau_2$
By definition of sub-basis:
:$\SS \subseteq \tau_1$
Hence:
:$\forall U \in \SS: f \sqbrk U \in \tau_2$
{{qed|lemma}} | Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $\SS \subseteq \powerset {X_1}$ be a [[Definition:Analytic Sub-Basis|sub-basis]] of $\tau_1$.
Let $f: X_1 \to X_2$ be an [[Definition:Injection|injection]].
Then:
:$f$ is an [[Definition:Open Mappi... | === Necessary Condition ===
Let $f$ be an [[Definition:Open Mapping|open mapping]].
By definition of [[Definition:Open Mapping|open mapping]]:
:$\forall U \in \tau_1 : f \sqbrk U \in \tau_2$
By definition of [[Definition:Analytic Sub-Basis|sub-basis]]:
:$\SS \subseteq \tau_1$
Hence:
:$\forall U \in \SS: f \sqbrk U ... | Injection is Open Mapping iff Image of Sub-Basis Set is Open | https://proofwiki.org/wiki/Injection_is_Open_Mapping_iff_Image_of_Sub-Basis_Set_is_Open | https://proofwiki.org/wiki/Injection_is_Open_Mapping_iff_Image_of_Sub-Basis_Set_is_Open | [
"Open Mappings"
] | [
"Definition:Topological Space",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Injection",
"Definition:Open Mapping"
] | [
"Definition:Open Mapping",
"Definition:Open Mapping",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Open Mapping"
] |
proofwiki-20574 | Reciprocal of 11 | :$\dfrac 1 {11} = 0 \cdotp \dot 0 \dot 9$ | Performing the calculation using long division:
<pre>
0.0909...
------------
11)1.000000000
99
----
100
99
---
...
</pre>
{{qed}}
Category:11
Category:Examples of Reciprocals
n9194mcn5vl7elqfpop63niwmfsm61o | :$\dfrac 1 {11} = 0 \cdotp \dot 0 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.0909...
------------
11)1.000000000
99
----
100
99
---
...
</pre>
{{qed}}
[[Category:11]]
[[Category:Examples of Reciprocals]]
n9194mcn5vl7elqfpop63niwmfsm61o | Reciprocal of 11 | https://proofwiki.org/wiki/Reciprocal_of_11 | https://proofwiki.org/wiki/Reciprocal_of_11 | [
"11",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:11",
"Category:Examples of Reciprocals"
] |
proofwiki-20575 | Reciprocal of 3 | :$\dfrac 1 3 = 0 \cdotp \dot 3$ | Performing the calculation using long division:
<pre>
0.33...
--------
3)1.00...
9
---
10
9
--
..
</pre>
{{qed}}
Category:3
Category:Examples of Reciprocals
ronv291wvqp44e48ldhjhsw2jd2zygn | :$\dfrac 1 3 = 0 \cdotp \dot 3$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.33...
--------
3)1.00...
9
---
10
9
--
..
</pre>
{{qed}}
[[Category:3]]
[[Category:Examples of Reciprocals]]
ronv291wvqp44e48ldhjhsw2jd2zygn | Reciprocal of 3 | https://proofwiki.org/wiki/Reciprocal_of_3 | https://proofwiki.org/wiki/Reciprocal_of_3 | [
"3",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:3",
"Category:Examples of Reciprocals"
] |
proofwiki-20576 | Period of Reciprocal of 43 is Odd | The decimal expansion of the reciprocal of $43$ has an odd period, that is, $21$:
:$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$ | From Reciprocal of $43$:
{{:Reciprocal of 43}}
Counting the digits, it is seen that this has a period of recurrence of $21$.
Hence the result.
{{qed}}
Category:43
Category:Examples of Reciprocals
snxig1txup7n228s8gghkmlpcy7pg1l | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $43$ has an [[Definition:Odd Integer|odd]] [[Definition:Period of Recurrence|period]], that is, $21$:
:$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$ | From [[Reciprocal of 43|Reciprocal of $43$]]:
{{:Reciprocal of 43}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $21$.
Hence the result.
