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-19000 | Hahn-Banach Theorem/Complex Vector Space | Let $X$ be a vector space over $\C$.
Let $p : X \to \R$ be a seminorm on $X$.
Let $X_0$ be a linear subspace of $X$.
Let $f_0 : X_0 \to \C$ be a linear functional such that:
:$\cmod {\map {f_0} x} \le \map p x$ for each $x \in X_0$.
Then there exists a linear functional $f$ defined on the whole space $X$ which exte... | {{Explain|Missing reference to the use of Boolean Prime Ideal Theorem/Ultrafilter Lemma or Axiom of Choice/Zorn's Lemma}}
Define $g_0 : X_0 \to \R$ by:
:$\map {g_0} x = \map \Re {\map {f_0} x}$
for each $x \in X_0$.
By Real Part of Linear Functional is Linear Functional, $g_0$ is an $\R$-linear functional.
Also, defin... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\C$.
Let $p : X \to \R$ be a [[Definition:Seminorm|seminorm]] on $X$.
Let $X_0$ be a [[Definition:Linear Subspace|linear subspace]] of $X$.
Let $f_0 : X_0 \to \C$ be a [[Definition:Linear Functional|linear functional]] such that:
:$\cmod {\map {f_0} x}... | {{Explain|Missing reference to the use of Boolean Prime Ideal Theorem/Ultrafilter Lemma or Axiom of Choice/Zorn's Lemma}}
Define $g_0 : X_0 \to \R$ by:
:$\map {g_0} x = \map \Re {\map {f_0} x}$
for each $x \in X_0$.
By [[Real Part of Linear Functional is Linear Functional]], $g_0$ is an [[Definition:Linear Function... | Hahn-Banach Theorem/Complex Vector Space | https://proofwiki.org/wiki/Hahn-Banach_Theorem/Complex_Vector_Space | https://proofwiki.org/wiki/Hahn-Banach_Theorem/Complex_Vector_Space | [
"Hahn-Banach Theorem"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Linear Subspace",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Vector Space",
"Definition:Extension of Mapping",
"Definition:Linear Functional"
] | [
"Real Part of Linear Functional is Linear Functional",
"Definition:Linear Functional",
"Imaginary Part of Linear Functional is Linear Functional",
"Definition:Linear Functional",
"Definition:Realification of Complex Vector Space",
"Hahn-Banach Theorem/Real Vector Space",
"Definition:Linear Functional",
... |
proofwiki-19001 | Fourier Transform of Derivative of Tempered Distribution | Let $T \in \map {\SS'} \R$ be a tempered distribution.
Let $\xi \in \R$ be a real number.
Let the hat denote the Fourier transform.
Then in the distributional sense it holds that:
:$\hat {\paren{T'} } = 2 \pi i \xi \hat T$ | Let $\phi \in \map \SS \R$ be a Schwartz test function.
Then:
{{begin-eqn}}
{{eqn | l = \map {\hat {\paren {T'} } } {\map \phi x}
| r = \map {T'} {\map {\hat \phi} x}
| c = {{Defof|Fourier Transform of Tempered Distribution}}
}}
{{eqn | r = \map {T'} {\int_{-\infty}^\infty \map \phi \xi e^{-2\pi i \xi x} }
... | Let $T \in \map {\SS'} \R$ be a [[Definition:Tempered Distribution|tempered distribution]].
Let $\xi \in \R$ be a [[Definition:Real Number|real number]].
Let the hat denote the [[Definition:Fourier Transform of Tempered Distribution|Fourier transform]].
Then in the [[Definition:Tempered Distribution|distributional]... | Let $\phi \in \map \SS \R$ be a [[Definition:Schwartz Test Function|Schwartz test function]].
Then:
{{begin-eqn}}
{{eqn | l = \map {\hat {\paren {T'} } } {\map \phi x}
| r = \map {T'} {\map {\hat \phi} x}
| c = {{Defof|Fourier Transform of Tempered Distribution}}
}}
{{eqn | r = \map {T'} {\int_{-\infty}^\... | Fourier Transform of Derivative of Tempered Distribution | https://proofwiki.org/wiki/Fourier_Transform_of_Derivative_of_Tempered_Distribution | https://proofwiki.org/wiki/Fourier_Transform_of_Derivative_of_Tempered_Distribution | [
"Tempered Distributions",
"Fourier Transforms"
] | [
"Definition:Tempered Distribution",
"Definition:Real Number",
"Definition:Fourier Transform of Tempered Distribution",
"Definition:Tempered Distribution"
] | [
"Definition:Schwartz Test Function"
] |
proofwiki-19002 | Existence of Support Functional | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\mathbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $X$.
Let $x \in X$.
Then there exists $f \in X^\ast$ such that:
:$(1): \quad$ $\norm f_{X^\ast} = 1$
:$(2): \quad$ $\map f x = \nor... | Let:
:$U = \span {\set x}$
Then $U$ consists precisely of the $u \in X$ of the form:
:$u = \alpha x$
for $\alpha \in \mathbb F$.
From Linear Span is Linear Subspace, we have:
:$U$ is a linear subspace of $X$.
Let $\struct {U^\ast, \norm \cdot_{U^\ast} }$ be the normed dual space of $U$.
Define $f_0 : U \to \R$ by: ... | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\mathbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $x \in X$.
Then there exists $f \in X^\ast$ such ... | Let:
:$U = \span {\set x}$
Then $U$ consists precisely of the $u \in X$ of the form:
:$u = \alpha x$
for $\alpha \in \mathbb F$.
From [[Linear Span is Linear Subspace]], we have:
:$U$ is a [[Definition:Linear Subspace|linear subspace]] of $X$.
Let $\struct {U^\ast, \norm \cdot_{U^\ast} }$ be the [[Definition... | Existence of Support Functional | https://proofwiki.org/wiki/Existence_of_Support_Functional | https://proofwiki.org/wiki/Existence_of_Support_Functional | [
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Support Functional"
] | [
"Linear Span is Linear Subspace",
"Definition:Linear Subspace",
"Definition:Normed Dual Space",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Definition:Norm/Bounded Linear Functional",
"Definition:Extension of Mapping",
"Definition:Extensio... |
proofwiki-19003 | Normed Dual Space Separates Points | Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $X$.
Then $X^\ast$ separates points.
That is, suppose that $x, y \in X$ are such that:
:$\map f x = \map f y$ for each $f \in X^\ast$.
Then $x = y$. | From Existence of Support Functional, there exists a $\phi \in X^\ast$ such that:
:$\map \phi {x - y} = \norm {x - y}$
Since $\phi$ is linear, we then have:
:$\map \phi x - \map \phi y = \norm {x - y}$
By hypothesis, we have:
:$\map \phi x = \map \phi y$
so:
:$\norm {x - y} = 0$
Since a norm is positive definite, we... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Then $X^\ast$ [[Definition:Mappings Separating Points|separates points]].
That is, suppose that $x, y \in X$... | From [[Existence of Support Functional]], there exists a $\phi \in X^\ast$ such that:
:$\map \phi {x - y} = \norm {x - y}$
Since $\phi$ is [[Definition:Linear Functional|linear]], we then have:
:$\map \phi x - \map \phi y = \norm {x - y}$
By hypothesis, we have:
:$\map \phi x = \map \phi y$
so:
:$\norm {x - y... | Normed Dual Space Separates Points | https://proofwiki.org/wiki/Normed_Dual_Space_Separates_Points | https://proofwiki.org/wiki/Normed_Dual_Space_Separates_Points | [
"Normed Dual Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Mappings Separating Points"
] | [
"Existence of Support Functional",
"Definition:Linear Functional",
"Definition:Norm/Vector Space",
"Definition:Positive Definite (Ring)"
] |
proofwiki-19004 | Scheffé's Lemma | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f_n$ be a sequence of $\mu$-integrable functions that converge almost everywhere to another $\mu$-integrable function $f$.
Then $f_n$ converges to $f$ in $L^1$ {{iff}} $\ds \int_X \size {f_n} \rd \mu$ converges to $\ds \int_X \size f \rd \mu$. | === Sufficient Condition ===
Let $f_n \to f$ in $L^1$.
Then:
{{begin-eqn}}
{{eqn | l = \size {\int_X \size f \rd \mu - \int_X \size {f_n} \rd \mu}
| o = \le
| r = \int_X \size {f - f_n} d\mu
| c = Reverse Triangle Inequality on $L^1$
}}
{{end-eqn}}
and so $f_n \to f$ in $L^1$ implies the {{RHS}} of th... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f_n$ be a sequence of $\mu$-integrable functions that [[Definition:Convergence Almost Everywhere|converge almost everywhere]] to another $\mu$-integrable function $f$.
Then $f_n$ [[Definition:Convergent Real Function|converges]] to $... | === Sufficient Condition ===
Let $f_n \to f$ in $L^1$.
Then:
{{begin-eqn}}
{{eqn | l = \size {\int_X \size f \rd \mu - \int_X \size {f_n} \rd \mu}
| o = \le
| r = \int_X \size {f - f_n} d\mu
| c = [[Reverse Triangle Inequality]] on $L^1$
}}
{{end-eqn}}
and so $f_n \to f$ in $L^1$ implies the {{RHS}... | Scheffé's Lemma | https://proofwiki.org/wiki/Scheffé's_Lemma | https://proofwiki.org/wiki/Scheffé's_Lemma | [
"Measure Theory",
"Scheffé's Lemma"
] | [
"Definition:Measure Space",
"Definition:Convergence Almost Everywhere",
"Definition:Convergent Mapping/Real Function",
"Definition:Convergent Mapping/Real Function"
] | [
"Reverse Triangle Inequality",
"Triangle Inequality"
] |
proofwiki-19005 | Positive Definite and Positive Homogeneous Map with Convex Closed Unit Ball is Norm | Let $\mathbb F$ be a subfield of $\C$.
Let $X$ be a vector space over $\mathbb F$.
Let $N : X \to \R_{\ge 0}$ be a positive definite and positive homogeneous function.
That is:
:$(1): \quad$ $\map N x = 0$ {{iff}} $x = 0$
:$(2): \quad$ $\map N {\lambda x} = \cmod \lambda \map N x$ for each $x \in X$ and $\lambda \in... | Note that $N$ satisfies {{NormAxiomVector|1}} and {{NormAxiomVector|2}}.
So we only need to verify {{NormAxiomVector|3}}.
That is, we want to show that:
:$\map N {x + y} \le \map N x + \map N y$
for each $x, y \in X$.
Fix $x, y \in X$.
If $\map N x = 0$, then $x = 0$ and we have:
{{begin-eqn}}
{{eqn | l = \map N {... | Let $\mathbb F$ be a [[Definition:Subfield|subfield]] of $\C$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\mathbb F$.
Let $N : X \to \R_{\ge 0}$ be a [[Definition:Positive Definite (Ring)|positive definite]] and [[Definition:Positive Homogeneous (Ring)|positive homogeneous]] function.
That is:
:... | Note that $N$ satisfies {{NormAxiomVector|1}} and {{NormAxiomVector|2}}.
So we only need to verify {{NormAxiomVector|3}}.
That is, we want to show that:
:$\map N {x + y} \le \map N x + \map N y$
for each $x, y \in X$.
Fix $x, y \in X$.
If $\map N x = 0$, then $x = 0$ and we have:
{{begin-eqn}}
{{eqn | l = \... | Positive Definite and Positive Homogeneous Map with Convex Closed Unit Ball is Norm | https://proofwiki.org/wiki/Positive_Definite_and_Positive_Homogeneous_Map_with_Convex_Closed_Unit_Ball_is_Norm | https://proofwiki.org/wiki/Positive_Definite_and_Positive_Homogeneous_Map_with_Convex_Closed_Unit_Ball_is_Norm | [
"Normed Vector Spaces",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Subfield",
"Definition:Vector Space",
"Definition:Positive Definite (Ring)",
"Definition:Positive Homogeneous (Ring)",
"Definition:Convex Set (Vector Space)",
"Definition:Norm/Vector Space"
] | [
"Definition:Positive Homogeneous (Ring)",
"Definition:Positive Homogeneous (Ring)",
"Definition:Convex Set (Vector Space)",
"Definition:Positive Homogeneous (Ring)"
] |
proofwiki-19006 | Characterization of Unit Open Balls of Norms of Euclidean Space | Let $\struct {\R^n, \norm \cdot}$ be the Euclidean $n$-space.
Let $K \subseteq \R^n$ be a non-empty open set of $\struct {\R^n, \norm \cdot}$.
Then there exists a norm $\norm \cdot_* : \R^n \to \R$ such that:
:$\set {x \in \R^n : \norm x_* < 1} = K$
{{iff}}:
:$(1): \quad$ $K$ is bounded in $\struct {\R^n, \norm \cdot... | === Necessary Condition === | Let $\struct {\R^n, \norm \cdot}$ be the [[Definition:Euclidean Space|Euclidean $n$-space]].
Let $K \subseteq \R^n$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set in Normed Vector Space|open set]] of $\struct {\R^n, \norm \cdot}$.
Then there exists a [[Definition:Norm on Vector Space|norm]] $\nor... | === Necessary Condition === | Characterization of Unit Open Balls of Norms of Euclidean Space | https://proofwiki.org/wiki/Characterization_of_Unit_Open_Balls_of_Norms_of_Euclidean_Space | https://proofwiki.org/wiki/Characterization_of_Unit_Open_Balls_of_Norms_of_Euclidean_Space | [
"Normed Vector Spaces"
] | [
"Definition:Euclidean Space",
"Definition:Non-Empty Set",
"Definition:Open Set/Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Bounded",
"Definition:Convex Set (Vector Space)",
"Definition:Symmetric Set/Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Unit Ball... | [] |
proofwiki-19007 | Polarization Identity/Real Vector Space | Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\R$.
Let $\norm \cdot$ be the inner product norm for $V$.
Then we have:
:$4 \innerprod x y = \norm {x + y}^2 - \norm {x - y}^2$
for all $x, y \in V$. | We have:
{{begin-eqn}}
{{eqn | l = \norm {x + y}^2 - \norm {x - y}^2
| r = \innerprod {x + y} {x + y} - \innerprod {x - y} {x - y}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \paren {\innerprod x {x + y} + \innerprod y {x + y} } - \paren {\innerprod x {x - y} - \innerprod y {x - y} }
| c = since an inner p... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]] over $\R$.
Let $\norm \cdot$ be the [[Definition:Inner Product Norm|inner product norm]] for $V$.
Then we have:
:$4 \innerprod x y = \norm {x + y}^2 - \norm {x - y}^2$
for all $x, y \in V$. | We have:
{{begin-eqn}}
{{eqn | l = \norm {x + y}^2 - \norm {x - y}^2
| r = \innerprod {x + y} {x + y} - \innerprod {x - y} {x - y}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \paren {\innerprod x {x + y} + \innerprod y {x + y} } - \paren {\innerprod x {x - y} - \innerprod y {x - y} }
| c = since an [[Defi... | Polarization Identity/Real Vector Space | https://proofwiki.org/wiki/Polarization_Identity/Real_Vector_Space | https://proofwiki.org/wiki/Polarization_Identity/Real_Vector_Space | [
"Polarization Identity"
] | [
"Definition:Inner Product Space",
"Definition:Inner Product Norm"
] | [
"Definition:Inner Product",
"Definition:Linear Transformation",
"Definition:Real Number",
"Definition:Inner Product",
"Definition:Symmetric Mapping (Linear Algebra)",
"Definition:Linear Transformation",
"Definition:Real Number",
"Definition:Inner Product",
"Definition:Symmetric Mapping (Linear Algeb... |
proofwiki-19008 | Polarization Identity/Complex Vector Space | Let $V$ be a vector space over $\C$.
Let $q : V \times V \to \C$ be a sesquilinear form.
Define $Q : V \to \C$ by:
:$\map Q x = \map q {x, x}$
Then, we have:
:$4 \map q {x, y} = \map Q {x + y} - \map Q {x - y} + i \map Q {x + i y} - i \map Q {x - i y}$
for each $x, y \in V$. | We write:
{{begin-eqn}}
{{eqn | l = \map Q {x + y} - \map Q {x - y} + i \map Q {x + i y} - i \map Q {x - i y}
| r = \sum_{n \mathop = 0}^3 i^n \map Q {x + i^n y}
}}
{{eqn | r = \sum_{n \mathop = 0}^3 i^n \map q {x + i^n y, x + i^n y}
| c = {{Defof|Inner Product Norm}}
}}
{{end-eqn}}
We then compute:
{{begin-eqn}}... | Let $V$ be a [[Definition:Vector Space|vector space]] over $\C$.
Let $q : V \times V \to \C$ be a [[Definition:Sesquilinear Form|sesquilinear form]].
Define $Q : V \to \C$ by:
:$\map Q x = \map q {x, x}$
Then, we have:
:$4 \map q {x, y} = \map Q {x + y} - \map Q {x - y} + i \map Q {x + i y} - i \map Q {x - i y}$... | We write:
{{begin-eqn}}
{{eqn | l = \map Q {x + y} - \map Q {x - y} + i \map Q {x + i y} - i \map Q {x - i y}
| r = \sum_{n \mathop = 0}^3 i^n \map Q {x + i^n y}
}}
{{eqn | r = \sum_{n \mathop = 0}^3 i^n \map q {x + i^n y, x + i^n y}
| c = {{Defof|Inner Product Norm}}
}}
{{end-eqn}}
We then compute:
{{begin-eq... | Polarization Identity/Complex Vector Space | https://proofwiki.org/wiki/Polarization_Identity/Complex_Vector_Space | https://proofwiki.org/wiki/Polarization_Identity/Complex_Vector_Space | [
"Polarization Identity"
] | [
"Definition:Vector Space",
"Definition:Sesquilinear Form"
] | [
"Inner Product is Sesquilinear",
"Inner Product is Sesquilinear",
"Power of Complex Conjugate is Complex Conjugate of Power"
] |
proofwiki-19009 | Existence of Distance Functional | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\mathbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $X$.
Let $Y$ be a proper closed linear subspace of $X$.
Let $x \in X \setminus Y$.
Let:
:$d = \map {\operatorname {dist} } {x, Y}$
w... | Consider the normed quotient vector space $X/Y$ with quotient mapping $\pi$.
From Kernel of Quotient Mapping, we have $\map \pi x \ne 0$.
So, from Existence of Support Functional, there exists $f \in \paren {X/Y}^\ast$ such that:
:$\norm f_{\paren {X/Y}^\ast} = 1$
and:
:$\map f {\map \pi x} = \norm {\map \pi x}_{X/Y}... | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\mathbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Let $Y$ be a [[Definition:Proper Subset|proper]] [[Defini... | Consider the [[Definition:Normed Quotient Vector Space|normed quotient vector space]] $X/Y$ with [[Definition:Quotient Mapping|quotient mapping]] $\pi$.
From [[Kernel of Quotient Mapping]], we have $\map \pi x \ne 0$.
So, from [[Existence of Support Functional]], there exists $f \in \paren {X/Y}^\ast$ such that:
:... | Existence of Distance Functional/Proof 2 | https://proofwiki.org/wiki/Existence_of_Distance_Functional | https://proofwiki.org/wiki/Existence_of_Distance_Functional/Proof_2 | [
"Normed Dual Spaces",
"Existence of Distance Functional"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Proper Subset",
"Definition:Closed Linear Subspace",
"Definition:Distance/Sets/Metric Spaces",
"Definition:Distance Functional"
] | [
"Definition:Normed Quotient Vector Space",
"Definition:Quotient Mapping",
"Kernel of Quotient Mapping",
"Existence of Support Functional",
"Definition:Quotient Norm",
"Normed Dual Space of Normed Quotient Vector Space is Isometrically Isomorphic to Annihilator",
"Definition:Linear Functional"
] |
proofwiki-19010 | Characterization of Separable Normed Vector Space | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
{{TFAE}}
:$(1): \quad$ $X$ is separable
:$(2): \quad$ $S_X = \set {x \in X : \norm x = 1}$ is separable
:$(3): \quad$ there exists a countable set $\set {x_n : n \in \N} \subseteq X$ such that the closed linear sp... | === $(1)$ implies $(2)$ ===
This is immediate from Subspace of Separable Metric Space is Separable.
{{qed|lemma}} | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
{{TFAE}}
:$(1): \quad$ $X$ is [[Definition:Separable Space|separable]]
:$(2): \quad$ $S_X = \set {x \in X : \norm x = 1}$ is [[Definition:Separable Space|separable]]
:$(3): \q... | === $(1)$ implies $(2)$ ===
This is immediate from [[Subspace of Separable Metric Space is Separable]].
{{qed|lemma}} | Characterization of Separable Normed Vector Space | https://proofwiki.org/wiki/Characterization_of_Separable_Normed_Vector_Space | https://proofwiki.org/wiki/Characterization_of_Separable_Normed_Vector_Space | [
"Separable Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Separable Space",
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Closed Linear Span"
] | [
"Subspace of Separable Metric Space is Separable"
] |
proofwiki-19011 | Subspace of Separable Metric Space is Separable | Let $\struct {X, d}$ be a separable metric space.
Let $\struct {Y, d_Y}$ be a metric subspace of $\struct {X, d}$.
Then $Y$ is separable. | If $Y = X$, we are done immediately.
Now consider the case $Y \ne X$.
Pick $x \in X \setminus Y$.
Let $\set {x_n : n \in \N}$ be a countable everywhere dense subset of $X$.
Iterate over the pairs $\struct {n, k}$ where $n, k \in \N$.
Check whether:
:$\map {B_{1/k} } {x_n} \cap Y \ne \O$
where $\map {B_{1/k} } {x_n}$... | Let $\struct {X, d}$ be a [[Definition:Separable Space|separable]] [[Definition:Metric Space|metric space]].
Let $\struct {Y, d_Y}$ be a [[Definition:Metric Subspace|metric subspace]] of $\struct {X, d}$.
Then $Y$ is [[Definition:Separable Space|separable]]. | If $Y = X$, we are done immediately.
Now consider the case $Y \ne X$.
Pick $x \in X \setminus Y$.
Let $\set {x_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Subset|subset]] of $X$.
Iterate over the pairs $\struct {n, k}$ where $n, k \in \N... | Subspace of Separable Metric Space is Separable | https://proofwiki.org/wiki/Subspace_of_Separable_Metric_Space_is_Separable | https://proofwiki.org/wiki/Subspace_of_Separable_Metric_Space_is_Separable | [
"Metric Spaces",
"Separable Spaces"
] | [
"Definition:Separable Space",
"Definition:Metric Space",
"Definition:Metric Subspace",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Subset",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Set Intersection",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Everywhere Dense",
"Definition:Everywhere... |
proofwiki-19012 | Normed Vector Space with Separable Dual is Separable | Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $X$.
Then, if $X^\ast$ is separable:
:$X$ is separable. | From Characterization of Separable Normed Vector Space, we have that:
:$S_{X^\ast} = \set {f \in X^\ast : \norm f_{X^\ast} = 1}$ is separable.
Let $\mathcal S_{X^\ast} = \set {f_n : n \in \N}$ be a countable everywhere dense subset of $S_{X^\ast}$.
For each $n \in \N$, pick $x_n \in X$ such that $\norm {x_n}_X = 1$ an... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $X$.
Then, if $X^\ast$ is [[Definition:Separable Space|separable]]:
:$X$ is [[Definition:Separable Space|separabl... | From [[Characterization of Separable Normed Vector Space]], we have that:
:$S_{X^\ast} = \set {f \in X^\ast : \norm f_{X^\ast} = 1}$ is [[Definition:Separable Space|separable]].