{{qed}}
[[Category:43]]
[[Category:Examples of Reciprocals]]
snxig1txup7n228s8gghkmlpcy7pg1l | Period of Reciprocal of 43 is Odd | https://proofwiki.org/wiki/Period_of_Reciprocal_of_43_is_Odd | https://proofwiki.org/wiki/Period_of_Reciprocal_of_43_is_Odd | [
"43",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Odd Integer",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 43",
"Definition:Basis Expansion/Recurrence/Period",
"Category:43",
"Category:Examples of Reciprocals"
] |
proofwiki-20577 | Long Period Prime/Examples/29 | The prime number $29$ is a long period prime:
:$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ | From Reciprocal of $29$:
{{:Reciprocal of 29}}
Counting the digits, it is seen that this has a period of recurrence of $28$.
Hence the result.
{{qed}}
Category:29
Category:Examples of Long Period Primes
cktz7jtmqcqpqhxb3a1g80a6fup7hpb | The [[Definition:Prime Number|prime number]] $29$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ | From [[Reciprocal of 29|Reciprocal of $29$]]:
{{:Reciprocal of 29}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $28$.
Hence the result.
{{qed}}
[[Category:29]]
[[Category:Examples of Long Period Primes]]
cktz7jtmqcqpqhxb3a1g80a6fup7hpb | Long Period Prime/Examples/29 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/29 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/29 | [
"29",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 29",
"Definition:Basis Expansion/Recurrence/Period",
"Category:29",
"Category:Examples of Long Period Primes"
] |
proofwiki-20578 | Period of Reciprocal of 79 is One Sixth of Maximal | The decimal expansion of the reciprocal of $79$ has $\dfrac 1 6$ the maximum period, that is: $13$:
:$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$ | From Reciprocal of $79$:
{{:Reciprocal of 79}}
Counting the digits, it is seen that this has a period of recurrence of $13$.
From Maximum Period of Reciprocal of Prime, the maximum period of recurrence of $\dfrac 1 p$ is $p - 1$.
We have that:
:$13 = \dfrac {79 - 1} 6$
{{qed}}
Category:79
Category:Examples of Reciproca... | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $79$ has $\dfrac 1 6$ the maximum [[Definition:Period of Recurrence|period]], that is: $13$:
:$\dfrac 1 {79} = 0 \cdotp \dot 01265 \, 82278 \, 48 \dot 1$ | From [[Reciprocal of 79|Reciprocal of $79$]]:
{{:Reciprocal of 79}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $13$.
From [[Maximum Period of Reciprocal of Prime]], the maximum [[Definition:Period of Recurrence|period of recurrence]] of $\dfrac 1 p$ is ... | Period of Reciprocal of 79 is One Sixth of Maximal | https://proofwiki.org/wiki/Period_of_Reciprocal_of_79_is_One_Sixth_of_Maximal | https://proofwiki.org/wiki/Period_of_Reciprocal_of_79_is_One_Sixth_of_Maximal | [
"79",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 79",
"Definition:Basis Expansion/Recurrence/Period",
"Maximum Period of Reciprocal of Prime",
"Definition:Basis Expansion/Recurrence/Period",
"Category:79",
"Category:Examples of Reciprocals"
] |
proofwiki-20579 | 101 is Smallest Number whose Period of Reciprocal is 4 | $101$ is the first positive integer the decimal expansion of whose reciprocal has a period of $4$:
:$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$ | {{tidy}}
Let the positive integer reciprocal be $\dfrac1 k$ for some $k \in \Z_{\ge 0}$.
For it to have a period of recurrence of $4$ in base $10$, it must be able to be expressed as $\dfrac a {10^4-1}$ for some $a \in \Z_{\ge 0}$.
{{begin-eqn}}
{{eqn | l = 10^4-1
| r = \paren { 10^2 - 1 } \paren { 10^2 + 1 }
... | $101$ is the first [[Definition:Positive Integer|positive integer]] the [[Definition:Decimal Expansion|decimal expansion]] of whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $4$:
:$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$ | {{tidy}}
Let the positive integer reciprocal be $\dfrac1 k$ for some $k \in \Z_{\ge 0}$.
For it to have a period of recurrence of $4$ in base $10$, it must be able to be expressed as $\dfrac a {10^4-1}$ for some $a \in \Z_{\ge 0}$.
{{begin-eqn}}
{{eqn | l = 10^4-1
| r = \paren { 10^2 - 1 } \paren { 10^2 + 1 }
... | 101 is Smallest Number whose Period of Reciprocal is 4 | https://proofwiki.org/wiki/101_is_Smallest_Number_whose_Period_of_Reciprocal_is_4 | https://proofwiki.org/wiki/101_is_Smallest_Number_whose_Period_of_Reciprocal_is_4 | [
"101",
"Examples of Reciprocals"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Difference of Two Squares",
"Category:101",
"Category:Examples of Reciprocals"
] |
proofwiki-20580 | Reciprocal of 83 has Prime Period | The decimal expansion of the reciprocal of $83$ has a prime period, that is $41$:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$ | From Reciprocal of $83$:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
Counting the digits, it is seen that this has a period of recurrence of $41$.
Indeed, $41$ is the $13$th prime number.
{{qed}}
Category:83
Category:Examples of Reciprocals
iqso3aznit6... | The [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] of $83$ has a [[Definition:Prime Number|prime]] [[Definition:Period of Recurrence|period]], that is $41$:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$ | From [[Reciprocal of 83|Reciprocal of $83$]]:
:$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $41$.
Indeed, $41$ is the $13$th [[Definition:Prime ... | Reciprocal of 83 has Prime Period | https://proofwiki.org/wiki/Reciprocal_of_83_has_Prime_Period | https://proofwiki.org/wiki/Reciprocal_of_83_has_Prime_Period | [
"83",
"Examples of Reciprocals"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Prime Number",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 83",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Prime Number",
"Category:83",
"Category:Examples of Reciprocals"
] |
proofwiki-20581 | Repunit 19 is Unique Period Prime with Period 19 | The repunit prime $R_{19}$ is a unique period prime whose reciprocal has a period of $19$:
:$\dfrac 1 {1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111} = 0 \cdotp \dot 00000 \, 00000 \, 00000 \, 000 \dot 9$ | The reciprocal of a repunit $R_n$ is of the form:
:$\dfrac 1 {R_n} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{n - 1} \dot 9$
{{TheoremWanted|Prove the above}}
Thus $\dfrac 1 {R_{19} }$ has a period of $19$.
From Period of Reciprocal of Prime, for prime numbers such that:
:$p \nmid 10$
we have that the period of such ... | The [[Definition:Repunit Prime|repunit prime]] $R_{19}$ is a [[Definition:Unique Period Prime|unique period prime]] whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $19$:
:$\dfrac 1 {1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111} = 0 \cdotp \dot 00000 \, 00000 \, 00000 \, 00... | The [[Definition:Reciprocal|reciprocal]] of a [[Definition:Repunit|repunit]] $R_n$ is of the form:
:$\dfrac 1 {R_n} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{n - 1} \dot 9$
{{TheoremWanted|Prove the above}}
Thus $\dfrac 1 {R_{19} }$ has a [[Definition:Period of Recurrence|period]] of $19$.
From [[Period of Recip... | Repunit 19 is Unique Period Prime with Period 19 | https://proofwiki.org/wiki/Repunit_19_is_Unique_Period_Prime_with_Period_19 | https://proofwiki.org/wiki/Repunit_19_is_Unique_Period_Prime_with_Period_19 | [
"1,111,111,111,111,111,111",
"Examples of Reciprocals",
"Examples of Unique Period Primes"
] | [
"Definition:Repunit Prime",
"Definition:Unique Period Prime",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Definition:Reciprocal",
"Definition:Repunit",
"Definition:Basis Expansion/Recurrence/Period",
"Period of Reciprocal of Prime",
"Definition:Prime Number",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Prime Number",
"Definition:Multiplicative Order of Integer",
"Definition:Integer",
... |
proofwiki-20582 | 73 is Smallest Number whose Period of Reciprocal is 8 | $73$ is the first positive integer the decimal expansion of whose reciprocal has a period of $8$:
:$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$ | From Reciprocal of $73$:
{{:Reciprocal of 73}}
Counting the digits, it is seen that this has a period of recurrence of $8$.
It remains to be shown that $73$ is the smallest positive integer which has this property.
{{ProofWanted}}
Category:73
Category:Examples of Reciprocals
grxidb2h35auwbwuvxfidb9pl7w2q0v | $73$ is the first [[Definition:Positive Integer|positive integer]] the [[Definition:Decimal Expansion|decimal expansion]] of whose [[Definition:Reciprocal|reciprocal]] has a [[Definition:Period of Recurrence|period]] of $8$:
:$\dfrac 1 {73} = 0 \cdotp \dot 01369 \, 86 \dot 3$ | From [[Reciprocal of 73|Reciprocal of $73$]]:
{{:Reciprocal of 73}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $8$.