Let $\mathcal S_{X^\ast} = \set {f_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dens... | Normed Vector Space with Separable Dual is Separable | https://proofwiki.org/wiki/Normed_Vector_Space_with_Separable_Dual_is_Separable | https://proofwiki.org/wiki/Normed_Vector_Space_with_Separable_Dual_is_Separable | [
"Separable Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Separable Space",
"Definition:Separable Space"
] | [
"Characterization of Separable Normed Vector Space",
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Characterization of Separable Normed Vector Space",
"Closed Linear Span is Closed Vector Subspace",
"Definition:Closed Vector Subspace",
"Existence of Distance... |
proofwiki-19013 | Pell's Equation/Examples/2/-1 | :$x^2 - 2 y^2 = -1$
has the positive integral solutions:
:<nowiki>$\begin {array} {r|r} x & y \\ \hline
1 & 1 \\
7 & 5 \\
41 & 29 \\
239 & 169 \\
1393 & 985 \\
\end {array}$</nowiki>
and so on. | From Continued Fraction Expansion of $\sqrt 2$:
:$\sqrt 2 = \sqbrk {1, \sequence 2}$
The cycle is of length is $1$.
By Solution of Pell's Equation, the only solutions of $x^2 - 2 y^2 = -1$ are:
:${p_r}^2 - 2 {q_r}^2 = \paren {-1}^r$
for $r = 1, 2, 3, \ldots$
From Convergents to Continued Fraction Expansion of $\sqrt 2$... | :$x^2 - 2 y^2 = -1$
has the [[Definition:Positive Integer|positive integral]] solutions:
:<nowiki>$\begin {array} {r|r} x & y \\ \hline
1 & 1 \\
7 & 5 \\
41 & 29 \\
239 & 169 \\
1393 & 985 \\
\end {array}$</nowiki>
and so on. | From [[Continued Fraction Expansion of Root 2|Continued Fraction Expansion of $\sqrt 2$]]:
:$\sqrt 2 = \sqbrk {1, \sequence 2}$
The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fraction|length]] is $1$.
By [[Solution of Pell's Equation]], the only sol... | Pell's Equation/Examples/2/-1 | https://proofwiki.org/wiki/Pell's_Equation/Examples/2/-1 | https://proofwiki.org/wiki/Pell's_Equation/Examples/2/-1 | [
"Pell's Equation",
"2"
] | [
"Definition:Positive/Integer"
] | [
"Continued Fraction Expansion of Irrational Square Root/Examples/2",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Solution to Pell's Equation",
"Continued Fraction Expansion of Irrational Square Root/Examples/2/Convergents",
"Definition:Convergent... |
proofwiki-19014 | Pell's Equation/Examples/2 | :$x^2 - 2 y^2 = 1$
has the positive integral solutions:
:<nowiki>$\begin {array} {r|r} x & y \\ \hline
3 & 2 \\
17 & 12 \\
99 & 70 \\
577 & 408 \\
3363 & 2378 \\
\end {array}$</nowiki>
and so on. | From Continued Fraction Expansion of $\sqrt 2$:
:$\sqrt 2 = \sqbrk {1, \sequence 2}$
The cycle is of length is $1$.
By Solution of Pell's Equation, the only solutions of $x^2 - 2 y^2 = -1$ are:
:${p_r}^2 - 2 {q_r}^2 = \paren {-1}^r$
for $r = 1, 2, 3, \ldots$
From Convergents to Continued Fraction Expansion of $\sqrt 2$... | :$x^2 - 2 y^2 = 1$
has the [[Definition:Positive Integer|positive integral]] solutions:
:<nowiki>$\begin {array} {r|r} x & y \\ \hline
3 & 2 \\
17 & 12 \\
99 & 70 \\
577 & 408 \\
3363 & 2378 \\
\end {array}$</nowiki>
and so on. | From [[Continued Fraction Expansion of Root 2|Continued Fraction Expansion of $\sqrt 2$]]:
:$\sqrt 2 = \sqbrk {1, \sequence 2}$
The [[Definition:Cycle of Periodic Continued Fraction|cycle]] is of [[Definition:Cycle Length of Periodic Continued Fraction|length]] is $1$.
By [[Solution of Pell's Equation]], the only sol... | Pell's Equation/Examples/2 | https://proofwiki.org/wiki/Pell's_Equation/Examples/2 | https://proofwiki.org/wiki/Pell's_Equation/Examples/2 | [
"Pell's Equation",
"2"
] | [
"Definition:Positive/Integer"
] | [
"Continued Fraction Expansion of Irrational Square Root/Examples/2",
"Definition:Periodic Continued Fraction/Cycle",
"Definition:Periodic Continued Fraction/Cycle/Length",
"Solution to Pell's Equation",
"Continued Fraction Expansion of Irrational Square Root/Examples/2/Convergents",
"Definition:Convergent... |
proofwiki-19015 | Norm satisfying Parallelogram Law induced by Inner Product/Real Case | Let $V$ be a vector space over $\R$.
Let $\norm \cdot : V \to \R$ be a norm on $V$ such that:
:$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$
for each $x, y \in V$.
Then the function $\innerprod \cdot \cdot : V \times V \to \R$ defined by:
:$\ds \innerprod x y = \frac {\norm {x + y}^2 - \norm ... | We first verify symmetry.
Let $x, y \in V$.
Then we have:
{{begin-eqn}}
{{eqn | l = \innerprod y x
| r = \frac {\norm {y + x}^2 - \norm {y - x}^2} 4
}}
{{eqn | r = \frac {\norm {x + y}^2 - \norm {-\paren {x - y} }^2} 4
}}
{{eqn | r = \frac {\norm {x + y}^2 - \norm {x - y}^2} 4
| c = {{NormAxiomVector|2}}
}}
{{eq... | Let $V$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $\norm \cdot : V \to \R$ be a [[Definition:Norm on Vector Space|norm]] on $V$ such that:
:$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$
for each $x, y \in V$.
Then the [[Definition:Function|function]] $\innerprod \cdot \... | We first verify [[Definition:Symmetric Mapping (Linear Algebra)|symmetry]].
Let $x, y \in V$.
Then we have:
{{begin-eqn}}
{{eqn | l = \innerprod y x
| r = \frac {\norm {y + x}^2 - \norm {y - x}^2} 4
}}
{{eqn | r = \frac {\norm {x + y}^2 - \norm {-\paren {x - y} }^2} 4
}}
{{eqn | r = \frac {\norm {x + y}^2 - \no... | Norm satisfying Parallelogram Law induced by Inner Product/Real Case | https://proofwiki.org/wiki/Norm_satisfying_Parallelogram_Law_induced_by_Inner_Product/Real_Case | https://proofwiki.org/wiki/Norm_satisfying_Parallelogram_Law_induced_by_Inner_Product/Real_Case | [
"Norm satisfying Parallelogram Law induced by Inner Product",
"Normed Vector Spaces",
"Inner Product Spaces"
] | [
"Definition:Vector Space",
"Definition:Norm/Vector Space",
"Definition:Function",
"Definition:Inner Product",
"Definition:Inner Product Norm"
] | [
"Definition:Symmetric Mapping (Linear Algebra)",
"Definition:Non-Negative Definite Mapping",
"Definition:Positiveness",
"Definition:Positive Homogeneous (Ring)",
"Definition:Norm/Vector Space",
"Definition:Norm/Vector Space",
"Definition:Positive Definite (Ring)",
"Definition:Linear Transformation",
... |
proofwiki-19016 | Convergence in Measure Implies Convergence a.e. of Subsequence | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $D \in \Sigma$.
Let $f : D \to \R$ be a $\Sigma$-measurable function.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n : D \to \R$ such that:
:$\sequence {f_n}_{n \mathop \in \N}$ converges in measure to $f$.
Then ther... | For each $n, k \ge 1$, define:
:$\ds B_{n, k} = \set {x \in X : \size {\map {f_n} x - \map f x} > \frac 1 k}$
Since $\sequence {f_n}_{n \mathop \in \N}$ converges in measure to $f$, we have:
:$\ds \lim_{n \mathop \to \infty} \map \mu {\set {x \in X : \size {\map {f_n} x - \map f x} > \frac 1 k} } = 0$
Then from the de... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $D \in \Sigma$.
Let $f : D \to \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Measurable Function|$\Sigma$-mea... | For each $n, k \ge 1$, define:
:$\ds B_{n, k} = \set {x \in X : \size {\map {f_n} x - \map f x} > \frac 1 k}$
Since $\sequence {f_n}_{n \mathop \in \N}$ [[Definition:Convergence in Measure|converges in measure]] to $f$, we have:
:$\ds \lim_{n \mathop \to \infty} \map \mu {\set {x \in X : \size {\map {f_n} x - \map ... | Convergence in Measure Implies Convergence a.e. of Subsequence | https://proofwiki.org/wiki/Convergence_in_Measure_Implies_Convergence_a.e._of_Subsequence | https://proofwiki.org/wiki/Convergence_in_Measure_Implies_Convergence_a.e._of_Subsequence | [
"Convergence Almost Everywhere"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Sequence",
"Definition:Measurable Function",
"Definition:Convergence in Measure",
"Definition:Subsequence",
"Definition:Convergence Almost Everywhere"
] | [
"Definition:Convergence in Measure",
"Definition:Convergence",
"Sum of Infinite Geometric Sequence",
"Borel-Cantelli Lemma",
"Equivalence of Definitions of Limit Superior of Sequence of Sets",
"Squeeze Theorem",
"Definition:Convergence Almost Everywhere"
] |
proofwiki-19017 | Convergence of Product of Convergent Scalar Sequence and Convergent Vector Sequence in Normed Vector Space | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {V, \norm \cdot}$ be a normed vector space on $\Bbb F$.
Let $\alpha \in \R$.
Let $x \in V$.
Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a real sequence in $\Bbb F$ such that:
:$\alpha_n \to \alpha$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $V$ such that... | Let $\epsilon > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\alpha_n x_n - \alpha x}
| r = \norm {\alpha_n x_n - \alpha x + \alpha_n x - \alpha_n x}
}}
{{eqn | o = \le
| r = \norm {\alpha_n x_n - \alpha_n x} + \norm {\alpha_n x - \alpha x}
| c = using the triangle inequality for the norm $\norm \cdot$
}}
{{eqn... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {V, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] on $\Bbb F$.
Let $\alpha \in \R$.
Let $x \in V$.
Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] in $\Bbb F$ such that:
:$\alpha_n \to \alpha$
L... | Let $\epsilon > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \norm {\alpha_n x_n - \alpha x}
| r = \norm {\alpha_n x_n - \alpha x + \alpha_n x - \alpha_n x}
}}
{{eqn | o = \le
| r = \norm {\alpha_n x_n - \alpha_n x} + \norm {\alpha_n x - \alpha x}
| c = using the triangle inequality for the [[Definition:Norm on Vect... | Convergence of Product of Convergent Scalar Sequence and Convergent Vector Sequence in Normed Vector Space | https://proofwiki.org/wiki/Convergence_of_Product_of_Convergent_Scalar_Sequence_and_Convergent_Vector_Sequence_in_Normed_Vector_Space | https://proofwiki.org/wiki/Convergence_of_Product_of_Convergent_Scalar_Sequence_and_Convergent_Vector_Sequence_in_Normed_Vector_Space | [
"Convergent Sequences (Normed Vector Spaces)",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Real Sequence",
"Definition:Sequence"
] | [
"Definition:Norm/Vector Space",
"Definition:Positive Homogeneous (Ring)",
"Definition:Norm/Vector Space",
"Convergent Real Sequence is Bounded",
"Definition:Positive Definite (Ring)",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Conve... |
proofwiki-19018 | Weak Limit in Normed Vector Space is Unique | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $x, y \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ such that:
:$x_n \weakconv x$
and:
:$x_n \weakconv y$
where $\weakconv$ denotes weak convergence.
Then:
:$x = y$ | Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $\struct {X, \norm \cdot}$.
Since:
:$x_n \weakconv x$
we have:
:$\map f {x_n} \to \map f x$ for each $f \in X^\ast$.
Since:
:$x_n \weakconv y$
we have:
:$\map f {x_n} \to \map f y$ for each $f \in X^\ast$.
From Convergent Complex Sequence has... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $x, y \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ such that:
:$x_n \weakconv x$
and:
:$x_n \weakconv y$
where $\weakconv$ denotes [[Definition:Weak Convergence (Norme... | Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm \cdot}$.
Since:
:$x_n \weakconv x$
we have:
:$\map f {x_n} \to \map f x$ for each $f \in X^\ast$.
Since:
:$x_n \weakconv y$
we have:
:$\map f {x_n} \to \map f y$ for each $f \in X^\a... | Weak Limit in Normed Vector Space is Unique | https://proofwiki.org/wiki/Weak_Limit_in_Normed_Vector_Space_is_Unique | https://proofwiki.org/wiki/Weak_Limit_in_Normed_Vector_Space_is_Unique | [
"Weak Convergence (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Definition:Normed Dual Space",
"Convergent Complex Sequence has Unique Limit",
"Normed Dual Space Separates Points"
] |
proofwiki-19019 | Weak Convergence in Hilbert Space | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\HH$.
Let $x \in X$.
Then:
:$\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$
{{iff}}:
:$\innerprod {x_n} y \to \innerprod x y$ for each $y \in \HH$. | Let $\struct {\HH^\ast, \norm \cdot_{\HH^\ast} }$ be the normed dual space of $\HH$. | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\HH$.
Let $x \in X$.
Then:
:$\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Weak Convergence (Normed Vector Space)|converges weakly]] to... | Let $\struct {\HH^\ast, \norm \cdot_{\HH^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\HH$. | Weak Convergence in Hilbert Space | https://proofwiki.org/wiki/Weak_Convergence_in_Hilbert_Space | https://proofwiki.org/wiki/Weak_Convergence_in_Hilbert_Space | [
"Weak Convergence (Normed Vector Spaces)",
"Hilbert Spaces",
"Weak Convergence in Hilbert Space"
] | [
"Definition:Hilbert Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Definition:Normed Dual Space",
"Definition:Normed Dual Space"
] |
proofwiki-19020 | Orthonormal Sequence in Hilbert Space Converges Weakly to Zero | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\sequence {e_n}_{n \mathop \in \N}$ be a sequence in $\HH$ such that:
:$\innerprod {e_n} {e_m} = 0$ if $n \ne m$
and:
:$\norm {e_n} = 1$ for each $n \in \N$.
Then:
:$e_n \weakconv 0$
where $\rightharpoonup$ denotes weak convergence. | From Bessel's Inequality, we have:
:$\ds \sum_{n \mathop = 1}^\infty \size {\innerprod y {e_n} }^2$ converges for each $y \in \HH$.
So from Terms in Convergent Series Converge to Zero, we have:
:$\cmod {\innerprod y {e_n} }^2 \to 0$ for each $y \in \HH$.
We then have, from Complex Sequence is Null iff Positive Integ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\sequence {e_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\HH$ such that:
:$\innerprod {e_n} {e_m} = 0$ if $n \ne m$
and:
:$\norm {e_n} = 1$ for each $n \in \N$.
Then:
:$e_n \weakconv 0$
wher... | From [[Bessel's Inequality]], we have:
:$\ds \sum_{n \mathop = 1}^\infty \size {\innerprod y {e_n} }^2$ [[Definition:Convergent Series|converges]] for each $y \in \HH$.
So from [[Terms in Convergent Series Converge to Zero]], we have:
:$\cmod {\innerprod y {e_n} }^2 \to 0$ for each $y \in \HH$.
We then have, fro... | Orthonormal Sequence in Hilbert Space Converges Weakly to Zero | https://proofwiki.org/wiki/Orthonormal_Sequence_in_Hilbert_Space_Converges_Weakly_to_Zero | https://proofwiki.org/wiki/Orthonormal_Sequence_in_Hilbert_Space_Converges_Weakly_to_Zero | [
"Weak Convergence (Normed Vector Spaces)",
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Bessel's Inequality",
"Definition:Convergent Series",
"Terms in Convergent Series Converge to Zero",
"Complex Sequence is Null iff Positive Integer Powers of Sequence are Null",
"Complex Sequence is Null iff Modulus of Sequence is Null",
"Weak Convergence in Hilbert Space/Corollary"
] |
proofwiki-19021 | Complex Sequence is Null iff Positive Integer Powers of Sequence are Null | Let $\sequence {z_n}_{n \mathop \in \N}$ be a complex sequence.
Let $k \in \N$.
Then:
:$z_n \to 0$
{{iff}}:
:${z_n}^k \to 0$ | === Necessary Condition ===
Suppose that:
:$z_n \to 0$
Let $\epsilon > 0$.
Then from the definition of convergent sequence we can find $N \in \N$ such that for $n \ge N$ we have:
:$\cmod {z_n} < \epsilon^{1/k}$
then:
:$\cmod {z_n}^k < \epsilon$
From Power of Complex Modulus equals Complex Modulus of Power, this gi... | Let $\sequence {z_n}_{n \mathop \in \N}$ be a [[Definition:Complex Sequence|complex sequence]].
Let $k \in \N$.
Then:
:$z_n \to 0$
{{iff}}:
:${z_n}^k \to 0$ | === Necessary Condition ===
Suppose that:
:$z_n \to 0$
Let $\epsilon > 0$.
Then from the definition of [[Definition:Convergent Complex Sequence|convergent sequence]] we can find $N \in \N$ such that for $n \ge N$ we have:
:$\cmod {z_n} < \epsilon^{1/k}$
then:
:$\cmod {z_n}^k < \epsilon$
From [[Power of Com... | Complex Sequence is Null iff Positive Integer Powers of Sequence are Null | https://proofwiki.org/wiki/Complex_Sequence_is_Null_iff_Positive_Integer_Powers_of_Sequence_are_Null | https://proofwiki.org/wiki/Complex_Sequence_is_Null_iff_Positive_Integer_Powers_of_Sequence_are_Null | [
"Convergent Complex Sequences"
] | [
"Definition:Complex Sequence"
] | [
"Definition:Convergent Sequence/Complex Numbers",
"Power of Complex Modulus equals Complex Modulus of Power",
"Power of Complex Modulus equals Complex Modulus of Power",
"Definition:Convergent Sequence/Complex Numbers"
] |
proofwiki-19022 | Convergence of Complex Conjugate of Convergent Complex Sequence | Let $z \in \C$.
Let $\sequence {z_n}_{n \mathop \in \N}$ be a complex sequence converging to $z$.
Then:
:$\overline {z_n} \to \overline z$ | Let $\epsilon > 0$.
Since $z_n \to z$, from the definition of convergence, we can find $N \in \N$ such that:
:$\cmod {z_n - z} < \epsilon$
From Complex Modulus equals Complex Modulus of Conjugate, we have:
:$\cmod {\overline {z_n - z} } = \cmod {z_n - z}$
From Difference of Complex Conjugates, we have:
:$\cmod {z_n... | Let $z \in \C$.
Let $\sequence {z_n}_{n \mathop \in \N}$ be a [[Definition:Complex Sequence|complex sequence]] [[Definition:Convergent Complex Sequence|converging]] to $z$.
Then:
:$\overline {z_n} \to \overline z$ | Let $\epsilon > 0$.
Since $z_n \to z$, from the definition of [[Definition:Convergent Complex Sequence|convergence]], we can find $N \in \N$ such that:
:$\cmod {z_n - z} < \epsilon$
From [[Complex Modulus equals Complex Modulus of Conjugate]], we have:
:$\cmod {\overline {z_n - z} } = \cmod {z_n - z}$
From [[Di... | Convergence of Complex Conjugate of Convergent Complex Sequence | https://proofwiki.org/wiki/Convergence_of_Complex_Conjugate_of_Convergent_Complex_Sequence | https://proofwiki.org/wiki/Convergence_of_Complex_Conjugate_of_Convergent_Complex_Sequence | [
"Complex Conjugates",
"Convergent Complex Sequences"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers"
] | [
"Definition:Convergent Sequence/Complex Numbers",
"Complex Modulus equals Complex Modulus of Conjugate",
"Difference of Complex Conjugates",
"Category:Complex Conjugates",
"Category:Convergent Complex Sequences"
] |
proofwiki-19023 | Rectangular Delta Sequence | thumb300pxThe graph of the rectangular delta sequence. As $n$ grows, the rectangle becomes thinner and taller. The area of each rectangle is equal to $1$.
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:<nowiki>$\map {\delta_n} x := \begin {cases}
0 & : x < - \dfrac 1 {2 n} \\ \\
n & : - \dfrac 1 {2 n} \l... | Let $\phi \in \map \DD \R$ be a test function.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x
| r = \lim_{n \mathop \to \infty} n \int_{-\frac 1 {2 n} }^{\frac 1 {2 n} } \map \phi x \rd x
}}
{{eqn | r = \lim_{n \mathop \to \infty} n \map \phi ... | [[File:RectangularDeltaSequence.png|thumb|300px|The graph of the rectangular delta sequence. As $n$ grows, the rectangle becomes thinner and taller. The area of each rectangle is equal to $1$.]]
Let $\sequence {\map {\delta_n} x}$ be a [[Definition:Sequence|sequence]] such that:
:<nowiki>$\map {\delta_n} x := \begin ... | Let $\phi \in \map \DD \R$ be a [[Definition:Test Function|test function]].
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} \int_{-\infty}^\infty \map {\delta_n} x \map \phi x \rd x
| r = \lim_{n \mathop \to \infty} n \int_{-\frac 1 {2 n} }^{\frac 1 {2 n} } \map \phi x \rd x
}}
{{eqn | r = \lim_{n \... | Rectangular Delta Sequence | https://proofwiki.org/wiki/Rectangular_Delta_Sequence | https://proofwiki.org/wiki/Rectangular_Delta_Sequence | [
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:RectangularDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Definition:Test Function",
"Mean Value Theorem for Integrals",
"Limit of Image of Sequence/Real Number Line",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-19024 | Minkowski Functional of Open Convex Set in Normed Vector Space is Well-Defined | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
Then, for each $x \in X$:
:$\set {t > 0 : t^{-1} x \in C} \ne \O$
and so the Minkowski functional of $C$ is well-defined. | If $x = 0$, then:
:$t^{-1} x \in C$
for all $t > 0$, so:
:$\set {t > 0 : t^{-1} x \in C} = \openint 0 \infty$
Now take $x \ne 0$.
Since $C$ is open, there exists $\delta > 0$ such that for all $x \in X$ with:
:$\norm x < \delta$
we have $x \in C$.
Note that we have, from positive homogeneity:
:$\ds \norm {\frac \de... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $C$ be an [[Definition:Open Set in Normed Vector Space|open]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of $X$ with $0 \in C$.
Then, for e... | If $x = 0$, then:
:$t^{-1} x \in C$
for all $t > 0$, so:
:$\set {t > 0 : t^{-1} x \in C} = \openint 0 \infty$
Now take $x \ne 0$.
Since $C$ is [[Definition:Open Set in Normed Vector Space|open]], there exists $\delta > 0$ such that for all $x \in X$ with:
:$\norm x < \delta$
we have $x \in C$.
Note that we h... | Minkowski Functional of Open Convex Set in Normed Vector Space is Well-Defined | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_is_Well-Defined | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_is_Well-Defined | [
"Minkowski Functionals in Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Minkowski Functional/Normed Vector Space"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Positive Homogeneous (Ring)",
"Continuum Property",
"Definition:Infimum of Set/Real Numbers",
"Definition:Minkowski Functional/Normed Vector Space",
"Category:Minkowski Functionals in Normed Vector Spaces"
] |
proofwiki-19025 | Minkowski Functional of Open Convex Set in Normed Vector Space is Bounded | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
Let $p_C$ be the Minkowski functional for $C$.
Then there exists a real number $c > 0$ such that:
:$0 \le \map {p_C} x \le c \norm x$
for each $x \in X$. | From the definition of the Minkowski functional, we have:
:$\map {p_C} x = \inf \set {t > 0 : t^{-1} x \in C}$
We have:
:$t \ge 0$ for every $t \in \set {t > 0 : t^{-1} x \in C}$
so:
:$\inf \set {t > 0 : t^{-1} x \in C} \ge 0$
from the definition of infimum.
If $x = 0$, we have:
:$t^{-1} x \in C$
for each $t > 0$, s... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $C$ be an [[Definition:Open Set in Normed Vector Space|open]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of $X$ with $0 \in C$.