It remains to be shown that $73$ is the smallest [[Definition:Positive Integer|positive integer]] which has this property.
{{ProofWanted}... | 73 is Smallest Number whose Period of Reciprocal is 8 | https://proofwiki.org/wiki/73_is_Smallest_Number_whose_Period_of_Reciprocal_is_8 | https://proofwiki.org/wiki/73_is_Smallest_Number_whose_Period_of_Reciprocal_is_8 | [
"73",
"Examples of Reciprocals"
] | [
"Definition:Positive/Integer",
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Basis Expansion/Recurrence/Period"
] | [
"Reciprocal of 73",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Positive/Integer",
"Category:73",
"Category:Examples of Reciprocals"
] |
proofwiki-20583 | Reciprocal of 9801 | :$\dfrac 1 {9801} = 0 \cdotp \dot 00010 \, 20304 \, 05060 \, 70809 \, 10111 \, 21314 \, ... \, 94959 \, 6979 \dot 9$ | {{ProofWanted}}
Category:9801
Category:Examples of Reciprocals
bpiqtpid8jh6p42qhqmr66jfo2afpjn | :$\dfrac 1 {9801} = 0 \cdotp \dot 00010 \, 20304 \, 05060 \, 70809 \, 10111 \, 21314 \, ... \, 94959 \, 6979 \dot 9$ | {{ProofWanted}}
[[Category:9801]]
[[Category:Examples of Reciprocals]]
bpiqtpid8jh6p42qhqmr66jfo2afpjn | Reciprocal of 9801 | https://proofwiki.org/wiki/Reciprocal_of_9801 | https://proofwiki.org/wiki/Reciprocal_of_9801 | [
"9801",
"Examples of Reciprocals"
] | [] | [
"Category:9801",
"Category:Examples of Reciprocals"
] |
proofwiki-20584 | Reciprocal of 909,091 | :$\dfrac 1 {909 \, 091} = 0 \cdotp \dot 00000 \, 10999 \, 998 \dot 9$ | Performing the calculation using long division:
<pre>
0.00000109999989000001...
--------------------------------------------
909091)1.00000000000000000000...
909091
------
9090900
8181819
-------
9090810
8181819
-------
... | :$\dfrac 1 {909 \, 091} = 0 \cdotp \dot 00000 \, 10999 \, 998 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.00000109999989000001...
--------------------------------------------
909091)1.00000000000000000000...
909091
------
9090900
8181819
-------
9090810
818... | Reciprocal of 909,091 | https://proofwiki.org/wiki/Reciprocal_of_909,091 | https://proofwiki.org/wiki/Reciprocal_of_909,091 | [
"909,091",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:909,091",
"Category:Examples of Reciprocals"
] |
proofwiki-20585 | Long Period Prime/Examples/47 | The prime number $47$ is a long period prime:
:$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$ | From Reciprocal of $47$:
{{:Reciprocal of 47}}
Counting the digits, it is seen that this has a period of recurrence of $46$.
Hence the result.
{{qed}}
Category:47
Category:Examples of Long Period Primes
jwxg7b83jx80qin8yzabynjlv90shp1 | The [[Definition:Prime Number|prime number]] $47$ is a [[Definition:Long Period Prime|long period prime]]:
:$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$ | From [[Reciprocal of 47|Reciprocal of $47$]]:
{{:Reciprocal of 47}}
Counting the digits, it is seen that this has a [[Definition:Period of Recurrence|period of recurrence]] of $46$.
Hence the result.
{{qed}}
[[Category:47]]
[[Category:Examples of Long Period Primes]]
jwxg7b83jx80qin8yzabynjlv90shp1 | Long Period Prime/Examples/47 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/47 | https://proofwiki.org/wiki/Long_Period_Prime/Examples/47 | [
"47",
"Examples of Long Period Primes"
] | [
"Definition:Prime Number",
"Definition:Long Period Prime"
] | [
"Reciprocal of 47",
"Definition:Basis Expansion/Recurrence/Period",
"Category:47",
"Category:Examples of Long Period Primes"
] |
proofwiki-20586 | Composite of Evaluation Mapping and Projection | Let $X$ be a topological space.
Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.
Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \... | By definition of projection:
{{begin-eqn}}
{{eqn | q = \forall x \in X, i \in I
| l = \map {\paren{pr_i \circ f} } x
| r = \map {pr_i} {\map f x}
| c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map {pr_i} {\family{\map {f_i} x} }
| c = {{Defof|Evaluation Mapping}}
}}
{{eqn | r = \map {f_i} x
... | Let $X$ be a [[Definition:Topological Space|topological space]].
Let $\family {Y_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] for some [[Definition:Indexing Set|indexing set]] $I$.
Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be ... | By definition of [[Definition:Projection (Mapping Theory)|projection]]:
{{begin-eqn}}
{{eqn | q = \forall x \in X, i \in I
| l = \map {\paren{pr_i \circ f} } x
| r = \map {pr_i} {\map f x}
| c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map {pr_i} {\family{\map {f_i} x} }
| c = {{Defof|Evaluat... | Composite of Evaluation Mapping and Projection | https://proofwiki.org/wiki/Composite_of_Evaluation_Mapping_and_Projection | https://proofwiki.org/wiki/Composite_of_Evaluation_Mapping_and_Projection | [
"Evaluation Mappings (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Continuous Mapping (Topology)",
"Definition:Product Space (Topology)",
"Definition:Evaluation Mapping (Topology)",
"Definition:Pr... | [
"Definition:Projection (Mapping Theory)",
"Equality of Mappings",
"Category:Evaluation Mappings (Topological Spaces)"
] |
proofwiki-20587 | Space of Bounded Linear Transformations forms Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $\map B {X, Y}$ be the space of bounded linear transformations from $X$ to $Y$.
Let $+_B$ and $\circ_B$ be pointwise addition and pointwise scalar multiplication on $Y... | Let $\map L {X, Y}$ be the space of linear transformations between $X$ and $Y$.
From Linear Mappings between Vector Spaces form Vector Space, $\map L {X, Y}$ is a vector space over $\GF$ with pointwise addition and pointwise scalar multiplication.
It therefore suffices to show that $\map B {X, Y}$ is a vector subspace ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $\map B {X, Y}$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations]] from $X$ to ... | Let $\map L {X, Y}$ be the [[Definition:Set of All Linear Transformations/Vector Space|space of linear transformations between $X$ and $Y$]].
From [[Linear Mappings between Vector Spaces form Vector Space]], $\map L {X, Y}$ is a [[Definition:Vector Space|vector space]] over $\GF$ with [[Definition:Pointwise Addition o... | Space of Bounded Linear Transformations forms Vector Space | https://proofwiki.org/wiki/Space_of_Bounded_Linear_Transformations_forms_Vector_Space | https://proofwiki.org/wiki/Space_of_Bounded_Linear_Transformations_forms_Vector_Space | [
"Space of Bounded Linear Transformations"
] | [
"Definition:Normed Vector Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Pointwise Addition of Mappings",
"Definition:Pointwise Scalar Multiplication of Mappings",
"Definition:Vector Space"
] | [
"Definition:Set of All Linear Transformations/Vector Space",
"Linear Mappings between Vector Spaces form Vector Space",
"Definition:Vector Space",
"Definition:Pointwise Addition of Mappings",
"Definition:Pointwise Scalar Multiplication of Mappings",
"Definition:Vector Subspace",
"One-Step Vector Subspac... |
proofwiki-20588 | Norm on Space of Bounded Linear Transformations is Norm | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $\map B {X, Y}$ be the space of bounded linear transformations between $X$ and $Y$.
Let $\norm {\, \cdot \,}_{\map B {X, Y} }$ be the norm on the space of bounded lin... | From Norm on Bounded Linear Transformation is Finite, $\norm {\, \cdot \,}_{\map B {X, Y} }$ is real-valued. | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]] over $\GF$.