Let $p_C$ be the [[Definition:Minkowski Functional in Normed... | From the definition of the [[Definition:Minkowski Functional in Normed Vector Space|Minkowski functional]], we have:
:$\map {p_C} x = \inf \set {t > 0 : t^{-1} x \in C}$
We have:
:$t \ge 0$ for every $t \in \set {t > 0 : t^{-1} x \in C}$
so:
:$\inf \set {t > 0 : t^{-1} x \in C} \ge 0$
from the definition of [[De... | Minkowski Functional of Open Convex Set in Normed Vector Space is Bounded | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_is_Bounded | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_is_Bounded | [
"Minkowski Functionals in Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Minkowski Functional/Normed Vector Space",
"Definition:Real Number"
] | [
"Definition:Minkowski Functional/Normed Vector Space",
"Definition:Infimum of Set/Real Numbers",
"Definition:Real Number",
"Definition:Open Set/Normed Vector Space",
"Minkowski Functional of Open Convex Set in Normed Vector Space is Well-Defined",
"Definition:Infimum of Set/Real Numbers"
] |
proofwiki-19026 | Minkowski Functional of Open Convex Set in Normed Vector Space is Sublinear Functional | Let $\struct {X, \norm \cdot}$ be a normed vector space over $\R$.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
Let $p_C$ be the Minkowski functional for $C$.
Then $p_C$ is a sublinear functional. | We will show that:
:$(1): \quad \map {p_C} {\lambda x} = \lambda \map {p_C} x$ for each $x \in X$ and $\lambda \in \R_{\ge 0}$
:$(2): \quad \map {p_C} {x + y} \le \map {p_C} x + \map {p_C} y$ for each $x, y \in X$. | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$.
Let $C$ be an [[Definition:Open Set in Normed Vector Space|open]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of $X$ with $0 \in C$.
Let $p_C$ be the [[Definition:Minkowski Functional... | We will show that:
:$(1): \quad \map {p_C} {\lambda x} = \lambda \map {p_C} x$ for each $x \in X$ and $\lambda \in \R_{\ge 0}$
:$(2): \quad \map {p_C} {x + y} \le \map {p_C} x + \map {p_C} y$ for each $x, y \in X$. | Minkowski Functional of Open Convex Set in Normed Vector Space is Sublinear Functional | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_is_Sublinear_Functional | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_is_Sublinear_Functional | [
"Minkowski Functional of Open Convex Set in Normed Vector Space is Sublinear Functional",
"Minkowski Functionals in Normed Vector Spaces",
"Sublinear Functionals"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Minkowski Functional/Normed Vector Space",
"Definition:Sublinear Functional"
] | [] |
proofwiki-19027 | Minkowski Functional of Open Convex Set in Normed Vector Space recovers Set | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $C$ be an open convex subset of $X$ with $0 \in C$.
Let $p_C$ be the Minkowski functional for $C$.
Then:
:$C = \set {x \in X : \map {p_C} x < 1}$ | We first show that:
:$\set {x \in X : \map {p_C} x < 1} \subseteq C$
Suppose that:
:$x \in \set {x \in X : \map {p_C} x < 1}$
Then:
:$\inf \set {t > 0 : t^{-1} x \in C} < 1$
from the definition of a Minkowski functional.
So, there exists $0 < \alpha < 1$ such that:
:$\alpha^{-1} x \in C$
Since $C$ is convex and $0... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $C$ be an [[Definition:Open Set in Normed Vector Space|open]] [[Definition:Convex Set (Vector Space)|convex]] [[Definition:Subset|subset]] of $X$ with $0 \in C$.
Let $p_C$ be the [[Definition:Minkowski Functional in Normed... | We first show that:
:$\set {x \in X : \map {p_C} x < 1} \subseteq C$
Suppose that:
:$x \in \set {x \in X : \map {p_C} x < 1}$
Then:
:$\inf \set {t > 0 : t^{-1} x \in C} < 1$
from the definition of a [[Definition:Minkowski Functional in Normed Vector Space|Minkowski functional]].
So, there exists $0 < \alpha <... | Minkowski Functional of Open Convex Set in Normed Vector Space recovers Set | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_recovers_Set | https://proofwiki.org/wiki/Minkowski_Functional_of_Open_Convex_Set_in_Normed_Vector_Space_recovers_Set | [
"Minkowski Functionals in Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Minkowski Functional/Normed Vector Space"
] | [
"Definition:Minkowski Functional/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Open Set/Normed Vector Space",
"Definition:Infimum of Set/Real Numbers",
"Definition:Set Equality"
] |
proofwiki-19028 | True Weight from False Balance/Unequal Arms | Let $B$ be a body whose weight is $W$.
Let $B$ be weighed in a false balance with unequal arms.
Let the readings of the weight of $B$ be $a$ and $b$ when placed in opposite pans.
Then:
:$W = \sqrt {a b}$ | We have that the false balance has arms of different lengths.
Let the lengths of the arms of the false balance be $x$ and $y$.
{{WLOG}}, placing $B$ in the pan at the end of $x$ gives:
:$W x = a y$
and placing $B$ in the pan at the end of $y$ gives:
:$W y = b x$
{{TheoremWanted|We need to invoke the physics of couples ... | Let $B$ be a [[Definition:Body|body]] whose [[Definition:Weight (Physics)|weight]] is $W$.
Let $B$ be [[Definition:Weigh|weighed]] in a [[Definition:False Balance with Unequal Arms|false balance with unequal arms]].
Let the readings of the [[Definition:Weight (Physics)|weight]] of $B$ be $a$ and $b$ when placed in o... | We have that the [[Definition:False Balance with Unequal Arms|false balance]] has [[Definition:Arm of Balance|arms]] of different [[Definition:Length (Linear Measure)|lengths]].
Let the [[Definition:Length (Linear Measure)|lengths]] of the [[Definition:Arm of Balance|arms]] of the [[Definition:False Balance|false bala... | True Weight from False Balance/Unequal Arms | https://proofwiki.org/wiki/True_Weight_from_False_Balance/Unequal_Arms | https://proofwiki.org/wiki/True_Weight_from_False_Balance/Unequal_Arms | [
"False Balances"
] | [
"Definition:Body",
"Definition:Weight (Physics)",
"Definition:Weight (Physics)/Weigh",
"Definition:False Balance/Unequal Arms",
"Definition:Weight (Physics)",
"Definition:Balance/Pan"
] | [
"Definition:False Balance/Unequal Arms",
"Definition:Balance/Arm",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Definition:Balance/Arm",
"Definition:False Balance",
"Definition:Balance/Pan",
"Definition:Balance/Pan",
"Category:False Balances"
] |
proofwiki-19029 | True Weight from False Balance/Imbalanced Pans | Let $B$ be a body whose weight is $W$.
Let $B$ be weighed in a false balance with imbalanced pans.
Let the readings of the weight of $B$ be $a$ and $b$ when placed in opposite pans.
Then:
:$W = \dfrac {a + b} 2$ | We have that the false balance has pans such that one weighs more than the other.
Let the lengths of the arms of the false balance be $x$.
Let one of the pans weigh $m$ more than the other.
Placing $B$ in the lighter pan gives:
:$W x = \paren {a + m} x$
and placing $B$ in the heavier pan gives:
:$\paren {W + m} x = b x... | Let $B$ be a [[Definition:Body|body]] whose [[Definition:Weight (Physics)|weight]] is $W$.
Let $B$ be [[Definition:Weigh|weighed]] in a [[Definition:False Balance with Imbalanced Pans|false balance with imbalanced pans]].
Let the readings of the [[Definition:Weight (Physics)|weight]] of $B$ be $a$ and $b$ when placed... | We have that the [[Definition:False Balance with Imbalanced Pans|false balance]] has [[Definition:Pan of Balance|pans]] such that one [[Definition:Weight (Physics)|weighs]] more than the other.
Let the [[Definition:Length (Linear Measure)|lengths]] of the [[Definition:Arm of Balance|arms]] of the [[Definition:False Ba... | True Weight from False Balance/Imbalanced Pans | https://proofwiki.org/wiki/True_Weight_from_False_Balance/Imbalanced_Pans | https://proofwiki.org/wiki/True_Weight_from_False_Balance/Imbalanced_Pans | [
"False Balances"
] | [
"Definition:Body",
"Definition:Weight (Physics)",
"Definition:Weight (Physics)/Weigh",
"Definition:False Balance/Imbalanced Pans",
"Definition:Weight (Physics)",
"Definition:Balance/Pan"
] | [
"Definition:False Balance/Imbalanced Pans",
"Definition:Balance/Pan",
"Definition:Weight (Physics)",
"Definition:Linear Measure/Length",
"Definition:Balance/Arm",
"Definition:False Balance",
"Definition:Balance/Pan",
"Definition:Balance/Pan",
"Definition:Balance/Pan",
"Category:False Balances"
] |
proofwiki-19030 | Seminorm is Sublinear Functional | Let $X$ be a vector space over $\R$.
Let $p : X \to \R$ be a seminorm on $X$.
Then:
:$p$ is a sublinear functional. | Since $p$ is a seminorm, we have:
:$\map p {x + y} \le \map p x + \map p y$ for each $x, y \in X$
We also have:
:$\map p {\lambda x} = \cmod \lambda \map p x$ for each $\lambda \in \R$ and $x \in X$.
and in particular:
:$\map p {\lambda x} = \lambda \map p x$ for each $\lambda \in \R_{\ge 0}$ and $x \in X$.
So:
:$p$ ... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $p : X \to \R$ be a [[Definition:Seminorm|seminorm]] on $X$.
Then:
:$p$ is a [[Definition:Sublinear Functional|sublinear functional]]. | Since $p$ is a [[Definition:Seminorm|seminorm]], we have:
:$\map p {x + y} \le \map p x + \map p y$ for each $x, y \in X$
We also have:
:$\map p {\lambda x} = \cmod \lambda \map p x$ for each $\lambda \in \R$ and $x \in X$.
and in particular:
:$\map p {\lambda x} = \lambda \map p x$ for each $\lambda \in \R_{\ge... | Seminorm is Sublinear Functional | https://proofwiki.org/wiki/Seminorm_is_Sublinear_Functional | https://proofwiki.org/wiki/Seminorm_is_Sublinear_Functional | [
"Seminorms",
"Sublinear Functionals"
] | [
"Definition:Vector Space",
"Definition:Seminorm",
"Definition:Sublinear Functional"
] | [
"Definition:Seminorm",
"Definition:Sublinear Functional",
"Category:Seminorms",
"Category:Sublinear Functionals"
] |
proofwiki-19031 | Weak-* Limit in Normed Dual Space is Unique | Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space for $\struct {X, \norm \cdot}$.
Let $f, g \in X^\ast$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence in $X^\ast$ such that:
:$f_n \weakstarconv f$
and:
:$f_n \weakstar... | By the definition of weak-$\ast$ convergence, we have:
:$\map {f_n} x \to \map f x$ for each $x \in X$
and:
:$\map {f_n} x \to \map g x$ for each $x \in X$.
Then, from Convergent Complex Sequence has Unique Limit we have:
:$\map f x = \map g x$ for each $x \in X$.
That is:
:$f = g$
{{qed}} | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] for $\struct {X, \norm \cdot}$.
Let $f, g \in X^\ast$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Def... | By the definition of [[Definition:Weak-* Convergence (Normed Vector Space)|weak-$\ast$ convergence]], we have:
:$\map {f_n} x \to \map f x$ for each $x \in X$
and:
:$\map {f_n} x \to \map g x$ for each $x \in X$.
Then, from [[Convergent Complex Sequence has Unique Limit]] we have:
:$\map f x = \map g x$ for each... | Weak-* Limit in Normed Dual Space is Unique | https://proofwiki.org/wiki/Weak-*_Limit_in_Normed_Dual_Space_is_Unique | https://proofwiki.org/wiki/Weak-*_Limit_in_Normed_Dual_Space_is_Unique | [
"Weak-* Convergence (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Sequence",
"Definition:Weak-* Convergence (Normed Vector Space)"
] | [
"Definition:Weak-* Convergence (Normed Vector Space)",
"Convergent Complex Sequence has Unique Limit"
] |
proofwiki-19032 | Limit Inferior of Norm of Weakly Convergent Sequence is Bounded Below by Norm of Weak Limit | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging weakly to $x$.
Then, we have:
:$\ds \norm x \le \liminf_{n \mathop \to \infty} \norm {x_n}$ | From Existence of Support Functional, there exists $f \in X^\ast$ such that:
:$\map f x = \norm x$
and:
:$\norm f_{X^\ast} = 1$
Since:
:$x_n \weakconv x$
we have:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = \map f x$
That is:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = \norm x$
From Absolute Value of Limit... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ [[Definition:Weak Convergence (Normed Vector Space)|converging weakly]] to $x$.
Then, we have:
:$\ds \norm x \le \limin... | From [[Existence of Support Functional]], there exists $f \in X^\ast$ such that:
:$\map f x = \norm x$
and:
:$\norm f_{X^\ast} = 1$
Since:
:$x_n \weakconv x$
we have:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = \map f x$
That is:
:$\ds \lim_{n \mathop \to \infty} \map f {x_n} = \norm x$
From [[Absolut... | Limit Inferior of Norm of Weakly Convergent Sequence is Bounded Below by Norm of Weak Limit | https://proofwiki.org/wiki/Limit_Inferior_of_Norm_of_Weakly_Convergent_Sequence_is_Bounded_Below_by_Norm_of_Weak_Limit | https://proofwiki.org/wiki/Limit_Inferior_of_Norm_of_Weakly_Convergent_Sequence_is_Bounded_Below_by_Norm_of_Weak_Limit | [
"Weak Convergence (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Existence of Support Functional",
"Modulus of Limit",
"Convergence of Limsup and Liminf",
"Fundamental Property of Norm on Bounded Linear Functional",
"Inequality Rule for Real Sequences"
] |
proofwiki-19033 | Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$ {{iff}}:
:$\norm {x_n - x} \to 0$ | From the definition of a convergent sequence in a normed vector space, we have that:
:$x_n$ converges to $x$
{{iff}}:
:for each $\epsilon > 0$ there exists $N \in \N$ such that $\norm {x_n - x} < \epsilon$.
From the definition of a convergent real sequence, we have that:
:$\norm {x_n - x} \to 0$
{{iff}}:
:for each $\... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $x$ {{if... | From the definition of a [[Definition:Convergent Sequence in Normed Vector Space|convergent sequence in a normed vector space]], we have that:
:$x_n$ [[Definition:Convergent Sequence in Normed Vector Space|converges]] to $x$
{{iff}}:
:for each $\epsilon > 0$ there exists $N \in \N$ such that $\norm {x_n - x} < \eps... | Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence | https://proofwiki.org/wiki/Sequence_in_Normed_Vector_Space_Convergent_to_Limit_iff_Norm_of_Sequence_minus_Limit_is_Null_Sequence | https://proofwiki.org/wiki/Sequence_in_Normed_Vector_Space_Convergent_to_Limit_iff_Norm_of_Sequence_minus_Limit_is_Null_Sequence | [
"Convergent Sequences (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space"
] | [
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Norm/Vector Space",
"Category:Convergent Sequences (Normed Vector Spaces)"
] |
proofwiki-19034 | Weakly Convergent Sequence in Hilbert Space with Convergent Norm is Convergent | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\HH$.
Let $x \in X$ be such that:
:$x_n \weakconv x$
and:
:$\norm {x_n} \to \norm x$
where $\weakconv$ denotes weak convergence.
Then:
:$x_n \to x$ | Let $\norm \cdot$ be the inner product norm for $\struct {\HH, \innerprod \cdot \cdot}$.
From Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence, we have:
:$x_n \to x$
{{iff}}:
:$\norm {x_n - x} \to 0$
From Complex Sequence is Null iff Positive Integer Powers of Seque... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\HH$.
Let $x \in X$ be such that:
:$x_n \weakconv x$
and:
:$\norm {x_n} \to \norm x$
where $\weakconv$ denotes [[Definition:Weak Converg... | Let $\norm \cdot$ be the [[Definition:Inner Product Norm|inner product norm]] for $\struct {\HH, \innerprod \cdot \cdot}$.
From [[Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence]], we have:
:$x_n \to x$
{{iff}}:
:$\norm {x_n - x} \to 0$
From [[Complex Sequence... | Weakly Convergent Sequence in Hilbert Space with Convergent Norm is Convergent | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Hilbert_Space_with_Convergent_Norm_is_Convergent | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Hilbert_Space_with_Convergent_Norm_is_Convergent | [
"Weak Convergence (Normed Vector Spaces)",
"Hilbert Spaces"
] | [
"Definition:Hilbert Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Definition:Inner Product Norm",
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Complex Sequence is Null iff Positive Integer Powers of Sequence are Null",
"Inner Product is Sesquilinear",
"Weak Convergence in Hilbert Space",
"Definition:Weak Conv... |
proofwiki-19035 | Convergent Sequence in Normed Vector Space is Weakly Convergent | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging to $x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$. | Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $\struct {X, \norm \cdot}$.
Let $f \in X^\ast$.
If $\norm f_{X^\ast} = 0$, then $f = 0$ and:
:$\map f {x_n} \to \map f x$
Take $f \ne 0$, so that $\norm f_{X^\ast} \ne 0$.
Let $\epsilon > 0$.
We then have:
{{begin-eqn}}
{{eqn | l = \cmod {\ma... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ [[Definition:Convergent Sequence in Normed Vector Space|converging]] to $x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ [[... | Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm \cdot}$.
Let $f \in X^\ast$.
If $\norm f_{X^\ast} = 0$, then $f = 0$ and:
:$\map f {x_n} \to \map f x$
Take $f \ne 0$, so that $\norm f_{X^\ast} \ne 0$.
Let $\epsilon > 0$.
We then have... | Convergent Sequence in Normed Vector Space is Weakly Convergent/Proof 1 | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_is_Weakly_Convergent | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_is_Weakly_Convergent/Proof_1 | [
"Weak Convergence (Normed Vector Spaces)",
"Convergent Sequences (Normed Vector Spaces)",
"Convergent Sequence in Normed Vector Space is Weakly Convergent"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Definition:Normed Dual Space",
"Fundamental Property of Norm on Bounded Linear Functional",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Weak Convergence (Normed Vector Space)"
] |
proofwiki-19036 | Convergent Sequence in Normed Vector Space is Weakly Convergent | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging to $x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$. | Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $\struct {X, \norm \cdot}$.
Then, for each $f \in X^\ast$:
{{begin-eqn}}
{{eqn | l = \size {\map f {x_n} - \map f x}
| o = \le
| r = \norm f_{X^\ast} \norm {x_n - x}_X
| c = Fundamental Property of Norm on Bounded Linear Functio... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $x \in X$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ [[Definition:Convergent Sequence in Normed Vector Space|converging]] to $x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ [[... | Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm \cdot}$.
Then, for each $f \in X^\ast$:
{{begin-eqn}}
{{eqn | l = \size {\map f {x_n} - \map f x}
| o = \le
| r = \norm f_{X^\ast} \norm {x_n - x}_X
| c = [[Fundamental Proper... | Convergent Sequence in Normed Vector Space is Weakly Convergent/Proof 2 | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_is_Weakly_Convergent | https://proofwiki.org/wiki/Convergent_Sequence_in_Normed_Vector_Space_is_Weakly_Convergent/Proof_2 | [
"Weak Convergence (Normed Vector Spaces)",
"Convergent Sequences (Normed Vector Spaces)",
"Convergent Sequence in Normed Vector Space is Weakly Convergent"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Definition:Normed Dual Space",
"Fundamental Property of Norm on Bounded Linear Functional"
] |
proofwiki-19037 | Linear Combination of Weakly Convergent Sequences is Weakly Convergent | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\Bbb F$.
Let $x, y \in X$.
Let $\alpha, \beta \in \Bbb F$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging weakly to $x$.
Let $\sequence {y_n}_{n \mathop \in \N}$ be a sequence in $X$ conve... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,} }$.
Since:
:$x_n \weakconv x$
We have:
:$\map f {x_n} \to \map f x$
for each $f \in X^\ast$.
Similarly, since:
:$y_n \weakconv y$
we have:
:$\map f {y_n} \to \map f y$
From Combined Sum Rule for Rea... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $x, y \in X$.
Let $\alpha, \beta \in \Bbb F$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $X$ [[Definition:Weak Convergence (... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual space]] of $\struct {X, \norm {\, \cdot \,} }$.
Since:
:$x_n \weakconv x$
We have:
:$\map f {x_n} \to \map f x$
for each $f \in X^\ast$.
Similarly, since:
:$y_n \weakconv y$
we have:
:$\map f {y_n} \to... | Linear Combination of Weakly Convergent Sequences is Weakly Convergent | https://proofwiki.org/wiki/Linear_Combination_of_Weakly_Convergent_Sequences_is_Weakly_Convergent | https://proofwiki.org/wiki/Linear_Combination_of_Weakly_Convergent_Sequences_is_Weakly_Convergent | [
"Weak Convergence (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Weak Convergence (Normed Vector Space)"
] | [
"Definition:Normed Dual Space",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Definition:Linear Functional",
"Definition:Weak Convergence (Normed Vector Space)",
"Category:Weak Convergence (Normed Vector Spaces)"
] |
proofwiki-19038 | Largest Rectangle Contained in Triangle | Let $T$ be a triangle.
Let $R$ be a rectangle contained within $T$.
Let $R$ have the largest area possible for the conditions given.
Then:
:$(1): \quad$ One side of $R$ is coincident with part of one side of $T$, and hence two vertices lie on that side of $T$
:$(2): \quad$ The other two vertices of $R$ bisect the other... | Note that a rectangle is a parallelogram.
By Largest Parallelogram Contained in Triangle, the area of $R$ cannot exceed half the area of $T$.
Hence we only need to show that when the first two conditions above are satisfied, the area of $R$ is exactly half the area of $T$.
Consider the diagram below.
:400px
Since $AD =... | Let $T$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $R$ be a [[Definition:Rectangle|rectangle]] contained within $T$.
Let $R$ have the largest [[Definition:Area|area]] possible for the conditions given.
Then:
:$(1): \quad$ One [[Definition:Side of Polygon|side]] of $R$ is coincident with part of one [[De... | Note that a [[Definition:Rectangle|rectangle]] is a [[Definition:Parallelogram|parallelogram]].
By [[Largest Parallelogram Contained in Triangle]], the [[Definition:Area|area]] of $R$ cannot exceed half the [[Definition:Area|area]] of $T$.
Hence we only need to show that when the first two conditions above are satisf... | Largest Rectangle Contained in Triangle | https://proofwiki.org/wiki/Largest_Rectangle_Contained_in_Triangle | https://proofwiki.org/wiki/Largest_Rectangle_Contained_in_Triangle | [
"Triangles",
"Rectangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Quadrilateral/Rectangle",
"Definition:Area",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Vertex",
"Definition:Polygon/Side",
"Definition:Polygon/Vertex",
"Definition:Bisection",
"Definition:Polygon/Side",
"Definition:Are... | [
"Definition:Quadrilateral/Rectangle",
"Definition:Quadrilateral/Parallelogram",
"Largest Parallelogram Contained in Triangle",
"Definition:Area",
"Definition:Area",
"Definition:Area",
"File:Largest-rectangle-in-triangle.png",
"Definition:Area",
"Definition:Area",
"Definition:Area",
"Category:Tri... |
proofwiki-19039 | Characterization of Hyperplanes | Let $\Bbb F$ be a field.
Let $X$ be a vector space over $\Bbb F$.
Let $U$ be a subspace of $X$.
{{TFAE}}
:$(1): \quad$ $U$ is a hyperplane
:$(2): \quad$ $U \ne X$, and for any $x \in X \setminus U$ we have $\map \span {U \cup \set x} = X$
:$(3): \quad$ there exists a non-zero linear functional $\phi : X \to \Bbb F$ su... | === $(1)$ implies $(2)$ ===
Suppose that:
:$U$ is a hyperplane.
Let $x \in X \setminus U$.
Then from Linear Span is Linear Subspace, we have:
:$\map \span {U \cup \set x}$ is a subspace of $X$.
We have that:
:$U \subseteq \map \span {U \cup \set x}$
So either:
:$U = \map \span {U \cup \set x}$
or:
:$X = \map \spa... | Let $\Bbb F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $\Bbb F$.
Let $U$ be a [[Definition:Subspace|subspace]] of $X$.
{{TFAE}}
:$(1): \quad$ $U$ is a [[Definition:Hyperplane|hyperplane]]
:$(2): \quad$ $U \ne X$, and for any $x \in X \setminus U... | === $(1)$ implies $(2)$ ===
Suppose that:
:$U$ is a [[Definition:Hyperplane|hyperplane]].
Let $x \in X \setminus U$.
Then from [[Linear Span is Linear Subspace]], we have:
:$\map \span {U \cup \set x}$ is a [[Definition:Linear Subspace|subspace]] of $X$.
We have that:
:$U \subseteq \map \span {U \cup \set x}... | Characterization of Hyperplanes | https://proofwiki.org/wiki/Characterization_of_Hyperplanes | https://proofwiki.org/wiki/Characterization_of_Hyperplanes | [
"Hyperplanes"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Subspace",
"Definition:Hyperplane",
"Definition:Linear Functional"
] | [
"Definition:Hyperplane",
"Linear Span is Linear Subspace",
"Definition:Linear Subspace",
"Definition:Linear Subspace",
"Definition:Hyperplane",
"Definition:Hyperplane",
"Definition:Linear Subspace"
] |
proofwiki-19040 | Hyperplane in Normed Vector Space generated by Unbounded Linear Functional is Everywhere Dense | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $\phi : X \to \Bbb F$ be a linear functional that is not bounded.
Let $U$ be a hyperplane in $X$ given by:
:$U = \map \ker \phi$
Then:
:$U$ is everywhere dense in $X$. | Since $\phi$ is not bounded, for each $n \in \N$ we can pick $v_n \ne 0$ such that:
:$\cmod {\map \phi {v_n} } \ge n \norm {v_n}$
Set:
:$\ds x_n = \frac {v_n} {\norm {v_n} }$
for each $n \in \N$.
Then for each $n \in \N$, we have:
:$\cmod {\map \phi {x_n} } \ge n$
from linearity with:
:$\norm {x_n} = 1$
Fix $x \in X$ ... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\phi : X \to \Bbb F$ be a [[Definition:Linear Functional|linear functional]] that is not [[Definition:Bounded Linear Functional|bounded]].
Let $U$ be a [[Definition:Hyperplan... | Since $\phi$ is not [[Definition:Bounded Linear Functional|bounded]], for each $n \in \N$ we can pick $v_n \ne 0$ such that:
:$\cmod {\map \phi {v_n} } \ge n \norm {v_n}$
Set:
:$\ds x_n = \frac {v_n} {\norm {v_n} }$
for each $n \in \N$.
Then for each $n \in \N$, we have:
:$\cmod {\map \phi {x_n} } \ge n$
from [[... | Hyperplane in Normed Vector Space generated by Unbounded Linear Functional is Everywhere Dense | https://proofwiki.org/wiki/Hyperplane_in_Normed_Vector_Space_generated_by_Unbounded_Linear_Functional_is_Everywhere_Dense | https://proofwiki.org/wiki/Hyperplane_in_Normed_Vector_Space_generated_by_Unbounded_Linear_Functional_is_Everywhere_Dense | [
"Hyperplanes",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Definition:Hyperplane",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Definition:Bounded Linear Functional",
"Definition:Linear Functional",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Norm/Vector Space",
"Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence",
"Definition:Sequen... |
proofwiki-19041 | Preimage of Maximum of Real Part of Continuous Linear Functional on Extreme Set in Convex Compact Set is Extreme Set | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ with its standard topology.
Let $X^\ast$ be the topological dual space of $X$.
Let $K$ be a compact convex subset of $X$.
Let $M \subseteq K$ be an extreme set in $K$.
Let $f \in X^\ast$.
Let:
:$\ds M^f = \set {x \in M : \map \R... | Note that from Continuous Function on Compact Space is Bounded we have that:
:$\map \Re f$ attains its supremum
and so $M^f$ is well-defined.
From Mapping is Continuous iff Inverse Images of Open Sets are Open: Corollary, we have:
:$M^f$ is closed.
Let $x, y \in K$ and $t \in \openint 0 1$ be such that:
:$t x + \par... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $X^\ast$ be the [[Definition:Topological Dual Space|topological dual space]] of $X$.
Let $K$ be a [[Defini... | Note that from [[Continuous Function on Compact Space is Bounded]] we have that:
:$\map \Re f$ attains its [[Definition:Supremum of Real-Valued Function|supremum]]
and so $M^f$ is well-defined.
From [[Mapping is Continuous iff Inverse Images of Open Sets are Open/Corollary|Mapping is Continuous iff Inverse Images o... | Preimage of Maximum of Real Part of Continuous Linear Functional on Extreme Set in Convex Compact Set is Extreme Set | https://proofwiki.org/wiki/Preimage_of_Maximum_of_Real_Part_of_Continuous_Linear_Functional_on_Extreme_Set_in_Convex_Compact_Set_is_Extreme_Set | https://proofwiki.org/wiki/Preimage_of_Maximum_of_Real_Part_of_Continuous_Linear_Functional_on_Extreme_Set_in_Convex_Compact_Set_is_Extreme_Set | [
"Extreme Sets"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Topological Dual Space",
"Definition:Compact Space/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Extreme Set",
"Definition:Extreme Set"
] | [
"Continuous Function on Compact Space is Bounded",
"Definition:Supremum of Mapping/Real-Valued Function",
"Mapping is Continuous iff Inverse Images of Open Sets are Open/Corollary",
"Definition:Closed Set/Normed Vector Space",
"Definition:Extreme Set",
"Definition:Maximum Value of Real Function/Absolute",... |
proofwiki-19042 | Point in Convex Set is Extreme Point iff Singleton is Extreme Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $K$ be a convex subset of $X$.
Let $a \in K$.
Then $a$ is an extreme point of $K$ {{iff}}:
:$\set a$ is an extreme set in $K$. | We have:
:$a$ is an extreme point of $K$.
{{iff}}:
:whenever $a = t x + \paren {1 - t} y$ for some $x, y \in K$ and $t \in \openint 0 1$, we have $x = y = a$.
We can rewrite this:
:whenever $t x + \paren {1 - t} y \in \set a$ for some $x, y \in K$ and $t \in \openint 0 1$, we have $x, y \in \set a$.
This is precisel... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $K$ be a [[Definition:Convex Set (Vector Space)|convex subset]] of $X$.
Let $a \in K$.
Then $a$ is an [[Definition:Extreme Point of Convex Set|extreme point]] of $K$ {{iff}}:
:$\set a$ is an [[Definition:Extreme ... | We have:
:$a$ is an [[Definition:Extreme Point of Convex Set|extreme point]] of $K$.
{{iff}}:
:whenever $a = t x + \paren {1 - t} y$ for some $x, y \in K$ and $t \in \openint 0 1$, we have $x = y = a$.
We can rewrite this:
:whenever $t x + \paren {1 - t} y \in \set a$ for some $x, y \in K$ and $t \in \openint 0... | Point in Convex Set is Extreme Point iff Singleton is Extreme Set | https://proofwiki.org/wiki/Point_in_Convex_Set_is_Extreme_Point_iff_Singleton_is_Extreme_Set | https://proofwiki.org/wiki/Point_in_Convex_Set_is_Extreme_Point_iff_Singleton_is_Extreme_Set | [
"Extreme Sets",
"Extreme Points of Convex Sets"
] | [
"Definition:Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Extreme Point of Convex Set",
"Definition:Extreme Set"
] | [
"Definition:Extreme Point of Convex Set",
"Definition:Extreme Set"
] |
proofwiki-19043 | Convex Hull is Smallest Convex Set containing Set | Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $U \subseteq X$ be non-empty.
Let $\map {\operatorname {conv} } U$ be the convex hull of $U$.
Then $\map {\operatorname {conv} } U$ is the smallest convex subset of $X$ containing $U$ in the sense that:
:$\map {\operatorname {conv} } U$ is convex a... | We have:
:$\ds \map {\operatorname {conv} } U = \set {\sum_{j \mathop = 1}^n \lambda_j u_j : n \in \N, \, u_j \in U \text { and } \lambda_j \in \R_{> 0} \text { for each } j, \, \sum_{j \mathop = 1}^n \lambda_j = 1}$
Considering $n = 1$ and $\lambda_1 = 1$, we obtain:
:$u \in \map {\operatorname {conv} } U$ for each ... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $\GF$.
Let $U \subseteq X$ be [[Definition:Non-Empty Set|non-empty]].
Let $\map {\operatorname {conv} } U$ be the [[Definition:Convex Hull/Definition 1|convex hull]] of $U$.
Then $\map {\operatorname {conv} } U$ is the [[Defini... | We have:
:$\ds \map {\operatorname {conv} } U = \set {\sum_{j \mathop = 1}^n \lambda_j u_j : n \in \N, \, u_j \in U \text { and } \lambda_j \in \R_{> 0} \text { for each } j, \, \sum_{j \mathop = 1}^n \lambda_j = 1}$
Considering $n = 1$ and $\lambda_1 = 1$, we obtain:
:$u \in \map {\operatorname {conv} } U$ for ea... | Convex Hull is Smallest Convex Set containing Set | https://proofwiki.org/wiki/Convex_Hull_is_Smallest_Convex_Set_containing_Set | https://proofwiki.org/wiki/Convex_Hull_is_Smallest_Convex_Set_containing_Set | [
"Convex Hull is Smallest Convex Set containing Set",
"Convex Hulls",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Vector Space",
"Definition:Non-Empty Set",
"Definition:Convex Hull/Definition 1",
"Definition:Smallest Set by Set Inclusion",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Convex Combination contained in Convex Set"
] |
proofwiki-19044 | Krein-Milman Theorem | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a Hausdorff locally convex space over $\GF$.
Let $K$ be a non-empty compact convex subset of $X$.
Let $\map E K$ be the set of extreme points in $K$.
Then $K$ is the closed convex hull of $\map E K$. | Let $K'$ be the closed convex hull of $\map E K$.
We show that $K' = K$.
From Convex Hull is Smallest Convex Set containing Set, we have:
:$\map E K \subseteq \map {\operatorname {conv} } {\map E K}$
From Set is Subset of its Topological Closure, we therefore have:
:$\map E K \subseteq K'$
We also have, from Convex H... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Hausdorff Locally Convex Space|Hausdorff locally convex space]] over $\GF$.
Let $K$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Compact Subset of Normed Vector Space|compact]] [[Definition:Convex Set (Vector Space)|convex]] [[Definitio... | Let $K'$ be the [[Definition:Closed Convex Hull|closed convex hull]] of $\map E K$.
We show that $K' = K$.
From [[Convex Hull is Smallest Convex Set containing Set]], we have:
:$\map E K \subseteq \map {\operatorname {conv} } {\map E K}$
From [[Set is Subset of its Topological Closure]], we therefore have:
:$\ma... | Krein-Milman Theorem | https://proofwiki.org/wiki/Krein-Milman_Theorem | https://proofwiki.org/wiki/Krein-Milman_Theorem | [
"Functional Analysis",
"Closed Convex Hulls"
] | [
"Definition:Locally Convex Space/Hausdorff",
"Definition:Non-Empty Set",
"Definition:Compact Space/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Set",
"Definition:Extreme Point of Convex Set",
"Definition:Closed Convex Hull"
] | [
"Definition:Closed Convex Hull",
"Convex Hull is Smallest Convex Set containing Set",
"Set is Subset of its Topological Closure",
"Convex Hull preserves Subsets",
"Convex Hull is Smallest Convex Set containing Set/Corollary",
"Set Closure Preserves Set Inclusion",
"Definition:Closure/Normed Vector Space... |
proofwiki-19045 | Extreme Set in Compact Convex Set contains Extreme Point | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a locally convex space over $\GF$ with its standard topology.
Let $K$ be a non-empty compact convex subset of $X$.
Let $E$ be an extreme set of $K$.
Then $E$ contains an extreme point of $K$. | Let $P$ be the set of extreme sets in $K$ that are contained in $E$.
Since $E \in P$, $P$ is certainly non-empty.
Define a relation $\preceq$ on $P$ by $A \preceq B$ {{iff}} $B \subseteq A$.
From Subset Relation is Ordering and Dual Ordering is Ordering, we have:
:$\struct {P, \preceq}$ is an ordered set.
We show tha... | Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \PP}$ be a [[Definition:Locally Convex Space|locally convex space]] over $\GF$ with its [[Definition:Locally Convex Space/Standard Topology|standard topology]].
Let $K$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Compact Subset of Normed Vector Space|compact]... | Let $P$ be the [[Definition:Set|set]] of [[Definition:Extreme Set|extreme sets]] in $K$ that are contained in $E$.
Since $E \in P$, $P$ is certainly [[Definition:Non-Empty Set|non-empty]].
Define a [[Definition:Relation|relation]] $\preceq$ on $P$ by $A \preceq B$ {{iff}} $B \subseteq A$.
From [[Subset Relation is... | Extreme Set in Compact Convex Set contains Extreme Point | https://proofwiki.org/wiki/Extreme_Set_in_Compact_Convex_Set_contains_Extreme_Point | https://proofwiki.org/wiki/Extreme_Set_in_Compact_Convex_Set_contains_Extreme_Point | [
"Convex Sets (Vector Spaces)",
"Extreme Sets",
"Extreme Set in Compact Convex Set contains Extreme Point",
"Extreme Points of Convex Sets"
] | [
"Definition:Locally Convex Space",
"Definition:Locally Convex Space/Standard Topology",
"Definition:Non-Empty Set",
"Definition:Compact Space/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Subset",
"Definition:Extreme Set",
"Definition:Extreme Point of Convex Set"
] | [
"Definition:Set",
"Definition:Extreme Set",
"Definition:Non-Empty Set",
"Definition:Relation",
"Subset Relation is Ordering",
"Dual Ordering is Ordering",
"Definition:Ordered Set",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)",
"Definition:Upper Bound of Set",
"Zorn's Lemma",
"... |
proofwiki-19046 | Invertibility of Identity Minus Operator | Let $\GF \in \set {\R, \C}$.
Let $X$ be a Banach space over $\GF$.
Let $T : X \to X$ be a bounded linear operator such that:
:$\norm T_{\map \BB X} < 1$
where $\norm {\, \cdot \,}_{\map \BB X}$ denotes the norm of a bounded linear operator.
Then $I - T$ is invertible as a bounded linear operator.
In particular:
:$... | For each $n \in \N$, define:
:$\ds S_n = \sum_{k \mathop = 0}^n T^k$
We argue first that $\sequence {S_n}_{n \mathop \in \N}$ is convergent.
Since $X$ is a Banach space, it is enough to show that $\sequence {S_n}_{n \mathop \in \N}$ is Cauchy.
Let $\epsilon > 0$.
Let $m, n \in \N$ with $n > m$.
Then we have:
{{be... | Let $\GF \in \set {\R, \C}$.
Let $X$ be a [[Definition:Banach Space|Banach space]] over $\GF$.
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]] such that:
:$\norm T_{\map \BB X} < 1$
where $\norm {\, \cdot \,}_{\map \BB X}$ denotes the [[Definition:Norm on Bounded Linear Tran... | For each $n \in \N$, define:
:$\ds S_n = \sum_{k \mathop = 0}^n T^k$
We argue first that $\sequence {S_n}_{n \mathop \in \N}$ is [[Definition:Convergent Sequence|convergent]].
Since $X$ is a [[Definition:Banach Space|Banach space]], it is enough to show that $\sequence {S_n}_{n \mathop \in \N}$ is [[Definition:Cau... | Invertibility of Identity Minus Operator | https://proofwiki.org/wiki/Invertibility_of_Identity_Minus_Operator | https://proofwiki.org/wiki/Invertibility_of_Identity_Minus_Operator | [
"Bounded Linear Operators",
"Invertibility of Identity Minus Operator"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Invertible Bounded Linear Operator"
] | [
"Definition:Convergent Sequence",
"Definition:Banach Space",
"Definition:Cauchy Sequence",
"Norm on Bounded Linear Transformation is Submultiplicative/Corollary",
"Sum of Infinite Geometric Sequence",
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence",
"Norm... |
proofwiki-19047 | Resolvent Mapping is Continuous/Bounded Linear Operator | Let $B$ be a Banach space.
Let $\mathfrak{L}(B, B)$ be the set of bounded linear operators from $B$ to itself.
Let $T \in \mathfrak{L}(B, B)$.
Let $\rho(T)$ be the resolvent set of $T$ in the complex plane.
Then the resolvent mapping $f : \rho(T) \to \mathfrak{L}(B,B)$ given by:
:$f(z) = (T - zI)^{-1}$
is continuous in... | Pick $z\in\rho(T)$. Since $z\in\rho(T)$, the operator $R_z = (T - zI)^{-1}$ exists and has finite norm $C \geq 0$.
Since Resolvent Set is Open, $z+h\in\rho(T)$ for any $h\in \Bbb C$ smaller than some $\delta > 0$. For such $h$,
{{begin-eqn}}
{{eqn | l = \norm{ f(z+h) - f(z) }_*
| r = \norm{ (T-(z+h)I)^{-1} - (T-z... | Let $B$ be a [[Definition:Banach Space|Banach space]].
Let $\mathfrak{L}(B, B)$ be the [[Definition:Set|set]] of [[Definition:Bounded Linear Operator|bounded linear operators]] from $B$ to itself.
Let $T \in \mathfrak{L}(B, B)$.
Let $\rho(T)$ be the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent set... | Pick $z\in\rho(T)$. Since $z\in\rho(T)$, the operator $R_z = (T - zI)^{-1}$ exists and has finite norm $C \geq 0$.
Since [[Resolvent Set is Open]], $z+h\in\rho(T)$ for any $h\in \Bbb C$ smaller than some $\delta > 0$. For such $h$,
{{begin-eqn}}
{{eqn | l = \norm{ f(z+h) - f(z) }_*
| r = \norm{ (T-(z+h)I)^{-1} ... | Resolvent Mapping is Continuous/Bounded Linear Operator | https://proofwiki.org/wiki/Resolvent_Mapping_is_Continuous/Bounded_Linear_Operator | https://proofwiki.org/wiki/Resolvent_Mapping_is_Continuous/Bounded_Linear_Operator | [
"Bounded Linear Operators",
"Resolvent Mapping is Continuous"
] | [
"Definition:Banach Space",
"Definition:Set",
"Definition:Bounded Linear Operator",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Resolvent Set is Open",
"Norm on Bounded Linear Transformation is Submultiplicative",
"Operator Norm is Norm",
"Invertibility of Identity Minus Operator",
"Norm on Bounded Linear Transformation is Submultiplicative",
"Triangle Inequality",
"Norm on Bounded Linear Transformation is Submultiplicative",
... |
proofwiki-19048 | Resolvent Mapping is Analytic/Bounded Linear Operator | Let $B$ be a Banach space.
Let $\map \LL {B, B}$ be the set of bounded linear operators from $B$ to itself.
Let $T \in \map \LL {B, B}$.
Let $\map \rho T$ be the resolvent set of $T$ in the complex plane.
Then the resolvent mapping $f : \map \rho T \to \map \LL {B, B}$ given by $\map f z = \paren {T - z I}^{-1}$ is ana... | For $a \in \map \rho T$, define:
:$R_a = \paren {T - a I}^{-1}$
Then we have:
{{begin-eqn}}
{{eqn | l = \frac {\norm {\map f {z + h} - \map f z - \paren {T - z I}^{-2} h }_*} {\size h}
| r = \frac {\norm {R_{z + h} - R_z - R_z^2 h}_*} {\size h}
}}
{{eqn | r = \frac {\norm {h R_{z + h} R_z - R_z^2 h }_*} {\size h}... | Let $B$ be a [[Definition:Banach Space|Banach space]].
Let $\map \LL {B, B}$ be the set of [[Definition:Bounded Linear Operator on Normed Vector Space|bounded linear operators]] from $B$ to itself.
Let $T \in \map \LL {B, B}$.
Let $\map \rho T$ be the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent s... | For $a \in \map \rho T$, define:
:$R_a = \paren {T - a I}^{-1}$
Then we have:
{{begin-eqn}}
{{eqn | l = \frac {\norm {\map f {z + h} - \map f z - \paren {T - z I}^{-2} h }_*} {\size h}
| r = \frac {\norm {R_{z + h} - R_z - R_z^2 h}_*} {\size h}
}}
{{eqn | r = \frac {\norm {h R_{z + h} R_z - R_z^2 h }_*} {\size ... | Resolvent Mapping is Analytic/Bounded Linear Operator/Proof 1 | https://proofwiki.org/wiki/Resolvent_Mapping_is_Analytic/Bounded_Linear_Operator | https://proofwiki.org/wiki/Resolvent_Mapping_is_Analytic/Bounded_Linear_Operator/Proof_1 | [
"Resolvent Mapping is Analytic",
"Functional Analysis"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator/Normed Vector Space",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Derivative"
] | [
"Resolvent Identity",
"Operator Norm is Norm",
"Resolvent Mapping is Continuous",
"Norm is Continuous"
] |
proofwiki-19049 | Resolvent Mapping is Analytic/Bounded Linear Operator | Let $B$ be a Banach space.
Let $\map \LL {B, B}$ be the set of bounded linear operators from $B$ to itself.
Let $T \in \map \LL {B, B}$.
Let $\map \rho T$ be the resolvent set of $T$ in the complex plane.
Then the resolvent mapping $f : \map \rho T \to \map \LL {B, B}$ given by $\map f z = \paren {T - z I}^{-1}$ is ana... | If $B=\set {\mathbf 0_B}$, the statement is trivial, since $\map f z = \mathbf 0_{\map \LL {B, B}}$ for all $z \in \C$.
We assume that $B \ne \set {\mathbf 0_B}$.
Especially, for all $z \in \map \rho T$:
:$\map f z \ne \mathbf 0_{\map \LL {B, B}}$
since:
:$\paren {T - z I} \map f z = I \ne \mathbf 0_{\map \LL {B, B}}$
... | Let $B$ be a [[Definition:Banach Space|Banach space]].
Let $\map \LL {B, B}$ be the set of [[Definition:Bounded Linear Operator on Normed Vector Space|bounded linear operators]] from $B$ to itself.
Let $T \in \map \LL {B, B}$.
Let $\map \rho T$ be the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent s... | If $B=\set {\mathbf 0_B}$, the statement is trivial, since $\map f z = \mathbf 0_{\map \LL {B, B}}$ for all $z \in \C$.
We assume that $B \ne \set {\mathbf 0_B}$.