Let $\map B {X, Y}$ be the [[Definition:Space of Bounded Linear Transformations|space of bounded linear transformations between $X$ an... | From [[Norm on Bounded Linear Transformation is Finite]], $\norm {\, \cdot \,}_{\map B {X, Y} }$ is [[Definition:Real Number|real-valued]]. | Norm on Space of Bounded Linear Transformations is Norm | https://proofwiki.org/wiki/Norm_on_Space_of_Bounded_Linear_Transformations_is_Norm | https://proofwiki.org/wiki/Norm_on_Space_of_Bounded_Linear_Transformations_is_Norm | [
"Space of Bounded Linear Transformations",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Space of Bounded Linear Transformations",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Norm/Vector Space"
] | [
"Norm on Bounded Linear Transformation is Finite",
"Definition:Real Number"
] |
proofwiki-20589 | Reciprocal of 47 | :$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$ | Performing the calculation using long division:
<pre>
0.021276595744680851063829787234042553191489361702...
------------------------------------------------------
47)1.000000000000000000000000000000000000000000000000...
94
----
60 270 400 140 160 150 440
47 235 ... | :$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.021276595744680851063829787234042553191489361702...
------------------------------------------------------
47)1.000000000000000000000000000000000000000000000000...
94
----
60 270 400 140 16... | Reciprocal of 47 | https://proofwiki.org/wiki/Reciprocal_of_47 | https://proofwiki.org/wiki/Reciprocal_of_47 | [
"47",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:47",
"Category:Examples of Reciprocals"
] |
proofwiki-20590 | Normed Dual Space is Banach Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.
Then $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ is a Banach space. | By definition, we have:
:$X^\ast = \map B {X, \GF}$
and:
:$\norm {\, \cdot \,}_{X^\ast} = \norm {\, \cdot \,}_{\map B {X, \GF} }$
From Real Number Line is Banach Space and Complex Plane is Banach Space, $\GF$ is a Banach space.
So from Space of Bounded Linear Transformations is Banach Space, $\struct {X, \norm {\, \cdo... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual]] of $\struct {X, \norm {\, \cdot \,}_X}$.
Then $\struct {X^\ast, \n... | By definition, we have:
:$X^\ast = \map B {X, \GF}$
and:
:$\norm {\, \cdot \,}_{X^\ast} = \norm {\, \cdot \,}_{\map B {X, \GF} }$
From [[Real Number Line is Banach Space]] and [[Complex Plane is Banach Space]], $\GF$ is a [[Definition:Banach Space|Banach space]].
So from [[Space of Bounded Linear Transformations is B... | Normed Dual Space is Banach Space | https://proofwiki.org/wiki/Normed_Dual_Space_is_Banach_Space | https://proofwiki.org/wiki/Normed_Dual_Space_is_Banach_Space | [
"Normed Dual Spaces",
"Banach Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Banach Space"
] | [
"Real Number Line is Banach Space",
"Complex Plane is Banach Space",
"Definition:Banach Space",
"Space of Bounded Linear Transformations is Banach Space",
"Definition:Banach Space",
"Category:Normed Dual Spaces",
"Category:Banach Spaces"
] |
proofwiki-20591 | Bound on Norm of Power of Element in Normed Algebra | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.
Let $x \in A$ and $n \in \N$.
Then:
:$\norm {x^n} \le \norm x^n$ | The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\norm {x^n} \le \norm x^n$ | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$.
Let $x \in A$ and $n \in \N$.
Then:
:$\norm {x^n} \le \norm x^n$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\norm {x^n} \le \norm x^n$ | Bound on Norm of Power of Element in Normed Algebra | https://proofwiki.org/wiki/Bound_on_Norm_of_Power_of_Element_in_Normed_Algebra | https://proofwiki.org/wiki/Bound_on_Norm_of_Power_of_Element_in_Normed_Algebra | [
"Normed Algebras"
] | [
"Definition:Normed Algebra"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-20592 | Product Rule for Sequence in Normed Algebra | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a normed algebra over $\GF$.
Let $\sequence {a_n}_{n \in \N}$ and $\sequence {b_n}_{n \in \N}$ be sequences in $A$ converging to $a$ and $b$ respectively.
Then:
:$a_n b_n \to a b$ | From Convergent Sequence in Normed Vector Space is Bounded, there exists $M > 0$ such that:
:$\norm {a_n} \le M$ for each $n \in \N$.
We have for $n \in \N$:
{{begin-eqn}}
{{eqn | l = \norm {a_n b_n - a b}
| r = \norm {a_n b_n - a_n b + a_n b - a b}
}}
{{eqn | r = \norm {a_n \paren {b_n - b} + b \paren {a_n - a} }
... | Let $\GF \in \set {\R, \C}$.