Especially, for all $z \in \map \rho T$:
:$\map f z \ne \mathbf 0_{\map \LL {B, B}}$
since:
:$\paren {T - z I} \map f z = I \ne \mathbf 0_{\map \LL {B, B}}... | Resolvent Mapping is Analytic/Bounded Linear Operator/Proof 2 | https://proofwiki.org/wiki/Resolvent_Mapping_is_Analytic/Bounded_Linear_Operator | https://proofwiki.org/wiki/Resolvent_Mapping_is_Analytic/Bounded_Linear_Operator/Proof_2 | [
"Resolvent Mapping is Analytic",
"Functional Analysis"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator/Normed Vector Space",
"Definition:Resolvent Set/Bounded Linear Operator",
"Definition:Analytic Function/Banach Space Valued Function",
"Definition:Derivative"
] | [
"Definition:Complex Disk/Open",
"Neumann Series",
"Inverse of Product",
"Definition:Neighborhood (Complex Analysis)",
"Definition:Analytic Function/Banach Space Valued Function",
"Derivative of Power Series/Banach Valued Function",
"Definition:Neighborhood (Complex Analysis)"
] |
proofwiki-19050 | Numbers for which Euler Phi Function equals Product of Digits | The sequence of positive integers $n$ for which $\map \phi n$ is equal to the product of the digits of $n$ begins:
:$1, 24, 26, 87, 168, 388, 594, 666, 1998, 2688, 5698, 5978, 6786, 7888, 68 \, 796$
{{OEIS|A058627}}
It is known that this sequence is finite, but it is unknown whether a $16^{\text {th}}$ term exists. | {{begin-eqn}}
{{eqn | l = \map \phi 1
| r = 1
| c = {{EulerPhiLink|1}}
}}
{{eqn | l = \map \phi {24}
| r = 8
| c = {{EulerPhiLink|24}}
}}
{{eqn | r = 2 \times 4
}}
{{eqn | l = \map \phi {26}
| r = 12
| c = {{EulerPhiLink|26}}
}}
{{eqn | r = 2 \times 6
}}
{{eqn | l = \map \phi {87}
... | The [[Definition:Integer Sequence|sequence]] of [[Definition:Positive Integer|positive integers]] $n$ for which $\map \phi n$ is equal to the [[Definition:Multiplication|product]] of the [[Definition:Digit|digits]] of $n$ begins:
:$1, 24, 26, 87, 168, 388, 594, 666, 1998, 2688, 5698, 5978, 6786, 7888, 68 \, 796$
{{OEIS... | {{begin-eqn}}
{{eqn | l = \map \phi 1
| r = 1
| c = {{EulerPhiLink|1}}
}}
{{eqn | l = \map \phi {24}
| r = 8
| c = {{EulerPhiLink|24}}
}}
{{eqn | r = 2 \times 4
}}
{{eqn | l = \map \phi {26}
| r = 12
| c = {{EulerPhiLink|26}}
}}
{{eqn | r = 2 \times 6
}}
{{eqn | l = \map \phi {87}
... | Numbers for which Euler Phi Function equals Product of Digits | https://proofwiki.org/wiki/Numbers_for_which_Euler_Phi_Function_equals_Product_of_Digits | https://proofwiki.org/wiki/Numbers_for_which_Euler_Phi_Function_equals_Product_of_Digits | [
"Euler Phi Function"
] | [
"Definition:Integer Sequence",
"Definition:Positive/Integer",
"Definition:Multiplication",
"Definition:Digit",
"Definition:Integer Sequence",
"Definition:Finite Sequence"
] | [
"Category:Euler Phi Function"
] |
proofwiki-19051 | Linear Isometry is Injective | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.
Let $T : X \to Y$ be a linear isometry.
Then $T$ is injective. | Let $x, y \in X$.
We have:
:$\norm {\map T {x - y} }_Y = \norm {T x - T y}_Y$
from the definition of a linear transformation.
Since $T$ is a linear isometry, we have:
:$\norm {\map T {x - y} }_Y = \norm {x - y}_X$
So:
:$\norm {T x - T y}_Y = 0$
{{iff}}:
:$\norm {x - y}_X = 0$
Since the norm is positive definite, thi... | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $T : X \to Y$ be a [[Definition:Linear Isometry|linear isometry]].
Then $T$ is [[Definition:Injection|injective]]. | Let $x, y \in X$.
We have:
:$\norm {\map T {x - y} }_Y = \norm {T x - T y}_Y$
from the definition of a [[Definition:Linear Transformation|linear transformation]].
Since $T$ is a [[Definition:Linear Isometry|linear isometry]], we have:
:$\norm {\map T {x - y} }_Y = \norm {x - y}_X$
So:
:$\norm {T x - T y}_Y = ... | Linear Isometry is Injective | https://proofwiki.org/wiki/Linear_Isometry_is_Injective | https://proofwiki.org/wiki/Linear_Isometry_is_Injective | [
"Linear Isometries",
"Linear Isometry is Injective"
] | [
"Definition:Normed Vector Space",
"Definition:Linear Isometry",
"Definition:Injection"
] | [
"Definition:Linear Transformation",
"Definition:Linear Isometry",
"Definition:Norm/Vector Space",
"Definition:Positive Definite (Ring)",
"Definition:Injection",
"Category:Linear Isometries",
"Category:Linear Isometry is Injective"
] |
proofwiki-19052 | Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Open Convex Set and Convex Set | Let $A \subseteq X$ be an open convex set.
Let $B \subseteq X$ be a convex set disjoint from $A$.
Then there exists $f \in X^\ast$ and $c \in \R$ such that:
:$A \subseteq \set {x \in X : \map f x < c}$
and:
:$B \subseteq \set {x \in X : \map f x \ge c}$
That is:
:there exists $f \in X^\ast$ and $c \in \R$ such that ... | Let $a_0 \in A$ and $b_0 \in B$.
Let:
:$v_0 = b_0 - a_0$
Let:
:$C = v_0 + A - B = \set {v_0 + a - b : a \in A, \, b \in B}$ | Let $A \subseteq X$ be an [[Definition:Open Set in Normed Vector Space|open]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $B \subseteq X$ be a [[Definition:Convex Set (Vector Space)|convex set]] [[Definition:Disjoint Sets|disjoint]] from $A$.
Then there exists $f \in X^\ast$ and $c \in \R$ such that:
... | Let $a_0 \in A$ and $b_0 \in B$.
Let:
:$v_0 = b_0 - a_0$
Let:
:$C = v_0 + A - B = \set {v_0 + a - b : a \in A, \, b \in B}$ | Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Open Convex Set and Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Real_Case/Open_Convex_Set_and_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Real_Case/Open_Convex_Set_and_Convex_Set | [
"Hahn-Banach Separation Theorem",
"Convex Sets (Vector Spaces)"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [] |
proofwiki-19053 | Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Compact Convex Set and Closed Convex Set | Let $A$ be a compact convex set.
Let $B$ be a closed convex set disjoint from $A$.
Then there exists $f \in X^\ast$, $c \in \R$ and $\epsilon > 0$ such that:
:$A \subseteq \set {x \in X : \map f x \le c - \epsilon}$
and:
:$B \subseteq \set {x \in X : \map f x \ge c + \epsilon}$
That is:
:there exists $f \in X^\ast$, ... | Let:
:$\delta = \dfrac 1 4 \inf \set {\norm {a - b} : a \in A, \, b \in B}$
so that:
:$\map d {A, B} = 4 \delta$
where $d$ is the metric induced by $\norm {\, \cdot \,}$ and $\map d {A, B}$ is the $d$-distance between $A$ and $B$.
From Distance between Disjoint Compact Set and Closed Set in Metric Space is Positive,... | Let $A$ be a [[Definition:Compact Subset of Normed Vector Space|compact]] [[Definition:Convex Set (Vector Space)|convex set]].
Let $B$ be a [[Definition:Closed Set in Normed Vector Space|closed]] [[Definition:Convex Set (Vector Space)|convex set]] [[Definition:Disjoint Sets|disjoint]] from $A$.
Then there exists $f ... | Let:
:$\delta = \dfrac 1 4 \inf \set {\norm {a - b} : a \in A, \, b \in B}$
so that:
:$\map d {A, B} = 4 \delta$
where $d$ is the [[Definition:Metric Induced by Norm|metric induced by $\norm {\, \cdot \,}$]] and $\map d {A, B}$ is the [[Definition:Distance between Element and Subset of Metric Space|$d$-distance ... | Hahn-Banach Separation Theorem/Normed Vector Space/Real Case/Compact Convex Set and Closed Convex Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Real_Case/Compact_Convex_Set_and_Closed_Convex_Set | https://proofwiki.org/wiki/Hahn-Banach_Separation_Theorem/Normed_Vector_Space/Real_Case/Compact_Convex_Set_and_Closed_Convex_Set | [
"Hahn-Banach Separation Theorem",
"Convex Sets (Vector Spaces)",
"Compact Normed Vector Spaces"
] | [
"Definition:Compact Space/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Closed Set/Normed Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Disjoint Sets"
] | [
"Definition:Metric Induced by Norm",
"Definition:Distance/Sets/Metric Spaces",
"Distance between Disjoint Compact Set and Closed Set in Metric Space is Positive",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Center",
"Definition:Open Ball/Radius",
"Sum of Set and Open Set in Topolog... |
proofwiki-19054 | Translation of Open Set in Normed Vector Space is Open | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $U \subseteq X$ be an open set.
Let $x \in X$.
Then:
:$U + x$ is open. | Let:
:$v \in U + x$
then:
:$v = u + x$
for some $u \in U$.
So:
:$v - x \in U$
Since $U$ is open, there exists $\epsilon > 0$ such that whenever $v' \in X$ and:
:$\norm {\paren {v - x} - \paren {v' - x} } < \epsilon$
we have $v' - x \in U$.
That is:
:$v' \in U + x$
Note that:
:$\norm {\paren {v - x} - \paren {v' - ... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $U \subseteq X$ be an [[Definition:Open Set in Normed Vector Space|open set]].
Let $x \in X$.
Then:
:$U + x$ is [[Definition:Open Set in Normed Vector Space|open]]. | Let:
:$v \in U + x$
then:
:$v = u + x$
for some $u \in U$.
So:
:$v - x \in U$
Since $U$ is [[Definition:Open Set in Normed Vector Space|open]], there exists $\epsilon > 0$ such that whenever $v' \in X$ and:
:$\norm {\paren {v - x} - \paren {v' - x} } < \epsilon$
we have $v' - x \in U$.
That is:
:$v' \in ... | Translation of Open Set in Normed Vector Space is Open | https://proofwiki.org/wiki/Translation_of_Open_Set_in_Normed_Vector_Space_is_Open | https://proofwiki.org/wiki/Translation_of_Open_Set_in_Normed_Vector_Space_is_Open | [
"Open Sets (Normed Vector Spaces)",
"Translation of Subsets of Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Category:Open Sets (Normed Vector Spaces)",
"Category:Translation of Subsets of Vector Spaces"
] |
proofwiki-19055 | Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below/Lemma 2 | :$\ds \forall r \in \N_{>0}: \tuple {n, m} = \tuple {n + r, m + r} + \sum_{k \mathop = 1}^r \tuple {n + k, m + k - 1}$
That is, each number in the Leibniz harmonic triangle is equal to the sum of the number below it, $\paren {r - 1}$ numbers diagonally below that number, and the number to the right of the last number. | Proof by induction:
For all $r \in \N_{>0}$, let $\map P r$ be the proposition:
:$\ds \tuple {n, m} = \tuple {n + r, m + r} + \sum_{k \mathop = 1}^r \tuple {n + k, m + k - 1}$ | :$\ds \forall r \in \N_{>0}: \tuple {n, m} = \tuple {n + r, m + r} + \sum_{k \mathop = 1}^r \tuple {n + k, m + k - 1}$
That is, each number in the [[Definition:Leibniz Harmonic Triangle|Leibniz harmonic triangle]] is equal to the sum of the number below it, $\paren {r - 1}$ numbers diagonally below that number, and th... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $r \in \N_{>0}$, let $\map P r$ be the [[Definition:Proposition|proposition]]:
:$\ds \tuple {n, m} = \tuple {n + r, m + r} + \sum_{k \mathop = 1}^r \tuple {n + k, m + k - 1}$ | Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below/Lemma 2 | https://proofwiki.org/wiki/Element_of_Leibniz_Harmonic_Triangle_as_Sum_of_Elements_on_Diagonal_from_Below/Lemma_2 | https://proofwiki.org/wiki/Element_of_Leibniz_Harmonic_Triangle_as_Sum_of_Elements_on_Diagonal_from_Below/Lemma_2 | [
"Leibniz Harmonic Triangle",
"Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below"
] | [
"Definition:Leibniz Harmonic Triangle"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-19056 | Convex Hull is Smallest Convex Set containing Set/Corollary | Let $K \subseteq X$ be non-empty.
Then:
:$K$ is convex {{iff}} $\map {\operatorname {conv} } K = K$ | === Sufficient Condition ===
Suppose that:
:$\map {\operatorname {conv} } K = K$
From Convex Hull is Smallest Convex Set containing Set, we have:
:$\map {\operatorname {conv} } K$ is convex.
So:
:$K$ is convex.
{{qed|lemma}} | Let $K \subseteq X$ be [[Definition:Non-Empty Set|non-empty]].
Then:
:$K$ is [[Definition:Convex Set (Vector Space)|convex]] {{iff}} $\map {\operatorname {conv} } K = K$ | === Sufficient Condition ===
Suppose that:
:$\map {\operatorname {conv} } K = K$
From [[Convex Hull is Smallest Convex Set containing Set]], we have:
:$\map {\operatorname {conv} } K$ is [[Definition:Convex Set (Vector Space)|convex]].
So:
:$K$ is [[Definition:Convex Set (Vector Space)|convex]].
{{qed|lemma}} | Convex Hull is Smallest Convex Set containing Set/Corollary | https://proofwiki.org/wiki/Convex_Hull_is_Smallest_Convex_Set_containing_Set/Corollary | https://proofwiki.org/wiki/Convex_Hull_is_Smallest_Convex_Set_containing_Set/Corollary | [
"Convex Hull is Smallest Convex Set containing Set"
] | [
"Definition:Non-Empty Set",
"Definition:Convex Set (Vector Space)"
] | [
"Convex Hull is Smallest Convex Set containing Set",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Convex Hull is Smallest Convex Set containing Set",
"Definition:Convex Set (Vector Space)",
"Convex Hull is Smallest Convex Set co... |
proofwiki-19057 | Convex Hull preserves Subsets | Let $X$ be a vector space over $\R$.
Let $U, V \subseteq X$ be non-empty with $U \subseteq V$.
Then:
:$\map {\operatorname {conv} } U \subseteq \map {\operatorname {conv} } V$
where $\operatorname {conv}$ denotes the convex hull. | Let:
:$u \in \map {\operatorname {conv} } U$
From the definition of the convex hull, there exists $u_1, u_2, \ldots, u_n \in U$ and $\lambda_1, \lambda_2, \ldots, \lambda_n \in \R_{> 0}$ with:
:$\ds \sum_{i \mathop = 1}^n \lambda_i = 1$
such that:
:$\ds u = \sum_{i \mathop = 1}^n \lambda_i u_i$
Since $U \subseteq V$... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$.
Let $U, V \subseteq X$ be [[Definition:Non-Empty Set|non-empty]] with $U \subseteq V$.
Then:
:$\map {\operatorname {conv} } U \subseteq \map {\operatorname {conv} } V$
where $\operatorname {conv}$ denotes the [[Definition:Convex Hull|convex hull]]. | Let:
:$u \in \map {\operatorname {conv} } U$
From the definition of the [[Definition:Convex Hull|convex hull]], there exists $u_1, u_2, \ldots, u_n \in U$ and $\lambda_1, \lambda_2, \ldots, \lambda_n \in \R_{> 0}$ with:
:$\ds \sum_{i \mathop = 1}^n \lambda_i = 1$
such that:
:$\ds u = \sum_{i \mathop = 1}^n \lam... | Convex Hull preserves Subsets | https://proofwiki.org/wiki/Convex_Hull_preserves_Subsets | https://proofwiki.org/wiki/Convex_Hull_preserves_Subsets | [
"Convex Hulls"
] | [
"Definition:Vector Space",
"Definition:Non-Empty Set",
"Definition:Convex Hull"
] | [
"Definition:Convex Hull",
"Definition:Subset",
"Category:Convex Hulls"
] |
proofwiki-19058 | Evaluation Linear Transformation on Normed Vector Space is Linear Isometry | Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm \cdot_X}$.
Let $J : X \to X^{\ast \ast}$ be the evaluation linear transformation for $X$.
Then:
:$J$ is a linear isometry. | From Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual, we have that:
:$J$ is a linear transformation.
It remains to show that:
:$\norm {\map J x}_{X^{\ast \ast} } = \norm x_X$
for each $x \in X$.
For each $x \in X$, we denote:
:$\map J x = x^\wedge$
We ... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm \cdot_X}$.
Let $J : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Tr... | From [[Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual]], we have that:
:$J$ is a [[Definition:Linear Transformation|linear transformation]].
It remains to show that:
:$\norm {\map J x}_{X^{\ast \ast} } = \norm x_X$
for each $x \in X$.
For each $x ... | Evaluation Linear Transformation on Normed Vector Space is Linear Isometry | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Linear_Isometry | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Linear_Isometry | [
"Linear Isometries",
"Evaluation Linear Transformations (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Linear Isometry"
] | [
"Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual",
"Definition:Linear Transformation",
"Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual",
"Definition:Norm/Bounded Linear Function... |
proofwiki-19059 | Pointwise Limit of Sequence of Bounded Linear Transformations from Banach Space to Normed Vector Space is Bounded Linear Transformation | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\Bbb F$.
Let $\struct {Y, \norm \cdot_Y}$ be a normed vector space over $\Bbb F$.
Let $T : X \to Y$ be a function.
Let $\sequence {T_n}_{n \mathop \in \N}$ be a sequence of bounded linear transformations such that:
:$\ds T x = \l... | We first show that $T$ is a linear transformation.
Let $\alpha, \beta \in \Bbb F$ and $x, y \in X$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map T {\alpha x + \beta y}
| r = \lim_{n \mathop \to \infty} \map {T_n} {\alpha x + \beta y}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren {\alpha T_n x + \beta T_n y}
... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Banach Space|Banach space]] over $\Bbb F$.
Let $\struct {Y, \norm \cdot_Y}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $T : X \to Y$ be a [[Definition:Function|function]].
Let $\sequence {T_n}_{n ... | We first show that $T$ is a [[Definition:Linear Transformation|linear transformation]].
Let $\alpha, \beta \in \Bbb F$ and $x, y \in X$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map T {\alpha x + \beta y}
| r = \lim_{n \mathop \to \infty} \map {T_n} {\alpha x + \beta y}
}}
{{eqn | r = \lim_{n \mathop \to \infty... | Pointwise Limit of Sequence of Bounded Linear Transformations from Banach Space to Normed Vector Space is Bounded Linear Transformation | https://proofwiki.org/wiki/Pointwise_Limit_of_Sequence_of_Bounded_Linear_Transformations_from_Banach_Space_to_Normed_Vector_Space_is_Bounded_Linear_Transformation | https://proofwiki.org/wiki/Pointwise_Limit_of_Sequence_of_Bounded_Linear_Transformations_from_Banach_Space_to_Normed_Vector_Space_is_Bounded_Linear_Transformation | [
"Bounded Linear Transformations",
"Linear Transformations on Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Normed Vector Space",
"Definition:Function",
"Definition:Sequence",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Bounded Linear Transformation",
"Banach-Steinhaus Theorem",
"Modulus of Limit/Normed Vector Space",
"Convergent Real Sequence is Bounded",
"Definition:Bounded Sequence/Complex",
"Definition:Finite Extended Real Number"... |
proofwiki-19060 | Principle of Condensation of Singularities | Let $\struct {X, \norm {\,\cdot\,}_X}$ be a Banach space.
Let $\struct {Y, \norm {\,\cdot\,}_Y}$ be a normed vector space.
Let $\family {T_\alpha: X \to Y}_{\alpha \mathop \in A}$ be an $A$-indexed family of bounded linear transformations from $X$ to $Y$ such that:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha} = \infty$
... | {{AimForCont}} that:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha x} < \infty$ for each $x \in X$.
From the Banach-Steinhaus Theorem, we have:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha} < \infty$.
contradicting that:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha} = \infty$
So, we have:
:$\ds \sup_{\alpha \in A} \norm {T_\... | Let $\struct {X, \norm {\,\cdot\,}_X}$ be a [[Definition:Banach Space|Banach space]].
Let $\struct {Y, \norm {\,\cdot\,}_Y}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\family {T_\alpha: X \to Y}_{\alpha \mathop \in A}$ be an [[Definition:Indexed Family|$A$-indexed family]] of [[Definition:Boun... | {{AimForCont}} that:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha x} < \infty$ for each $x \in X$.
From the [[Banach-Steinhaus Theorem]], we have:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha} < \infty$.
contradicting that:
:$\ds \sup_{\alpha \in A} \norm {T_\alpha} = \infty$
So, we have:
:$\ds \sup_{\alpha \in A}... | Principle of Condensation of Singularities | https://proofwiki.org/wiki/Principle_of_Condensation_of_Singularities | https://proofwiki.org/wiki/Principle_of_Condensation_of_Singularities | [
"Linear Transformations on Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Normed Vector Space",
"Definition:Indexing Set/Family",
"Definition:Bounded Linear Transformation"
] | [
"Banach-Steinhaus Theorem"
] |
proofwiki-19061 | Banach-Schauder Theorem | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be Banach spaces.
Let $T : X \to Y$ be a surjective bounded linear transformation.
Then $T$ is an open mapping. | For each $x \in X$ and $r > 0$, let $\map {B_X} {x, r}$ be the open ball in $\struct {X, \norm \cdot_X}$ with centre $x$ and radius $r$.
For each $y \in Y$ and $r > 0$, let $\map {B_Y} {y, r}$ be the open ball in $\struct {Y, \norm \cdot_Y}$ with centre $y$ and radius $r$.
Note that we can write:
:$\ds X = \bigcup_{n ... | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be [[Definition:Banach Space|Banach spaces]].
Let $T : X \to Y$ be a [[Definition:Surjection|surjective]] [[Definition:Bounded Linear Transformation|bounded linear transformation]].
Then $T$ is an [[Definition:Open Mapping|open mapping]]. | For each $x \in X$ and $r > 0$, let $\map {B_X} {x, r}$ be the [[Definition:Open Ball in Normed Vector Space|open ball]] in $\struct {X, \norm \cdot_X}$ with centre $x$ and radius $r$.
For each $y \in Y$ and $r > 0$, let $\map {B_Y} {y, r}$ be the [[Definition:Open Ball in Normed Vector Space|open ball]] in $\struct {... | Banach-Schauder Theorem | https://proofwiki.org/wiki/Banach-Schauder_Theorem | https://proofwiki.org/wiki/Banach-Schauder_Theorem | [
"Linear Transformations on Banach Spaces",
"Banach-Schauder Theorem",
"Open Mappings"
] | [
"Definition:Banach Space",
"Definition:Surjection",
"Definition:Bounded Linear Transformation",
"Definition:Open Mapping"
] | [
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Image of Union under Mapping/General Result",
"Definition:Surjection",
"Definition:Banach Space",
"Baire Category Theorem",
"Definition:Baire Space (Topology)",
"Baire Space is Non-Meager",
"Definition:Meager Sp... |
proofwiki-19062 | Sine n x over Pi x Delta Sequence | thumb500pxThe graph of the $\ds \frac {\map \sin {nx}} {\pi x}$ delta sequence. As $n$ grows, the central peak becomes thinner and taller. The area between each curve and the horizontal axis is equal to $1$. Note that that the area below the line contribute negatively.
Let $\sequence {\map {\delta_n} x}$ be a sequence ... | From Integral to Infinity of Sine p x over x for $n \in \N_{> 0}$ we have that:
:$\ds \int_0^\infty \frac {\map \sin {n x} } x \rd x = \frac \pi 2$
Furthermore:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\map \sin {n x} } x \rd x
| r = \int_0^{-\infty} \frac {\map \sin {-n y} } {-y} \paren {- \rd y}
| c... | [[File:SinnxOverPixDeltaSequence.png|thumb|500px|The graph of the $\ds \frac {\map \sin {nx}} {\pi x}$ delta sequence. As $n$ grows, the central peak becomes thinner and taller. The area between each curve and the horizontal axis is equal to $1$. Note that that the area below the line contribute negatively.]]
Let $\seq... | From [[Integral to Infinity of Sine p x over x]] for $n \in \N_{> 0}$ we have that:
:$\ds \int_0^\infty \frac {\map \sin {n x} } x \rd x = \frac \pi 2$
Furthermore:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\map \sin {n x} } x \rd x
| r = \int_0^{-\infty} \frac {\map \sin {-n y} } {-y} \paren {- \rd y}
... | Sine n x over Pi x Delta Sequence | https://proofwiki.org/wiki/Sine_n_x_over_Pi_x_Delta_Sequence | https://proofwiki.org/wiki/Sine_n_x_over_Pi_x_Delta_Sequence | [
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:SinnxOverPixDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Integral to Infinity of Sine p x over x",
"Integration by Substitution/Definite Integral",
"Integration by Substitution/Definite Integral",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Mean Value Theorem for Integra... |
proofwiki-19063 | Sylvester's Law of Inertia | Let $\struct {V, q}$ be the scalar product space.
Let $V^*$ be the algebraic dual of $V$.
Suppose there exists a basis $\tuple {\beta^i}$ for $V^*$ such that $q$ is expressible as:
:$q = \paren {\beta^1}^2 + \ldots + \paren {\beta^r}^2 - \paren {\beta^{r + 1} }^2 - \ldots - \paren {\beta^{r + s} }^2$
where:
:$r, s \in ... | {{ProofWanted}}
{{Namedfor|James Joseph Sylvester|cat = Sylvester}} | Let $\struct {V, q}$ be the [[Definition:Scalar Product Space|scalar product space]].
Let $V^*$ be the [[Definition:Vector Algebraic Dual|algebraic dual]] of $V$.
Suppose there exists a [[Definition:Basis of Vector Space|basis]] $\tuple {\beta^i}$ for $V^*$ such that $q$ is expressible as:
:$q = \paren {\beta^1}^2 +... | {{ProofWanted}}
{{Namedfor|James Joseph Sylvester|cat = Sylvester}} | Sylvester's Law of Inertia | https://proofwiki.org/wiki/Sylvester's_Law_of_Inertia | https://proofwiki.org/wiki/Sylvester's_Law_of_Inertia | [
"Sylvester's Law of Inertia",
"Scalar Product Spaces"
] | [
"Definition:Scalar Product Space",
"Definition:Algebraic Dual/Vector Space",
"Definition:Basis of Vector Space",
"Definition:Greatest Element",
"Definition:Dimension (Linear Algebra)",
"Definition:Vector Subspace",
"Definition:Restriction/Mapping",
"Definition:Positive Definite Matrix",
"Definition:... | [] |
proofwiki-19064 | Closed Graph Theorem/Banach Space | Let $\struct {X, \norm {\,\cdot\,}_X}$ and $\struct {Y, \norm {\,\cdot\,}_Y}$ be Banach spaces.
Let $T : X \to Y$ be a linear transformation.
Let $G_T \subseteq X \times Y$ be the graph of $T$.
Suppose that:
:$G_T$ is closed in the direct product $X \times Y$ equipped with the direct product norm $\norm \cdot_{X \tim... | Let $\norm {\,\cdot\,}_{G_T}$ be the restriction of $\norm \cdot_{X \times Y}$ to $G_T$.
By Closed Subspace of Banach Space forms Banach Space, $\struct {G_T, \norm \cdot_{G_T} }$ is a Banach space.
Define $\pi_X : G_T \to X$ by:
:$\map {\pi_X} {x, y} = x$ for each $\tuple {x, y} \in X \times Y$.
Note that by the de... | Let $\struct {X, \norm {\,\cdot\,}_X}$ and $\struct {Y, \norm {\,\cdot\,}_Y}$ be [[Definition:Banach Space|Banach spaces]].
Let $T : X \to Y$ be a [[Definition:Linear Transformation|linear transformation]].
Let $G_T \subseteq X \times Y$ be the [[Definition:Graph of Mapping|graph]] of $T$.
Suppose that:
:$G_T$ is... | Let $\norm {\,\cdot\,}_{G_T}$ be the [[Definition:Restriction of Mapping|restriction]] of $\norm \cdot_{X \times Y}$ to $G_T$.
By [[Closed Subspace of Banach Space forms Banach Space]], $\struct {G_T, \norm \cdot_{G_T} }$ is a [[Definition:Banach Space|Banach space]].
Define $\pi_X : G_T \to X$ by:
:$\map {\pi_X} ... | Closed Graph Theorem/Banach Space | https://proofwiki.org/wiki/Closed_Graph_Theorem/Banach_Space | https://proofwiki.org/wiki/Closed_Graph_Theorem/Banach_Space | [
"Linear Transformations on Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Linear Transformation",
"Definition:Graph of Mapping",
"Definition:Closed Set/Normed Vector Space",
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Restriction/Mapping",
"Closed Subspace of Banach Space forms Banach Space",
"Definition:Banach Space",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Projections on Direct Product of Normed Vector Spaces define Bounded Linear Transformations",
"Definition:Bounded Linear Transformatio... |
proofwiki-19065 | Hellinger-Toeplitz Theorem | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $T : \HH \to \HH$ be a Hermitian operator.
That is:
:$\innerprod {T x} y = \innerprod x {T y}$ for each $x, y \in \HH$.
Then:
:$T$ is bounded. | Let $\struct {\HH \times \HH, \norm \cdot_{\HH \times \HH} }$ be the direct product of $\HH$ with itself, with the direct product norm.
From the Closed Graph Theorem, it suffices to show that:
:$G_T = \set {\tuple {x, T x} \in \HH \times \HH : x \in \HH}$
is closed in $\struct {\HH \times \HH, \norm \cdot_{\HH \times ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $T : \HH \to \HH$ be a [[Definition:Hermitian Operator|Hermitian operator]].
That is:
:$\innerprod {T x} y = \innerprod x {T y}$ for each $x, y \in \HH$.
Then:
:$T$ is [[Definition:Bounded Linear Transformation|boun... | Let $\struct {\HH \times \HH, \norm \cdot_{\HH \times \HH} }$ be the [[Definition:Direct Product of Vector Spaces|direct product]] of $\HH$ with itself, with the [[Definition:Direct Product Norm|direct product norm]].
From the [[Closed Graph Theorem]], it suffices to show that:
:$G_T = \set {\tuple {x, T x} \in \HH ... | Hellinger-Toeplitz Theorem/Proof 1 | https://proofwiki.org/wiki/Hellinger-Toeplitz_Theorem | https://proofwiki.org/wiki/Hellinger-Toeplitz_Theorem/Proof_1 | [
"Linear Transformations on Hilbert Spaces",
"Hermitian Operators",
"Hermitian Operators",
"Hellinger-Toeplitz Theorem"
] | [
"Definition:Hilbert Space",
"Definition:Hermitian Operator",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Direct Product of Vector Spaces",
"Definition:Direct Product Norm",
"Closed Graph Theorem",
"Definition:Closed Set/Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Convergence in Direct Product Norm",
"Definition:Convergent Sequence/Normed... |
proofwiki-19066 | Hellinger-Toeplitz Theorem | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.
Let $T : \HH \to \HH$ be a Hermitian operator.
That is:
:$\innerprod {T x} y = \innerprod x {T y}$ for each $x, y \in \HH$.
Then:
:$T$ is bounded. | {{AimForCont}} suppose that $T$ is not bounded.
Then there does not exist $C > 0$ such that:
:$\norm {T x}_\HH \le C$ for each $x \in \HH$ with $\norm x_\HH = 1$.
That is, for each $n \in \N$ there exists $y_n \in \HH$ such that:
:$\norm {T y_n}_\HH \ge n$
with $\norm {y_n}_\HH = 1$.
For each $n \in \N$, define the ... | Let $\struct {\HH, \innerprod \cdot \cdot}$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $T : \HH \to \HH$ be a [[Definition:Hermitian Operator|Hermitian operator]].
That is:
:$\innerprod {T x} y = \innerprod x {T y}$ for each $x, y \in \HH$.
Then:
:$T$ is [[Definition:Bounded Linear Transformation|boun... | {{AimForCont}} suppose that $T$ is not [[Definition:Bounded Linear Transformation|bounded]].
Then there does not exist $C > 0$ such that:
:$\norm {T x}_\HH \le C$ for each $x \in \HH$ with $\norm x_\HH = 1$.
That is, for each $n \in \N$ there exists $y_n \in \HH$ such that:
:$\norm {T y_n}_\HH \ge n$
with $\nor... | Hellinger-Toeplitz Theorem/Proof 2 | https://proofwiki.org/wiki/Hellinger-Toeplitz_Theorem | https://proofwiki.org/wiki/Hellinger-Toeplitz_Theorem/Proof_2 | [
"Linear Transformations on Hilbert Spaces",
"Hermitian Operators",
"Hermitian Operators",
"Hellinger-Toeplitz Theorem"
] | [
"Definition:Hilbert Space",
"Definition:Hermitian Operator",
"Definition:Bounded Linear Transformation"
] | [
"Definition:Bounded Linear Transformation",
"Definition:Linear Functional",
"Riesz Representation Theorem (Hilbert Spaces)",
"Definition:Bounded Linear Functional",
"Principle of Condensation of Singularities",
"Definition:Hermitian Operator",
"Cauchy-Bunyakovsky-Schwarz Inequality",
"Definition:Contr... |
proofwiki-19067 | Characterisation of Terminal P-adic Expansion | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
Let $x \in \Q_p$.
Then:
:the $p$-adic expansion of $x$ terminates
{{iff}}
:$\exists a \in \N : \exists k \in \Z : x = \dfrac a {p^k}$ | === Necessary Condition ===
{{:Characterisation of Terminal P-adic Expansion/Necessary Condition}}{{qed|lemma}} | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] for some [[Definition:Prime Number|prime]] $p$.
Let $x \in \Q_p$.
Then:
:the [[Definition:P-adic Expansion|$p$-adic expansion]] of $x$ [[Definition:Terminal P-adic Expansion|terminates]]
{{iff}}
:$\exist... | === [[Characterisation of Terminal P-adic Expansion/Necessary Condition|Necessary Condition]] ===
{{:Characterisation of Terminal P-adic Expansion/Necessary Condition}}{{qed|lemma}} | Characterisation of Terminal P-adic Expansion | https://proofwiki.org/wiki/Characterisation_of_Terminal_P-adic_Expansion | https://proofwiki.org/wiki/Characterisation_of_Terminal_P-adic_Expansion | [
"P-adic Number Theory",
"Characterisation of Terminal P-adic Expansion"
] | [
"Definition:Valued Field of P-adic Numbers",
"Definition:Prime Number",
"Definition:P-adic Expansion",
"Definition:Terminal P-adic Expansion"
] | [
"Characterisation of Terminal P-adic Expansion/Necessary Condition"
] |
proofwiki-19068 | Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\Bbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual of $\struct {X, \norm \cdot_X}$.
Let $\map L {X^\ast, \Bbb F}$ be the set of linear functionals on $X^\ast$.
For each $x \in X$, define $x^\wedge : ... | We first show that $J$ is a linear transformation.
Let $x, y \in X$ and $\alpha, \beta \in \Bbb F$.
Then, we have, for each $f \in X^\ast$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map J {\alpha x + \beta y} } } f
| r = \map {\paren {\alpha x + \beta y}^\wedge} f
}}
{{eqn | r = \map f {\alpha x + \beta y}
}}
{{eqn | ... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual]] of $\struct {X, \norm \cdot_X}$.
Let $\map L {X^\ast, \Bbb F}$ be the set of [... | We first show that $J$ is a [[Definition:Linear Transformation|linear transformation]].
Let $x, y \in X$ and $\alpha, \beta \in \Bbb F$.
Then, we have, for each $f \in X^\ast$:
{{begin-eqn}}
{{eqn | l = \map {\paren {\map J {\alpha x + \beta y} } } f
| r = \map {\paren {\alpha x + \beta y}^\wedge} f
}}
{{eqn | r =... | Evaluation Linear Transformation on Normed Vector Space is Linear Transformation from Space to Second Normed Dual | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Linear_Transformation_from_Space_to_Second_Normed_Dual | https://proofwiki.org/wiki/Evaluation_Linear_Transformation_on_Normed_Vector_Space_is_Linear_Transformation_from_Space_to_Second_Normed_Dual | [
"Second Normed Duals",
"Evaluation Linear Transformations (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Normed Dual Space",
"Definition:Linear Functional",
"Definition:Linear Transformation",
"Definition:Second Normed Dual"
] | [
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Second Normed Dual",
"Definition:Linear Functional",
"Definition:Linear Functional",
"Definition:Bounded Linear Functional",
"Fundamental Property of Norm on Bounded Linear Functional",
"Definition:Bounded Linear Funct... |
proofwiki-19069 | Hensel's Lemma/P-adic Integers | Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a polynomial.
Let $\map {F'} X$ be the (formal) derivative of $F$.
Let $p\Z_p$ denote the principal ideal of $\Z_p$ generated by $p$.
For all $x,y \in \Z_p$, let:
:$x \equiv y \pmod {p\Z_p}$
denote congruence modulo the principa... | === Lemma 1 ===
{{:Hensel's Lemma/P-adic Integers/Lemma 1}}{{qed|lemma}}
Let:
:$\ds \alpha = \sum_{n \mathop = 0}^\infty d_n p^n$ | Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]] for some [[Definition:Prime Number|prime]] $p$.
Let $\map F X \in \Z_p \sqbrk X$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]].
Let $\map {F'} X$ be the [[Definition:Formal Derivative of Polynomial|(formal) derivative]] of $F$.
Let $p... | === [[Hensel's Lemma/P-adic Integers/Lemma 1|Lemma 1]] ===
{{:Hensel's Lemma/P-adic Integers/Lemma 1}}{{qed|lemma}}
Let:
:$\ds \alpha = \sum_{n \mathop = 0}^\infty d_n p^n$ | Hensel's Lemma/P-adic Integers | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers | https://proofwiki.org/wiki/Hensel's_Lemma/P-adic_Integers | [
"Hensel's Lemma"
] | [
"Definition:P-adic Integer",
"Definition:Prime Number",
"Definition:Polynomial over Ring",
"Definition:Formal Derivative of Polynomial",
"Definition:Principal Ideal of Ring",
"Definition:Congruence Modulo Ideal",
"Definition:Principal Ideal of Ring",
"Definition:P-adic Integer",
"Definition:Unique"
... | [
"Hensel's Lemma/P-adic Integers/Lemma 1"
] |
proofwiki-19070 | Normed Vector Space is Reflexive iff Surjective Evaluation Linear Transformation | Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm \cdot_X}$.
Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation.
Then $X$ is reflexive {{iff}}:
:$\iota$ is surjective.
That is:
:... | From the definition of a reflexive space, we have that $X$ is reflexive {{iff}}:
:$\iota$ is an isometric isomorphism.
From Evaluation Linear Transformation on Normed Vector Space is Linear Isometry, we have:
:$\iota$ is a linear isometry.
From Linear Isometry is Injective: Corollary, we then have:
:$\iota$ is an iso... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm \cdot_X}$.
Let $\iota : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linea... | From the definition of a [[Definition:Reflexive Space|reflexive space]], we have that $X$ is [[Definition:Reflexive Space|reflexive]] {{iff}}:
:$\iota$ is an [[Definition:Isometric Isomorphism on Normed Vector Space|isometric isomorphism]].
From [[Evaluation Linear Transformation on Normed Vector Space is Linear Isom... | Normed Vector Space is Reflexive iff Surjective Evaluation Linear Transformation | https://proofwiki.org/wiki/Normed_Vector_Space_is_Reflexive_iff_Surjective_Evaluation_Linear_Transformation | https://proofwiki.org/wiki/Normed_Vector_Space_is_Reflexive_iff_Surjective_Evaluation_Linear_Transformation | [
"Reflexive Spaces",
"Evaluation Linear Transformations (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Reflexive Space",
"Definition:Surjection"
] | [
"Definition:Reflexive Space",
"Definition:Reflexive Space",
"Definition:Isometric Isomorphism/Normed Vector Space",
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Linear Isometry",
"Linear Isometry is Injective/Corollary",
"Definition:Isometric Isomorphism/Norm... |
proofwiki-19071 | Image of Evaluation Linear Transformation on Banach Space is Closed | Let $\struct {X, \norm \cdot_X}$ be a Banach space.
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm \cdot_X}$.
Let $J : X \to X^{\ast \ast}$ be the evaluation linear transformation.
Then:
:$\map J X$ is closed in $X^{\ast \ast}$. | Let $L$ be the limit of a convergent sequence in $\map J X$.
Let $\sequence {j_n}_{n \mathop \in \N}$ be a sequence in $\map J X$ such that:
:$\sequence {j_n}_{n \mathop \in \N}$ converges to $L$.
Note that for each $n \in \N$ there exists $x_n \in X$ such that:
:$j_n = J x_n$
From Evaluation Linear Transformation on... | Let $\struct {X, \norm \cdot_X}$ be a [[Definition:Banach Space|Banach space]].
Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $\struct {X, \norm \cdot_X}$.
Let $J : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Linear Transformation o... | Let $L$ be the [[Definition:Limit of Sequence in Normed Vector Space|limit]] of a [[Definition:Convergent Sequence in Normed Vector Space|convergent sequence]] in $\map J X$.
Let $\sequence {j_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\map J X$ such that:
:$\sequence {j_n}_{n \mathop \in \N}$ ... | Image of Evaluation Linear Transformation on Banach Space is Closed | https://proofwiki.org/wiki/Image_of_Evaluation_Linear_Transformation_on_Banach_Space_is_Closed | https://proofwiki.org/wiki/Image_of_Evaluation_Linear_Transformation_on_Banach_Space_is_Closed | [
"Evaluation Linear Transformations (Normed Vector Spaces)",
"Linear Transformations on Banach Spaces"
] | [
"Definition:Banach Space",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space"
] | [
"Definition:Limit of Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Evaluation Linear Transformation on Normed Vector Space is Linear Isometry",
"Definition:Linear Isometry",
"Definition:B... |
proofwiki-19072 | Weakly Convergent Sequence in Normed Vector Space is Bounded | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a weakly convergent sequence in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is bounded. | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $X$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the second normed dual of $X$.
Let $J : X \to X^{\ast \ast}$ be the evaluation linear transformation on $X$.
Let:
:$x^\wedge = \map J x$
for each $x \in X$.
Consi... | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Weak Convergence (Normed Vector Space)|weakly convergent sequence]] in $X$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is [[Definition:Bounded Sequence in No... | Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the [[Definition:Normed Dual Space|normed dual]] of $X$.
Let $\struct {X^{\ast \ast}, \norm {\, \cdot \,}_{X^{\ast \ast} } }$ be the [[Definition:Second Normed Dual|second normed dual]] of $X$.
Let $J : X \to X^{\ast \ast}$ be the [[Definition:Evaluation Line... | Weakly Convergent Sequence in Normed Vector Space is Bounded | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Normed_Vector_Space_is_Bounded | https://proofwiki.org/wiki/Weakly_Convergent_Sequence_in_Normed_Vector_Space_is_Bounded | [
"Weak Convergence (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Bounded Sequence/Normed Vector Space"
] | [
"Definition:Normed Dual Space",
"Definition:Second Normed Dual",
"Definition:Evaluation Linear Transformation",
"Definition:Sequence",
"Definition:Weak Convergence (Normed Vector Space)",
"Definition:Convergent Sequence/Complex Numbers",
"Convergent Complex Sequence is Bounded",
"Definition:Bounded Se... |
proofwiki-19073 | Equivalence of Definitions of Schauder Basis | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $\set {e_n : n \in \N}$ be a countable subset of $X$.
{{TFAE|def = Schauder Basis}} | === Definition 1 implies Definition 2 ===
Suppose that:
:for each $x \in X$, there exists a unique sequence $\sequence {\map {\alpha_j} x}_{j \mathop \in \N}$ in $\Bbb F$ such that:
::$\ds x = \sum_{j \mathop = 1}^\infty \map {\alpha_j} x e_j$
Clearly $\set {e_n : n \in \N}$ then satisfies $(1)$ of Definition 2.
Supp... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\set {e_n : n \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $X$.
{{TFAE|def = Schauder Basis}} | === Definition 1 implies Definition 2 ===
Suppose that:
:for each $x \in X$, there exists a unique [[Definition:Sequence|sequence]] $\sequence {\map {\alpha_j} x}_{j \mathop \in \N}$ in $\Bbb F$ such that:
::$\ds x = \sum_{j \mathop = 1}^\infty \map {\alpha_j} x e_j$
Clearly $\set {e_n : n \in \N}$ then satisfies... | Equivalence of Definitions of Schauder Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Schauder_Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Schauder_Basis | [
"Schauder Bases"
] | [
"Definition:Normed Vector Space",
"Definition:Countable Set",
"Definition:Subset"
] | [
"Definition:Sequence",
"Definition:Schauder Basis/Definition 2",
"Definition:Sequence",
"Definition:Schauder Basis/Definition 2",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Sequence"
] |
proofwiki-19074 | Resolvent Mapping Converges to 0 at Infinity | Let $B$ be a Banach space.
Let $\map \LL {B, B}$ be the set of bounded linear operators from $B$ to itself.
Let $T \in \map \LL {B, B}$.
Let $\map \rho T$ be the resolvent set of $T$ in the complex plane.
Then the resolvent mapping $f : \map \rho T \to \map \LL {B, B}$ given by $\map f z = \paren {T - z I}^{-1}$ is suc... | Pick $z \in \Bbb C$ such that $\size z > 2 \norm T_*$.
By Operator Norm is Norm:
:$\norm {\dfrac T z}_* = \dfrac {\norm T_*} {\size z} < \dfrac 1 2$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {\map f z}_*
| r = \norm {\paren {T - z I}^{-1} }_*
}}
{{eqn | r = \norm {\paren {z \paren {\dfrac T z - I} }^{-1} }_*
}}
{{eq... | Let $B$ be a [[Definition:Banach Space|Banach space]].
Let $\map \LL {B, B}$ be the set of [[Definition:Bounded Linear Operator on Normed Vector Space|bounded linear operators]] from $B$ to itself.
Let $T \in \map \LL {B, B}$.
Let $\map \rho T$ be the [[Definition:Resolvent Set of Bounded Linear Operator|resolvent s... | Pick $z \in \Bbb C$ such that $\size z > 2 \norm T_*$.
By [[Operator Norm is Norm]]:
:$\norm {\dfrac T z}_* = \dfrac {\norm T_*} {\size z} < \dfrac 1 2$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {\map f z}_*
| r = \norm {\paren {T - z I}^{-1} }_*
}}
{{eqn | r = \norm {\paren {z \paren {\dfrac T z - I} }^{-1} }_*
... | Resolvent Mapping Converges to 0 at Infinity | https://proofwiki.org/wiki/Resolvent_Mapping_Converges_to_0_at_Infinity | https://proofwiki.org/wiki/Resolvent_Mapping_Converges_to_0_at_Infinity | [
"Functional Analysis"
] | [
"Definition:Banach Space",
"Definition:Bounded Linear Operator/Normed Vector Space",
"Definition:Resolvent Set/Bounded Linear Operator"
] | [
"Operator Norm is Norm",
"Norm of Inverse of Constant Times Operator",
"Operator Norm is Norm",
"Invertibility of Identity Minus Operator",
"Triangle Inequality",
"Norm on Bounded Linear Transformation is Submultiplicative",
"Category:Functional Analysis"
] |
proofwiki-19075 | Spectrum of Bounded Linear Operator is Non-Empty | Let $B$ be a Banach space over $\C$.
Let $\map {\mathfrak L} {B, B}$ be the set of bounded linear operators from $B$ to itself.
Let $T \in \map {\mathfrak L} {B, B}$.
Then the spectrum $\map \sigma T$ of $T$ is non-empty. | Let $f: \C \to \map {\mathfrak L} {B, B}$ be the resolvent mapping defined as $\map f z = \paren {T - z I}^{-1}$.
{{AimForCont}} the spectrum of $T$ is empty, so that $\map f z$ is well-defined for all $z \in \C$.
We first show that $\norm {\map f z}_*$ is uniformly bounded by some constant $C$.
Observe that:
:$(1): \... | Let $B$ be a [[Definition:Banach Space|Banach space]] over $\C$.
Let $\map {\mathfrak L} {B, B}$ be the [[Definition:Set|set]] of [[Definition:Bounded Linear Operator on Normed Vector Space|bounded linear operators]] from $B$ to itself.
Let $T \in \map {\mathfrak L} {B, B}$.
Then the [[Definition:Spectrum of Bounde... | Let $f: \C \to \map {\mathfrak L} {B, B}$ be the resolvent mapping defined as $\map f z = \paren {T - z I}^{-1}$.