Let $\struct {A, \norm {\, \cdot \,} }$ be a [[Definition:Normed Algebra|normed algebra]] over $\GF$.
Let $\sequence {a_n}_{n \in \N}$ and $\sequence {b_n}_{n \in \N}$ be [[Definition:Sequence|sequences]] in $A$ [[Definition:Convergent Sequence in Normed Vector Space|converging]] to $a$... | From [[Convergent Sequence in Normed Vector Space is Bounded]], there exists $M > 0$ such that:
:$\norm {a_n} \le M$ for each $n \in \N$.
We have for $n \in \N$:
{{begin-eqn}}
{{eqn | l = \norm {a_n b_n - a b}
| r = \norm {a_n b_n - a_n b + a_n b - a b}
}}
{{eqn | r = \norm {a_n \paren {b_n - b} + b \paren {a_n ... | Product Rule for Sequence in Normed Algebra | https://proofwiki.org/wiki/Product_Rule_for_Sequence_in_Normed_Algebra | https://proofwiki.org/wiki/Product_Rule_for_Sequence_in_Normed_Algebra | [
"Normed Algebras",
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Algebra",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Convergent Sequence in Normed Vector Space is Bounded",
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Category:Normed Algebras",
"Category:Convergent Sequences (Normed Vector Spaces)"
] |
proofwiki-20593 | Convergent Sequence in Normed Vector Space is Bounded | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\sequence {x_n}_{n \in \N}$ be a convergent sequence.
Then there exists $M > 0$ such that:
:$\norm {x_n} \le M$
for each $n \in \N$. | Suppose that $x_n \to x$.
From Convergent Sequence is Cauchy Sequence, $\sequence {x_n}_{n \in \N}$ is a Cauchy sequence.
So there exists $N \in \N$ such that:
:$\norm {x_n - x_m} < 1$
for each $n, m \ge N$.
That is:
:$\norm {x_n - x_N} < 1$
for $n \ge N$.
From Reverse Triangle Inequality: Normed Vector Space, we ... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}_{n \in \N}$ be a [[Definition:Convergent Sequence in Normed Vector Space|convergent sequence]].
Then there exists $M > 0$ such that:
:$\norm {x_n} \le M$
for e... | Suppose that $x_n \to x$.
From [[Convergent Sequence is Cauchy Sequence]], $\sequence {x_n}_{n \in \N}$ is a [[Definition:Cauchy Sequence|Cauchy sequence]].
So there exists $N \in \N$ such that:
:$\norm {x_n - x_m} < 1$
for each $n, m \ge N$.
That is:
:$\norm {x_n - x_N} < 1$
for $n \ge N$.
From [[Reverse ... | Convergent Sequence in Normed Vector Space is Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_is_Bounded | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_is_Bounded | [
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Convergent Sequence is Cauchy Sequence",
"Definition:Cauchy Sequence",
"Reverse Triangle Inequality/Normed Vector Space",
"Category:Convergent Sequences (Normed Vector Spaces)"
] |
proofwiki-20594 | Sum Rule for Sequence in Normed Vector Space | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.
Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be convergent sequences such that:
:$x_n \to x$
and:
:$y_n \to y$
Then:
:$x_n + y_n \to x + y$ | For each $n \in \N$, we have:
{{begin-eqn}}
{{eqn | l = \norm {\paren {x_n + y_n} - \paren {x + y} }
| r = \norm {\paren {x_n - x} + \paren {y_n - y} }
}}
{{eqn | o = \le
| r = \norm {x_n - x} + \norm {y_n - y}
| c = {{NormAxiomVector|3}}
}}
{{eqn | o = \to
| r = 0
}}
{{end-eqn}}
So from Sequence in Normed Vec... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\GF$.
Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be [[Definition:Convergent Sequence in Normed Vector Space|convergent sequences]] such that:
:$x_n \to x$
... | For each $n \in \N$, we have:
{{begin-eqn}}
{{eqn | l = \norm {\paren {x_n + y_n} - \paren {x + y} }
| r = \norm {\paren {x_n - x} + \paren {y_n - y} }
}}
{{eqn | o = \le
| r = \norm {x_n - x} + \norm {y_n - y}
| c = {{NormAxiomVector|3}}
}}
{{eqn | o = \to
| r = 0
}}
{{end-eqn}}
So from [[Sequence in Normed... | Sum Rule for Sequence in Normed Vector Space | https://proofwiki.org/wiki/Sum_Rule_for_Sequence_in_Normed_Vector_Space | https://proofwiki.org/wiki/Sum_Rule_for_Sequence_in_Normed_Vector_Space | [
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Category:Convergent Sequences (Normed Vector Spaces)"
] |
proofwiki-20595 | Reciprocal of 13 | :$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3 \ldots$ | Performing the calculation using long division:
<pre>
0.07692307...
-------------
13)1.00000000...
91
----
90
78
--
120
117
---
30
26
--
40
39
--
100
91
---
...
</... | :$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3 \ldots$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.07692307...
-------------
13)1.00000000...
91
----
90
78
--
120
117
---
30
26
--
40
39
--
100
91
... | Reciprocal of 13 | https://proofwiki.org/wiki/Reciprocal_of_13 | https://proofwiki.org/wiki/Reciprocal_of_13 | [
"13",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:13",
"Category:Examples of Reciprocals"
] |
proofwiki-20596 | Reciprocal of 17 | :$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ | Performing the calculation using long division:
<pre>
0.058823529411764705...
------------------------
17)1.000000000000000000000
85 68
-- --
150 20
136 17
--- --
140 30
136 17
--- --
40 130
34 11... | :$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.058823529411764705...
------------------------
17)1.000000000000000000000
85 68
-- --
150 20
136 17
--- --
140 30
136 17
--- --
... | Reciprocal of 17 | https://proofwiki.org/wiki/Reciprocal_of_17 | https://proofwiki.org/wiki/Reciprocal_of_17 | [
"17",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:17",
"Category:Examples of Reciprocals"
] |
proofwiki-20597 | Reciprocal of 27 | :$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$ | Performing the calculation using long division:
<pre>
0.037...
--------
27)1.00000
81
--
190
189
---
100
81
---
...
</pre>
It is to be noted that this is because $999 = 27 \times 37$.
{{qed}}
Category:27
Category:Examples of Reciprocals
dqrjepi3jc57oo9qvkiy5dr... | :$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.037...
--------
27)1.00000
81
--
190
189
---
100
81
---
...
</pre>
It is to be noted that this is because $999 = 27 \times 37$.
{{qed}}
[[Category:27]]
[[Category:Examples ... | Reciprocal of 27 | https://proofwiki.org/wiki/Reciprocal_of_27 | https://proofwiki.org/wiki/Reciprocal_of_27 | [
"27",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:27",
"Category:Examples of Reciprocals"
] |
proofwiki-20598 | Reciprocal of 101 | :$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$ | Performing the calculation using long division:
<pre>
0.00990099...
--------------
101)1.00000000...
909
---
910
909
---
1000
909
----
910
909
---
...
</pre>
{{qed}}
Category:101
Category:Examples of Reciproc... | :$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.00990099...
--------------
101)1.00000000...
909
---
910
909
---
1000
909
----
910
909
---
...
</pre>
{{qed}}
[[Category:... | Reciprocal of 101 | https://proofwiki.org/wiki/Reciprocal_of_101 | https://proofwiki.org/wiki/Reciprocal_of_101 | [
"101",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:101",
"Category:Examples of Reciprocals"
] |
proofwiki-20599 | Reciprocal of 19 | :$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ | Performing the calculation using long division:
<pre>
0.05263157894736842105...
------------------------
19)1.00000000000000000000000
95 152 76
-- --- --
50 180 40
38 171 38
-- --- --
120 90 20
114 76 19
--- -- ... | :$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ | Performing the calculation using [[Definition:Long Division|long division]]:
<pre>
0.05263157894736842105...
------------------------
19)1.00000000000000000000000
95 152 76
-- --- --
50 180 40
38 171 38
-- --- --
120 90 20
114 ... | Reciprocal of 19 | https://proofwiki.org/wiki/Reciprocal_of_19 | https://proofwiki.org/wiki/Reciprocal_of_19 | [
"19",
"Examples of Reciprocals"
] | [] | [
"Definition:Classical Algorithm/Division",
"Category:19",
"Category:Examples of Reciprocals"
] |
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