{{AimForCont}} the [[Definition:Spectrum of Bounded Linear Operator|spectrum]] of $T$ is empty, so that $\map f z$ is well-defined for all $z \in \C$.
We first show that $\norm {\map f z}_*$ is uniforml... | Spectrum of Bounded Linear Operator is Non-Empty | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_is_Non-Empty | https://proofwiki.org/wiki/Spectrum_of_Bounded_Linear_Operator_is_Non-Empty | [
"Bounded Linear Operators",
"Spectra (Bounded Linear Operators)"
] | [
"Definition:Banach Space",
"Definition:Set",
"Definition:Bounded Linear Operator/Normed Vector Space",
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Definition:Non-Empty Set"
] | [
"Definition:Spectrum (Spectral Theory)/Bounded Linear Operator",
"Operator Norm is Norm",
"Invertibility of Identity Minus Operator",
"Triangle Inequality",
"Norm on Bounded Linear Transformation is Submultiplicative",
"Definition:Complex Number/Complex Plane",
"Resolvent Mapping is Continuous",
"Norm... |
proofwiki-19076 | Inner Product Space is Uniformly Convex | Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Then $V$ is uniformly convex. | Let $\norm \cdot$ be the inner product norm for $\struct {V, \innerprod \cdot \cdot}$.
Let $\epsilon > 0$.
Let $x, y \in V$ be such that:
:$\norm x = \norm y = 1$
and:
:$\norm {x - y} > \epsilon$
Then from the Parallelogram Law (Inner Product Space), we have:
:$\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 ... | Let $\struct {V, \innerprod \cdot \cdot}$ be an [[Definition:Inner Product Space|inner product space]].
Then $V$ is [[Definition:Uniformly Convex Normed Vector Space|uniformly convex]]. | Let $\norm \cdot$ be the [[Definition:Inner Product Norm|inner product norm]] for $\struct {V, \innerprod \cdot \cdot}$.
Let $\epsilon > 0$.
Let $x, y \in V$ be such that:
:$\norm x = \norm y = 1$
and:
:$\norm {x - y} > \epsilon$
Then from the [[Parallelogram Law (Inner Product Space)]], we have:
:$\norm {x + ... | Inner Product Space is Uniformly Convex | https://proofwiki.org/wiki/Inner_Product_Space_is_Uniformly_Convex | https://proofwiki.org/wiki/Inner_Product_Space_is_Uniformly_Convex | [
"Inner Product Spaces",
"Uniformly Convex Normed Vector Spaces"
] | [
"Definition:Inner Product Space",
"Definition:Uniformly Convex Normed Vector Space"
] | [
"Definition:Inner Product Norm",
"Parallelogram Law (Inner Product Space)",
"Definition:Positive Homogeneous (Ring)",
"Definition:Vacuous Truth",
"Definition:Positive/Real Number",
"Definition:Uniformly Convex Normed Vector Space"
] |
proofwiki-19077 | Schauder Basis is Linearly Independent | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.
Let $\set {e_n : n \in \N}$ be a Schauder basis for $X$.
Then $\set {e_n : n \in \N}$ is linearly independent. | Suppose that:
:$\ds \sum_{k \mathop = 1}^n \alpha_{i_k} e_{i_k} = 0$
for some $n \in \N$, $i_1, \ldots, i_n \in \N$ and $\alpha_{i_1}, \ldots, \alpha_{i_n} \in \Bbb F$.
Define a sequence $\sequence {\alpha_j}_{j \mathop \in \N}$ in $\Bbb F$ by:
:$\ds \alpha_j = \begin{cases}\alpha_{i_k} & \text { if there exists } k ... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $\set {e_n : n \in \N}$ be a [[Definition:Schauder Basis|Schauder basis]] for $X$.
Then $\set {e_n : n \in \N}$ is [[Definition:Linearly Independent|linearly independent]]. | Suppose that:
:$\ds \sum_{k \mathop = 1}^n \alpha_{i_k} e_{i_k} = 0$
for some $n \in \N$, $i_1, \ldots, i_n \in \N$ and $\alpha_{i_1}, \ldots, \alpha_{i_n} \in \Bbb F$.
Define a [[Definition:Sequence|sequence]] $\sequence {\alpha_j}_{j \mathop \in \N}$ in $\Bbb F$ by:
:$\ds \alpha_j = \begin{cases}\alpha_{i_k} & ... | Schauder Basis is Linearly Independent | https://proofwiki.org/wiki/Schauder_Basis_is_Linearly_Independent | https://proofwiki.org/wiki/Schauder_Basis_is_Linearly_Independent | [
"Schauder Bases",
"Linear Independence"
] | [
"Definition:Normed Vector Space",
"Definition:Schauder Basis",
"Definition:Linearly Independent"
] | [
"Definition:Sequence",
"Definition:Schauder Basis",
"Definition:Linearly Independent"
] |
proofwiki-19078 | N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence | thumb600pxThe graph of the $\ds \frac n \pi \frac 1 {1 + n^2 x^2}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:$\map {\delta_n} x := \frac n \pi \frac 1 {1 + n^2 x^2}$
Then $\sequence {\map {... | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac n \pi \frac 1 {1 + n^2 x^2} \rd x
| r = \int_0^\infty \frac 1 \pi \frac 1 {1 + n^2 x^2} \rd \paren {n x}
}}
{{eqn | r = \int_0^\infty \frac 1 \pi \frac 1 {1 + y^2} \rd y
| c = $n x = y$, Integration by Substitution
}}
{{eqn | r = \frac 1 2
| c = {{Corollar... | [[File:LorDeltaSequence.png|thumb|600px|The graph of the $\ds \frac n \pi \frac 1 {1 + n^2 x^2}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.]]
Let $\sequence {\map {\delta_n} x}$ be a [[Definition:Sequence|sequence]] such that:
:$\map {\delta_n} x := \... | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac n \pi \frac 1 {1 + n^2 x^2} \rd x
| r = \int_0^\infty \frac 1 \pi \frac 1 {1 + n^2 x^2} \rd \paren {n x}
}}
{{eqn | r = \int_0^\infty \frac 1 \pi \frac 1 {1 + y^2} \rd y
| c = $n x = y$, [[Integration by Substitution/Definite Integral|Integration by Substitution... | N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence/Proof 1 | https://proofwiki.org/wiki/N_over_Pi_times_Reciprocal_of_1_Plus_n_Squared_x_Squared_Delta_Sequence | https://proofwiki.org/wiki/N_over_Pi_times_Reciprocal_of_1_Plus_n_Squared_x_Squared_Delta_Sequence/Proof_1 | [
"N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence",
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:LorDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Integration by Substitution/Definite Integral",
"Integration by Substitution/Definite Integral",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Mean Value Theorem for Integrals/Generalization",
"Definition:Arbitrary C... |
proofwiki-19079 | N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence | thumb600pxThe graph of the $\ds \frac n \pi \frac 1 {1 + n^2 x^2}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:$\map {\delta_n} x := \frac n \pi \frac 1 {1 + n^2 x^2}$
Then $\sequence {\map {... | Let $\map g x = \map \phi x - \map \phi 0$.
Then:
:$\ds \int_{- \infty}^\infty \map \phi x \map {\delta_n} x \rd x = \map \phi 0 + \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x$
Let $A \in \R_{> 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x
| r = \int_{- \i... | [[File:LorDeltaSequence.png|thumb|600px|The graph of the $\ds \frac n \pi \frac 1 {1 + n^2 x^2}$ delta sequence. As $n$ grows, the graph becomes thinner and taller. The area under each graph is equal to $1$.]]
Let $\sequence {\map {\delta_n} x}$ be a [[Definition:Sequence|sequence]] such that:
:$\map {\delta_n} x := \... | Let $\map g x = \map \phi x - \map \phi 0$.
Then:
:$\ds \int_{- \infty}^\infty \map \phi x \map {\delta_n} x \rd x = \map \phi 0 + \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x$
Let $A \in \R_{> 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x
| r = \int_... | N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence/Proof 2 | https://proofwiki.org/wiki/N_over_Pi_times_Reciprocal_of_1_Plus_n_Squared_x_Squared_Delta_Sequence | https://proofwiki.org/wiki/N_over_Pi_times_Reciprocal_of_1_Plus_n_Squared_x_Squared_Delta_Sequence/Proof_2 | [
"N over Pi times Reciprocal of 1 Plus n Squared x Squared Delta Sequence",
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:LorDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Definition:Test Function",
"Definition:Smooth Real Function",
"Differentiable Function is Continuous",
"Definition:Continuous Real Function/Point",
"Definition:Limit... |
proofwiki-19080 | Gaussian Delta Sequence | thumb500pxThe graph of the Gaussian delta sequence. As $n$ grows, the graph becomes thinner and taller. The area of under each Gaussian is equal to $1$.
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:$\map {\delta_n} x := \dfrac n {\sqrt \pi} e^{- n^2 x^2}$
Then $\sequence {\map {\delta_n} x}_{n \mathop ... | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac n {\sqrt \pi} e^{- n^2 x^2} \rd x
| r = \int_0^\infty \frac 1 {\sqrt \pi} e^{- \paren {n x}^2} \rd \paren {n x}
}}
{{eqn | r = \int_0^\infty \frac 1 {\sqrt \pi} e^{- y^2} \rd y
| c = $n x = y$, Integration by Substitution
}}
{{eqn | r = \frac 1 2
| c = Int... | [[File:GaussianDeltaSequence.png|thumb|500px|The graph of the Gaussian delta sequence. As $n$ grows, the graph becomes thinner and taller. The area of under each Gaussian is equal to $1$.]]
Let $\sequence {\map {\delta_n} x}$ be a [[Definition:Sequence|sequence]] such that:
:$\map {\delta_n} x := \dfrac n {\sqrt \pi} ... | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac n {\sqrt \pi} e^{- n^2 x^2} \rd x
| r = \int_0^\infty \frac 1 {\sqrt \pi} e^{- \paren {n x}^2} \rd \paren {n x}
}}
{{eqn | r = \int_0^\infty \frac 1 {\sqrt \pi} e^{- y^2} \rd y
| c = $n x = y$, [[Integration by Substitution/Definite Integral|Integration by Subst... | Gaussian Delta Sequence/Proof 1 | https://proofwiki.org/wiki/Gaussian_Delta_Sequence | https://proofwiki.org/wiki/Gaussian_Delta_Sequence/Proof_1 | [
"Gaussian Delta Sequence",
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:GaussianDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Integration by Substitution/Definite Integral",
"Integral to Infinity of Exponential of -t^2",
"Integration by Substitution/Definite Integral",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Mean Value Theorem for Int... |
proofwiki-19081 | Gaussian Delta Sequence | thumb500pxThe graph of the Gaussian delta sequence. As $n$ grows, the graph becomes thinner and taller. The area of under each Gaussian is equal to $1$.
Let $\sequence {\map {\delta_n} x}$ be a sequence such that:
:$\map {\delta_n} x := \dfrac n {\sqrt \pi} e^{- n^2 x^2}$
Then $\sequence {\map {\delta_n} x}_{n \mathop ... | Let $\map g x = \map \phi x - \map \phi 0$.
Then:
:$\ds \int_{- \infty}^\infty \map \phi x \map {\delta_n} x \rd x = \map \phi 0 + \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x$
Let $A \in \R_{> 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x
| r = \int_{- \i... | [[File:GaussianDeltaSequence.png|thumb|500px|The graph of the Gaussian delta sequence. As $n$ grows, the graph becomes thinner and taller. The area of under each Gaussian is equal to $1$.]]
Let $\sequence {\map {\delta_n} x}$ be a [[Definition:Sequence|sequence]] such that:
:$\map {\delta_n} x := \dfrac n {\sqrt \pi} ... | Let $\map g x = \map \phi x - \map \phi 0$.
Then:
:$\ds \int_{- \infty}^\infty \map \phi x \map {\delta_n} x \rd x = \map \phi 0 + \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x$
Let $A \in \R_{> 0}$.
Then:
{{begin-eqn}}
{{eqn | l = \int_{- \infty}^\infty \map g x \map {\delta_n} x \rd x
| r = \int_... | Gaussian Delta Sequence/Proof 2 | https://proofwiki.org/wiki/Gaussian_Delta_Sequence | https://proofwiki.org/wiki/Gaussian_Delta_Sequence/Proof_2 | [
"Gaussian Delta Sequence",
"Examples of Delta Sequences",
"Dirac Delta Distribution"
] | [
"File:GaussianDeltaSequence.png",
"Definition:Sequence",
"Definition:Delta Sequence",
"Definition:Schwartz Distribution",
"Definition:Test Function",
"Definition:Dirac Delta Distribution",
"Definition:Abuse of Notation"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Definition:Bounded Mapping/Real-Valued",
"Triangle Inequality/Real Numbers",
"Integration by Substitution/Definite Integral",
"Definition:Strictly Positive/Real Number",
"Integration by Substitution/Definite Integral",
"Definition:Stric... |
proofwiki-19082 | Inverse of Linear Transformation is Linear Transformation | Let $K$ be a field.
Let $V$ and $U$ be vector spaces over $K$.
Let $A : V \to U$ be an invertible (in the sense of a mapping) linear transformation with inverse mapping $A^{-1} : U \to V$.
Then $A^{-1}$ is a linear transformation. | We aim to show that:
:$\map {A^{-1} } {\alpha x + \beta y} = \alpha A^{-1} x + \beta A^{-1} y$
for all $x, y \in U$ and $\alpha, \beta \in K$.
Since $A$ is a linear transformation, we have:
:$\map A {\alpha u + \beta v} = \alpha A u + \beta A v$
for each $u, v \in V$.
Note that we have $A^{-1} x \in V$ and $A^{-1} y... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $V$ and $U$ be [[Definition:Vector Space|vector spaces]] over $K$.
Let $A : V \to U$ be an [[Definition:Invertible Mapping|invertible]] (in the sense of a [[Definition:Mapping|mapping]]) [[Definition:Linear Transformation|linear transformation]] with [[... | We aim to show that:
:$\map {A^{-1} } {\alpha x + \beta y} = \alpha A^{-1} x + \beta A^{-1} y$
for all $x, y \in U$ and $\alpha, \beta \in K$.
Since $A$ is a [[Definition:Linear Transformation|linear transformation]], we have:
:$\map A {\alpha u + \beta v} = \alpha A u + \beta A v$
for each $u, v \in V$.
Note ... | Inverse of Linear Transformation is Linear Transformation | https://proofwiki.org/wiki/Inverse_of_Linear_Transformation_is_Linear_Transformation | https://proofwiki.org/wiki/Inverse_of_Linear_Transformation_is_Linear_Transformation | [
"Linear Transformations"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Inverse Mapping",
"Definition:Mapping",
"Definition:Linear Transformation",
"Definition:Inverse Mapping",
"Definition:Linear Transformation"
] | [
"Definition:Linear Transformation",
"Definition:Composition of Mappings",
"Definition:Inverse Mapping",
"Definition:Linear Transformation",
"Category:Linear Transformations"
] |
proofwiki-19083 | Inverse of Linear Operator is Linear Operator | Let $X$ be a vector space.
Let $A : X \to X$ be an invertible (in the sense of a mapping) linear transformation with inverse mapping $A^{-1} : X \to X$.
Then $A^{-1}$ is a linear operator. | Applying Inverse of Linear Transformation is Linear Transformation in the case $U = V = X$ we have:
:$A^{-1}$ is a linear transformation.
Since $A^{-1}$ is a linear transformation $X \to X$, we have:
:$A^{-1}$ is a linear operator.
{{qed}}
Category:Linear Operators
tu3wscntho696ilz8bjsxkfo5tx6zme | Let $X$ be a [[Definition:Vector Space|vector space]].
Let $A : X \to X$ be an [[Definition:Invertible Mapping|invertible]] (in the sense of a mapping) [[Definition:Linear Transformation|linear transformation]] with [[Definition:Inverse Mapping|inverse mapping]] $A^{-1} : X \to X$.
Then $A^{-1}$ is a [[Definition:Li... | Applying [[Inverse of Linear Transformation is Linear Transformation]] in the case $U = V = X$ we have:
:$A^{-1}$ is a [[Definition:Linear Transformation|linear transformation]].
Since $A^{-1}$ is a [[Definition:Linear Transformation|linear transformation]] $X \to X$, we have:
:$A^{-1}$ is a [[Definition:Linear Op... | Inverse of Linear Operator is Linear Operator | https://proofwiki.org/wiki/Inverse_of_Linear_Operator_is_Linear_Operator | https://proofwiki.org/wiki/Inverse_of_Linear_Operator_is_Linear_Operator | [
"Linear Operators"
] | [
"Definition:Vector Space",
"Definition:Inverse Mapping",
"Definition:Linear Transformation",
"Definition:Inverse Mapping",
"Definition:Linear Operator"
] | [
"Inverse of Linear Transformation is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Operator",
"Category:Linear Operators"
] |
proofwiki-19084 | Largest Parallelogram Contained in Triangle | Let $T$ be a triangle.
Let $P$ be a parallelogram contained within $T$.
Let $P$ have the largest area possible for the conditions given.
Then:
:$(1): \quad$ One side of $P$ is coincident with part of one side of $T$, and hence two vertices lie on that side of $T$
:$(2): \quad$ The other two vertices of $P$ bisect the o... | We will first find the maximum area of $P$ when $(1)$ is satisfied, that is, when $P$ is inscribed in $T$. | Let $T$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $P$ be a [[Definition:Parallelogram|parallelogram]] contained within $T$.
Let $P$ have the largest [[Definition:Area|area]] possible for the conditions given.
Then:
:$(1): \quad$ One [[Definition:Side of Polygon|side]] of $P$ is coincident with part of ... | We will first find the maximum [[Definition:Area|area]] of $P$ when $(1)$ is satisfied, that is, when $P$ is [[Definition:Polygon Inscribed within Polygon|inscribed]] in $T$. | Largest Parallelogram Contained in Triangle | https://proofwiki.org/wiki/Largest_Parallelogram_Contained_in_Triangle | https://proofwiki.org/wiki/Largest_Parallelogram_Contained_in_Triangle | [
"Triangles",
"Parallelograms"
] | [
"Definition:Triangle (Geometry)",
"Definition:Quadrilateral/Parallelogram",
"Definition:Area",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Polygon/Vertex",
"Definition:Polygon/Side",
"Definition:Polygon/Vertex",
"Definition:Bisection",
"Definition:Polygon/Side",
"Definition... | [
"Definition:Area",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Area",
"Definition:Area",
"Definition:Area",
"Definition:Area",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Area",
"Definition:Ar... |
proofwiki-19085 | Relative Algebraic Closure with Algebraically Closed Extension is Algebraic Closure | Let $L$ be an algebraically closed field.
Let $L/K$ be a field extension.
Then the relative algebraic closure of $K$ contained in $L$ is an algebraic closure of $K$. | Let $K'$ denote the relative algebraic closure of $K$ contained in $L$.
By definition, $K' = \{\alpha \in L \mid \alpha \text{ is algebraic over } K\}$.
First we show that $K'$ is a field extension of $K$.
By Field is Algebraic over itself, $K \subseteq K'$.
Since $K$ is a field, it and (therefore) $K'$ are nonzero.
Le... | Let $L$ be an [[Definition:Algebraically Closed Field|algebraically closed field]].
Let $L/K$ be a [[Definition:Field Extension|field extension]].
Then the [[Definition:Relative Algebraic Closure|relative algebraic closure]] of $K$ contained in $L$ is an [[Definition:Algebraic Closure|algebraic closure]] of $K$. | Let $K'$ denote the [[Definition:Relative Algebraic Closure|relative algebraic closure]] of $K$ contained in $L$.
By definition, $K' = \{\alpha \in L \mid \alpha \text{ is algebraic over } K\}$.
First we show that $K'$ is a [[Definition:Field Extension|field extension]] of $K$.
By [[Field is Algebraic over itself]]... | Relative Algebraic Closure with Algebraically Closed Extension is Algebraic Closure | https://proofwiki.org/wiki/Relative_Algebraic_Closure_with_Algebraically_Closed_Extension_is_Algebraic_Closure | https://proofwiki.org/wiki/Relative_Algebraic_Closure_with_Algebraically_Closed_Extension_is_Algebraic_Closure | [
"Field Extensions"
] | [
"Definition:Algebraically Closed Field",
"Definition:Field Extension",
"Definition:Relative Algebraic Closure",
"Definition:Algebraic Closure"
] | [
"Definition:Relative Algebraic Closure",
"Definition:Field Extension",
"Field is Algebraic over itself",
"Definition:Field (Abstract Algebra)",
"Definition:Non-Null Ring",
"Definition:Generated Field Extension",
"Definition:Field (Abstract Algebra)",
"Finitely Generated Algebraic Extension is Finite",... |
proofwiki-19086 | Modulus of Eigenvalue of Bounded Linear Operator is Bounded Above by Operator Norm | Let $\struct {X, \norm \cdot}$ be a normed vector space.
Let $T : X \to X$ be a bounded linear operator.
Let $\lambda$ be a eigenvalue of $T$.
Then:
:$\cmod \lambda \le \norm T$
where $\norm T$ denotes the norm of $T$. | Since $\lambda$ is an eigenvalue of $T$, there exists $x \ne 0$ such that:
:$T x = \lambda x$
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {\map T {\frac x {\norm x} } }
| r = \norm {\frac 1 {\norm x} \map T x}
| c = {{Defof|Linear Transformation on Vector Space}}
}}
{{eqn | r = \frac {\norm {\map T x} } {\norm... | Let $\struct {X, \norm \cdot}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $T : X \to X$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\lambda$ be a [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $T$.
Then:
:$\cmod \lambda \le \norm T$
where $\norm T$... | Since $\lambda$ is an [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $T$, there exists $x \ne 0$ such that:
:$T x = \lambda x$
Then we have:
{{begin-eqn}}
{{eqn | l = \norm {\map T {\frac x {\norm x} } }
| r = \norm {\frac 1 {\norm x} \map T x}
| c = {{Defof|Linear Transformation on Vector Space}}... | Modulus of Eigenvalue of Bounded Linear Operator is Bounded Above by Operator Norm | https://proofwiki.org/wiki/Modulus_of_Eigenvalue_of_Bounded_Linear_Operator_is_Bounded_Above_by_Operator_Norm | https://proofwiki.org/wiki/Modulus_of_Eigenvalue_of_Bounded_Linear_Operator_is_Bounded_Above_by_Operator_Norm | [
"Eigenvalues of Linear Operators",
"Bounded Linear Operators"
] | [
"Definition:Normed Vector Space",
"Definition:Bounded Linear Operator",
"Definition:Eigenvalue/Linear Operator",
"Definition:Norm/Bounded Linear Transformation"
] | [
"Definition:Eigenvalue/Linear Operator",
"Definition:Supremum of Set/Real Numbers",
"Definition:Norm/Bounded Linear Transformation"
] |
proofwiki-19087 | Divisibility by 37 | Let $n$ be an integer which has at least $3$ digits when expressed in decimal notation.
Let the digits of $n$ be divided into groups of $3$, counting from the right, and those groups added.
Then the result is equal to a multiple of $37$ {{iff}} $n$ is divisible by $37$. | Write $n = \ds \sum_{i \mathop = 0}^k a_i 10^{3 i}$, where $0 \le a_i < 1000$.
This divides the digits of $n$ into groups of $3$.
Then the statement is equivalent to:
:$37 \divides n \iff 37 \divides \ds \sum_{i \mathop = 0}^k a_i$
Note that $1000 = 37 \times 27 + 1 \equiv 1 \pmod {37}$.
Hence:
{{begin-eqn}}
{{eqn | l ... | Let $n$ be an [[Definition:Integer|integer]] which has at least $3$ [[Definition:Digit|digits]] when expressed in [[Definition:Decimal Notation|decimal notation]].
Let the [[Definition:Digit|digits]] of $n$ be divided into groups of $3$, counting from the right, and those groups added.
Then the result is equal to a ... | Write $n = \ds \sum_{i \mathop = 0}^k a_i 10^{3 i}$, where $0 \le a_i < 1000$.
This divides the [[Definition:Digit|digits]] of $n$ into groups of $3$.
Then the statement is equivalent to:
:$37 \divides n \iff 37 \divides \ds \sum_{i \mathop = 0}^k a_i$
Note that $1000 = 37 \times 27 + 1 \equiv 1 \pmod {37}$.
Hence:... | Divisibility by 37 | https://proofwiki.org/wiki/Divisibility_by_37 | https://proofwiki.org/wiki/Divisibility_by_37 | [
"Divisibility Tests"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Decimal Notation",
"Definition:Digit",
"Definition:Multiple/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Digit",
"Congruence of Powers",
"Category:Divisibility Tests"
] |
proofwiki-19088 | Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse | :$l = \dfrac {a b c} {a b + c^2}$ | :400px
In the figure above, let $BC = a$ and $AC = b$.
$CD$ is drawn such that $AB \perp CD$.
Since $CD$ is the height of $\triangle ABC$:
:$CD = \dfrac {a b} c$
Note that $FH \parallel AB$.
Therefore $\triangle CFH \sim \triangle CAB$ by Equiangular Triangles are Similar.
Thus:
{{begin-eqn}}
{{eqn | l = \frac {CG} {CD... | :$l = \dfrac {a b c} {a b + c^2}$ | :[[File:Inscribed-square-h.png|400px]]
In the figure above, let $BC = a$ and $AC = b$.
$CD$ is drawn such that $AB \perp CD$.
Since $CD$ is the [[Definition:Height|height]] of $\triangle ABC$:
:$CD = \dfrac {a b} c$
Note that $FH \parallel AB$.
Therefore $\triangle CFH \sim \triangle CAB$ by [[Equiangular Triangl... | Inscribed Squares in Right-Angled Triangle/Side Lengths/Side Lies on Hypotenuse | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Side_Lengths/Side_Lies_on_Hypotenuse | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Side_Lengths/Side_Lies_on_Hypotenuse | [
"Inscribed Squares in Right-Angled Triangle"
] | [] | [
"File:Inscribed-square-h.png",
"Definition:Height",
"Equiangular Triangles are Similar",
"Category:Inscribed Squares in Right-Angled Triangle"
] |
proofwiki-19089 | Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle | :$l = \dfrac {a b} {a + b}$ | :200px
In the figure above, let $BC = a$ and $AC = b$.
Note that $DE \parallel CF$.
Therefore $\triangle BDE \sim \triangle BCA$ by Equiangular Triangles are Similar.
Thus:
{{begin-eqn}}
{{eqn | l = \frac {BD} {DE}
| r = \frac {BC} {CA}
| c = {{Defof|Similar Triangles}}
}}
{{eqn | l = \frac {a - l} l
... | :$l = \dfrac {a b} {a + b}$ | :[[File:Inscribed-square-r.png|200px]]
In the figure above, let $BC = a$ and $AC = b$.
Note that $DE \parallel CF$.
Therefore $\triangle BDE \sim \triangle BCA$ by [[Equiangular Triangles are Similar]].
Thus:
{{begin-eqn}}
{{eqn | l = \frac {BD} {DE}
| r = \frac {BC} {CA}
| c = {{Defof|Similar Triangles... | Inscribed Squares in Right-Angled Triangle/Side Lengths/Shared Right Angle | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Side_Lengths/Shared_Right_Angle | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Side_Lengths/Shared_Right_Angle | [
"Inscribed Squares in Right-Angled Triangle"
] | [] | [
"File:Inscribed-square-r.png",
"Equiangular Triangles are Similar"
] |
proofwiki-19090 | Inscribed Squares in Right-Angled Triangle | For any right-angled triangle, two squares can be inscribed inside it.
One square would share a vertex with the right-angled vertex of the right-angled triangle:
:200px
The other square would have a side lying on the hypotenuse of the right-angled triangle:
::400px | By definition of inscribed polygon, all four vertices of the inscribed square lies on the sides of the right-angled triangle.
By Pigeonhole Principle, at least two of the vertices must lie on the same side of the right-angled triangle.
The case where this side is the hypotenuse would be the second case above.
For the c... | For any [[Definition:Right-Angled Triangle|right-angled triangle]], two [[Definition:Square|squares]] can be [[Definition:Polygon Inscribed within Polygon|inscribed]] inside it.
One [[Definition:Square|square]] would share a [[Definition:Vertex of Polygon|vertex]] with the [[Definition:Right Angle|right-angled]] [[Def... | By definition of [[Definition:Polygon Inscribed within Polygon|inscribed polygon]], all four [[Definition:Vertex of Polygon|vertices]] of the [[Definition:Polygon Inscribed within Polygon|inscribed]] [[Definition:Square|square]] lies on the [[Definition:Side of Polygon|sides]] of the [[Definition:Right-Angled Triangle|... | Inscribed Squares in Right-Angled Triangle | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle | [
"Inscribed Squares in Right-Angled Triangle",
"Squares",
"Right Triangles"
] | [
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Square",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Square",
"Definition:Polygon/Vertex",
"Definition:Right Angle",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Right-Angled",
"File:Inscribed-square-r.png",
"Defi... | [
"Definition:Inscribe/Polygon in Polygon",
"Definition:Polygon/Vertex",
"Definition:Inscribe/Polygon in Polygon",
"Definition:Square",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Right-Angled",
"Dirichlet's Box Principle/Corollary",
"Definition:Polygon/Vertex",
"Definition:Polygon/Side... |
proofwiki-19091 | Inverse of Linear Isometry is Linear Isometry | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.
Let $T : X \to Y$ be an invertible (in the sense of a mapping) linear isometry with inverse $T^{-1} : Y \to X$.
Then $T^{-1}$ is a linear isometry. | From Inverse of Linear Transformation is Linear Transformation, we have:
:$T^{-1}$ is a linear transformation.
Since $T$ is a linear isometry, we have:
:$\norm {T x}_Y = \norm x_X$
for each $x \in X$.
Note that for each $y \in Y$, we have $T^{-1} y \in X$.
We then have:
{{begin-eqn}}
{{eqn | l = \norm {T^{-1} y}_X
... | Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $T : X \to Y$ be an [[Definition:Invertible Mapping|invertible]] (in the sense of a mapping) [[Definition:Linear Isometry|linear isometry]] with [[Definition:Inverse Mapping|inverse]] $T^{-... | From [[Inverse of Linear Transformation is Linear Transformation]], we have:
:$T^{-1}$ is a [[Definition:Linear Transformation|linear transformation]].
Since $T$ is a [[Definition:Linear Isometry|linear isometry]], we have:
:$\norm {T x}_Y = \norm x_X$
for each $x \in X$.
Note that for each $y \in Y$, we have $T... | Inverse of Linear Isometry is Linear Isometry | https://proofwiki.org/wiki/Inverse_of_Linear_Isometry_is_Linear_Isometry | https://proofwiki.org/wiki/Inverse_of_Linear_Isometry_is_Linear_Isometry | [
"Linear Isometries"
] | [
"Definition:Normed Vector Space",
"Definition:Inverse Mapping",
"Definition:Linear Isometry",
"Definition:Inverse Mapping",
"Definition:Linear Isometry"
] | [
"Inverse of Linear Transformation is Linear Transformation",
"Definition:Linear Transformation",
"Definition:Linear Isometry",
"Definition:Linear Transformation",
"Definition:Linear Isometry",
"Category:Linear Isometries"
] |
proofwiki-19092 | Greatest Area of Quadrilateral with Sides in Arithmetic Sequence | Let $Q$ be a quadrilateral whose sides $a$, $b$, $c$ and $d$ are in arithmetic sequence.
Let $\AA$ be the area of $Q$.
Let $Q$ be such that $\AA$ is the greatest area possible for one with sides $a$, $b$, $c$ and $d$.
Then:
:$\AA = \sqrt {a b c d}$ | We are given that $\AA$ is the greatest possible for a quadrilateral whose sides are $a$, $b$, $c$ and $d$.
From Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic, $Q$ is cyclic.
Hence $\AA$ can be found using Brahmagupta's Formula.
Let $s$ denote the semiperimeter of $Q$:
:$s = \dfrac {a ... | Let $Q$ be a [[Definition:Quadrilateral|quadrilateral]] whose [[Definition:Side of Polygon|sides]] $a$, $b$, $c$ and $d$ are in [[Definition:Arithmetic Sequence|arithmetic sequence]].
Let $\AA$ be the [[Definition:Area|area]] of $Q$.
Let $Q$ be such that $\AA$ is the greatest [[Definition:Area|area]] possible for one... | We are given that $\AA$ is the greatest possible for a [[Definition:Quadrilateral|quadrilateral]] whose [[Definition:Side of Polygon|sides]] are $a$, $b$, $c$ and $d$.
From [[Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic]], $Q$ is [[Definition:Cyclic Quadrilateral|cyclic]].
Hence $\A... | Greatest Area of Quadrilateral with Sides in Arithmetic Sequence | https://proofwiki.org/wiki/Greatest_Area_of_Quadrilateral_with_Sides_in_Arithmetic_Sequence | https://proofwiki.org/wiki/Greatest_Area_of_Quadrilateral_with_Sides_in_Arithmetic_Sequence | [
"Cyclic Quadrilaterals"
] | [
"Definition:Quadrilateral",
"Definition:Polygon/Side",
"Definition:Arithmetic Sequence",
"Definition:Area",
"Definition:Area",
"Definition:Polygon/Side"
] | [
"Definition:Quadrilateral",
"Definition:Polygon/Side",
"Area of Quadrilateral with Given Sides is Greatest when Quadrilateral is Cyclic",
"Definition:Cyclic Quadrilateral",
"Brahmagupta's Formula",
"Definition:Semiperimeter",
"Definition:Arithmetic Sequence",
"Definition:Arithmetic Sequence/Common Dif... |
proofwiki-19093 | Thomsen Graph is Toroidal | The Thomsen graph, that is, the complete bipartite graph $K_{3, 3}$, is a toroidal graph. | {{ProofWanted|More background work necessary}} | The [[Definition:Thomsen Graph|Thomsen graph]], that is, the [[Definition:Complete Bipartite Graph|complete bipartite graph]] $K_{3, 3}$, is a [[Definition:Toroidal Graph|toroidal graph]]. | {{ProofWanted|More background work necessary}} | Thomsen Graph is Toroidal | https://proofwiki.org/wiki/Thomsen_Graph_is_Toroidal | https://proofwiki.org/wiki/Thomsen_Graph_is_Toroidal | [
"Utilities Problem",
"Toroidal Graphs"
] | [
"Definition:Thomsen Graph",
"Definition:Complete Bipartite Graph",
"Definition:Toroidal Graph"
] | [] |
proofwiki-19094 | Seven Touching Cylinders | It is possible to arrange $7$ identical cylinders so that each one touches each of the others.
The cylinders must be such that their heights must be at least $\dfrac {7 \sqrt 3} 2$ of the diameters of their bases. | :600px
It remains to be proved that the heights of the cylinders must be at least $\dfrac {7 \sqrt 3} 2$ of the diameters of their bases.
{{ProofWanted|Prove the above}} | It is possible to arrange $7$ identical [[Definition:Right Circular Cylinder|cylinders]] so that each one touches each of the others.
The [[Definition:Right Circular Cylinder|cylinders]] must be such that their [[Definition:Height of Cylinder|heights]] must be at least $\dfrac {7 \sqrt 3} 2$ of the [[Definition:Diamet... | :[[File:Seven-Touching-Cylinders.png|600px]]
It remains to be proved that the [[Definition:Height of Cylinder|heights]] of the [[Definition:Right Circular Cylinder|cylinders]] must be at least $\dfrac {7 \sqrt 3} 2$ of the [[Definition:Diameter of Circle|diameters]] of their [[Definition:Base of Right Circular Cylinde... | Seven Touching Cylinders | https://proofwiki.org/wiki/Seven_Touching_Cylinders | https://proofwiki.org/wiki/Seven_Touching_Cylinders | [
"Right Circular Cylinders",
"Recreational Mathematics"
] | [
"Definition:Right Circular Cylinder",
"Definition:Right Circular Cylinder",
"Definition:Cylinder/Height",
"Definition:Circle/Diameter",
"Definition:Right Circular Cylinder/Base"
] | [
"File:Seven-Touching-Cylinders.png",
"Definition:Cylinder/Height",
"Definition:Right Circular Cylinder",
"Definition:Circle/Diameter",
"Definition:Right Circular Cylinder/Base"
] |
proofwiki-19095 | Inscribed Squares in Right-Angled Triangle/Compass and Straightedge Construction/Shared Right Angle | :200px
Let $\triangle ABC$ be a right-angled triangle, where $\angle C = 90^\circ$.
Construct the angle bisector of $\angle C$.
Let the point of intersection of this angle bisector and side $AB$ be $E$.
Construct perpedicular lines from $E$ to sides $AC$ and $BC$, and name their intersections $F$ and $D$ respectively.
... | Note that $\angle C = 90^\circ$ and $ED \perp BC$, $EF \perp AC$ by construction.
Therefore $CDEF$ is a rectangle.
By definition of an angle bisector, $\angle ECF = 45^\circ$.
Since $\angle CFE = 90^\circ$, by Sum of Angles of Triangle equals Two Right Angles:
:$\angle CEF + \angle CFE + \angle ECF = 180^\circ$
:$\ther... | :[[File:Inscribed_square_rc.png|200px]]
Let $\triangle ABC$ be a [[Definition:Right-Angled Triangle|right-angled triangle]], where $\angle C = 90^\circ$.
[[Bisection of Angle|Construct]] the [[Definition:Angle Bisector|angle bisector]] of $\angle C$.
Let the point of [[Definition:Intersection (Geometry)|intersection... | Note that $\angle C = 90^\circ$ and $ED \perp BC$, $EF \perp AC$ by construction.
Therefore $CDEF$ is a [[Definition:Rectangle|rectangle]].
By definition of an [[Definition:Angle Bisector|angle bisector]], $\angle ECF = 45^\circ$.
Since $\angle CFE = 90^\circ$, by [[Sum of Angles of Triangle equals Two Right Angles... | Inscribed Squares in Right-Angled Triangle/Compass and Straightedge Construction/Shared Right Angle | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Compass_and_Straightedge_Construction/Shared_Right_Angle | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Compass_and_Straightedge_Construction/Shared_Right_Angle | [
"Inscribed Squares in Right-Angled Triangle"
] | [
"File:Inscribed_square_rc.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Bisection of Angle",
"Definition:Angle Bisector",
"Definition:Intersection (Geometry)",
"Definition:Angle Bisector",
"Definition:Polygon/Side",
"Construction of Perpendicular Line",
"Definition:Right Angle/Perpendicular"... | [
"Definition:Quadrilateral/Rectangle",
"Definition:Angle Bisector",
"Sum of Angles of Triangle equals Two Right Angles",
"Triangle with Two Equal Angles is Isosceles",
"Definition:Quadrilateral/Rectangle",
"Definition:Polygon/Side",
"Definition:Square",
"Category:Inscribed Squares in Right-Angled Trian... |
proofwiki-19096 | Inscribed Squares in Right-Angled Triangle/Compass and Straightedge Construction/Side Lies on Hypotenuse | :400px
Let $\triangle ABC$ be a right-angled triangle, where $\angle C = 90^\circ$.
Construct a perpedicular line from $C$ to side $AB$, and name the intersection $D$.
Construct the angle bisector of $\angle ADC$.
Let the point of intersection of this angle bisector and side $AC$ be $E$.
Construct a line parallel to $A... | Note that $HG \perp HI$, $HG \perp GJ$ and $HG \parallel IJ$ by construction.
Therefore $GHIJ$ is a rectangle.
By definition of an angle bisector, $\angle FDE = 45^\circ$.
Since $\angle EFD = 90^\circ$ by construction, by Sum of Angles of Triangle equals Two Right Angles:
:$\angle DEF + \angle EFD + \angle FDE = 180^\c... | :[[File:Inscribed_square_hc.png|400px]]
Let $\triangle ABC$ be a [[Definition:Right-Angled Triangle|right-angled triangle]], where $\angle C = 90^\circ$.
[[Construction of Perpendicular Line|Construct]] a [[Definition:Perpendicular|perpedicular line]] from $C$ to [[Definition:Side of Polygon|side]] $AB$, and name the... | Note that $HG \perp HI$, $HG \perp GJ$ and $HG \parallel IJ$ by construction.
Therefore $GHIJ$ is a [[Definition:Rectangle|rectangle]].
By definition of an [[Definition:Angle Bisector|angle bisector]], $\angle FDE = 45^\circ$.
Since $\angle EFD = 90^\circ$ by construction, by [[Sum of Angles of Triangle equals Two ... | Inscribed Squares in Right-Angled Triangle/Compass and Straightedge Construction/Side Lies on Hypotenuse | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Compass_and_Straightedge_Construction/Side_Lies_on_Hypotenuse | https://proofwiki.org/wiki/Inscribed_Squares_in_Right-Angled_Triangle/Compass_and_Straightedge_Construction/Side_Lies_on_Hypotenuse | [
"Inscribed Squares in Right-Angled Triangle"
] | [
"File:Inscribed_square_hc.png",
"Definition:Triangle (Geometry)/Right-Angled",
"Construction of Perpendicular Line",
"Definition:Right Angle/Perpendicular",
"Definition:Polygon/Side",
"Definition:Intersection (Geometry)",
"Bisection of Angle",
"Definition:Angle Bisector",
"Definition:Intersection (G... | [
"Definition:Quadrilateral/Rectangle",
"Definition:Angle Bisector",
"Sum of Angles of Triangle equals Two Right Angles",
"Triangle with Two Equal Angles is Isosceles",
"Equiangular Triangles are Similar",
"Equiangular Triangles are Similar",
"Definition:Similar Triangles",
"Definition:Quadrilateral/Rec... |
proofwiki-19097 | Queen's Tour | Consider a chessboard $\CC$ of size $n \times n$ such that $n > 2$.
Then the shortest queen's tour on $\CC$ is of $2 n - 2$ moves.
For $n < 5$ it is necessary for the queen to move outside the boundary of the chessboard in order for this to happen. | First it is shown that at least $2 n - 2$ moves are needed.
Let there be $R$ rows and $S$ columns which have none of the given moves on them.
The $R \times S$-square segment of this chessboard has a $2 R + 2 S - 4$ squares on its edge if $R$ and $S$ are both greater than $1$.
Each diagonal moves covers at most $2$ of t... | Consider a [[Definition:Chessboard|chessboard]] $\CC$ of size $n \times n$ such that $n > 2$.
Then the shortest [[Definition:Queen's Tour|queen's tour]] on $\CC$ is of $2 n - 2$ [[Definition:Chess Move|moves]].
For $n < 5$ it is necessary for the [[Definition:Chess Queen|queen]] to move outside the boundary of the [... | First it is shown that at least $2 n - 2$ [[Definition:Chess Move|moves]] are needed.
Let there be $R$ rows and $S$ columns which have none of the given [[Definition:Chess Move|moves]] on them.
The $R \times S$-[[Definition:Square of Chessboard|square]] segment of this [[Definition:Chessboard|chessboard]] has a $2 R ... | Queen's Tour | https://proofwiki.org/wiki/Queen's_Tour | https://proofwiki.org/wiki/Queen's_Tour | [
"Queen's Tours"
] | [
"Definition:Chess/Chessboard",
"Definition:Queen's Tour",
"Definition:Chess/Move",
"Definition:Chess/Piece/Queen",
"Definition:Chess/Chessboard"
] | [
"Definition:Chess/Move",
"Definition:Chess/Move",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Chessboard",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Move",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Move",
"Definition:Chess/Move",
"Definition:Chess/Move",
"... |
proofwiki-19098 | Re-entrant Queen's Tour | Consider a chessboard $\CC$ of size $n \times n$ such that $n > 3$.
Then there exists a re-entrant queen's tour on $\CC$ of $2 n - 2$ moves.
For $n < 6$ it is necessary for the queen to move outside the boundary of the chessboard in order for this to happen. | First it is shown that at least $2 n - 2$ moves are needed.
Let there be $R$ rows and $S$ columns which have none of the given moves on them.
The $R \times S$-square segment of this chessboard has a $2 R + 2 S - 4$ squares on its edge if $R$ and $S$ are both greater than $1$.
Each diagonal moves covers at most $2$ of t... | Consider a [[Definition:Chessboard|chessboard]] $\CC$ of size $n \times n$ such that $n > 3$.
Then there exists a [[Definition:Re-entrant Queen's Tour|re-entrant queen's tour]] on $\CC$ of $2 n - 2$ [[Definition:Chess Move|moves]].
For $n < 6$ it is necessary for the [[Definition:Chess Queen|queen]] to move outside ... | First it is shown that at least $2 n - 2$ [[Definition:Chess Move|moves]] are needed.
Let there be $R$ rows and $S$ columns which have none of the given [[Definition:Chess Move|moves]] on them.
The $R \times S$-[[Definition:Square of Chessboard|square]] segment of this [[Definition:Chessboard|chessboard]] has a $2 R ... | Re-entrant Queen's Tour | https://proofwiki.org/wiki/Re-entrant_Queen's_Tour | https://proofwiki.org/wiki/Re-entrant_Queen's_Tour | [
"Queen's Tours"
] | [
"Definition:Chess/Chessboard",
"Definition:Queen's Tour/Re-entrant",
"Definition:Chess/Move",
"Definition:Chess/Piece/Queen",
"Definition:Chess/Chessboard"
] | [
"Definition:Chess/Move",
"Definition:Chess/Move",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Chessboard",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Move",
"Definition:Chess/Chessboard/Square",
"Definition:Chess/Move",
"Definition:Chess/Move",
"Definition:Chess/Move",
"... |
proofwiki-19099 | Coefficients of Polynomial add to 0 iff 1 is a Root | Let $\map E x$ be the equation in $x$ represented as:
:$\ds \sum_{j \mathop = 0}^n a_j x^j = 0$
where the $a_j$s are constants.
Then $1$ is a root of $\map E x$ {{iff}}:
:$\ds \sum_{j \mathop = 0}^n a_j = 0$
That is, $1$ is a root of $\map E x$ {{iff}} all the coefficients of the polynomial in $x$ sum to zero. | Letting $x = 1$ in $E$;
{{begin-eqn}}
{{eqn | l = x
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{j \mathop = 0}^n a_j \times 1^j
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{j \mathop = 0}^n a_j
| r = 0
| c =
}}
{{end-eqn}}
{{qed}}
Category:Alge... | Let $\map E x$ be the [[Definition:Equation|equation]] in $x$ represented as:
:$\ds \sum_{j \mathop = 0}^n a_j x^j = 0$
where the $a_j$s are [[Definition:Constant|constants]].
Then $1$ is a [[Definition:Root of Equation|root]] of $\map E x$ {{iff}}:
:$\ds \sum_{j \mathop = 0}^n a_j = 0$
That is, $1$ is a [[Definiti... | Letting $x = 1$ in $E$;
{{begin-eqn}}
{{eqn | l = x
| r = 1
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{j \mathop = 0}^n a_j \times 1^j
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{j \mathop = 0}^n a_j
| r = 0
| c =
}}
{{end-eqn}}
{{qed}}
[[Category:... | Coefficients of Polynomial add to 0 iff 1 is a Root | https://proofwiki.org/wiki/Coefficients_of_Polynomial_add_to_0_iff_1_is_a_Root | https://proofwiki.org/wiki/Coefficients_of_Polynomial_add_to_0_iff_1_is_a_Root | [
"Algebra"
] | [
"Definition:Equation",
"Definition:Constant",
"Definition:Root of Equation",
"Definition:Root of Equation",
"Definition:Coefficient of Polynomial",
"Definition:Polynomial",
"Definition:Addition/Integers",
"Definition:Zero (Number)"
] | [
"Category:Algebra"
] |
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