diff options
Diffstat (limited to 'pages')
28 files changed, 113 insertions, 109 deletions
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md index 3cf81f7..6ac0803 100644 --- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md +++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md @@ -66,19 +66,21 @@ $$ Note that the supremum is finite (and is attained), because $f$ is continuous and the circle is compact. Clearly, the integral evaluates to $2 \pi r / r^{n+1}$ -and the right hand side of the inequality reduces to the desired expression. +and the right-hand side of the inequality reduces to the desired expression. {% endproof %} --- -Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane. +Recall that an *entire* function is a holomorphic function +that is defined everywhere in the complex plane. {% theorem * Liouville's Theorem %} Every bounded entire function is constant. {% endtheorem %} {% proof %} -Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$. +Consider an entire function $f$ and +assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$. Since $f$ is holomorphic on the whole plane, we may make [Cauchy's Estimate](#cauchy-s-estimate) for all disks centered at any point $a \in \CC$ and with any radius $r>0$. diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md index 2445b8b..6d78e89 100644 --- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md +++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md @@ -14,7 +14,7 @@ If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then $$ \int_{\gamma_0} \! f(z) \, dz = -\int_{\gamma_1} \! f(z) \, dz +\int_{\gamma_1} \! f(z) \, dz $$ If $\gamma$ is a null-homotopic closed curve in $G$, then diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md index 31793ab..4876d30 100644 --- a/pages/complex-analysis/one-complex-variable/power-series.md +++ b/pages/complex-analysis/one-complex-variable/power-series.md @@ -21,7 +21,8 @@ $a$ is the *center* of the series. {% enddefinition %} {% lemma %} -Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$. +Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ +converges for $z = a + w$. Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$. {% endlemma %} @@ -35,7 +36,7 @@ Then either - the series converges only for $z=a$ (formally $R=0$), or - there exists a number $0<R<\infty$ such that - the series converges absolutely whenever $\abs{z-a} < R$ + the series converges absolutely whenever $\abs{z-a} < R$ and diverges whenever $\abs{z-a} > R$, or - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$). diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md index 3055c8f..08a3ab2 100644 --- a/pages/distribution-theory/index.md +++ b/pages/distribution-theory/index.md @@ -19,7 +19,7 @@ an unbounded linear operator $\Phi(f)$ in $\hilb{H}$ such that {: .my-0 } - there is a dense linear subspace $\mathcal{D}$ of $\hilb{H}$ that -is contained in the domain of all the $\Phi(f)$ +is contained in the domain of all the $\Phi(f)$ - for every fixed pair of vectors $\phi, \psi \in \hilb{D}$ the mapping $f \mapsto \innerp{\phi}{\Phi(f) \psi}$ is a tempered distribution. {% enddefinition %} diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md index 91906cd..0913776 100644 --- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md +++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md @@ -19,4 +19,6 @@ The {{ page.title }} is a special case of the following result: The polar of a neighborhood of zero in a locally convex space is weak\* compact. {% endtheorem %} -See [Alaoglu–Bourbaki Theorem]({% link pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.md %}) for more information. +See +[Alaoglu–Bourbaki Theorem](/pages/more-functional-analysis/locally-convex-spaces/alaoglu-bourbaki-theorem.html) +for more information. diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md index 781fb1f..58ca1d3 100644 --- a/pages/functional-analysis-basics/reflexive-spaces.md +++ b/pages/functional-analysis-basics/reflexive-spaces.md @@ -17,7 +17,7 @@ $$ where the functional $g_x$ on $X'$ is defined by $$ -g_x(f) = f(x) \quad \text{for $f \in X'$,} +g_x(f) = f(x) \quad \text{for $f \in X'$,} $$ is called the *canonical embedding* of $X$ into its bidual $X''$. @@ -79,7 +79,7 @@ C : X \longrightarrow X'', \quad C(x)(f) = f(x), \quad x \in X, f \in X', $$ is an isomorphism. -Therefore, the the dual map +Therefore, the dual map $$ C' : X''' \longrightarrow X', \quad C'(h)(x) = h(C(x)), \quad x \in X, h \in X''', diff --git a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md index f6a9783..e0ec62b 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md @@ -33,10 +33,11 @@ This shows that $\graph{T}$ is closed. Conversely, suppose that $\graph{T}$ is a closed subspace of $X \times Y$. Note that $X \times Y$ is a Banach space with norm $\norm{(x,y)} = \norm{x} + \norm{y}$. -Therefore $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$. +Therefore, $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$. The canonical projections $\pi_X : \graph{T} \to X$ and $\pi_Y : \graph{T} \to Y$ are bounded. -Clearly, $\pi_X$ is bijective, so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the +Clearly, $\pi_X$ is bijective, +so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the [Bounded Inverse Theorem](/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.html#bounded-inverse-theorem). -Consequently the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded. +Consequently, the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded. To complete the proof, observe that $\pi_Y \circ \pi_X^{-1} = T$. {% endproof %} diff --git a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md index 18cf64a..a2602ac 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md @@ -6,7 +6,6 @@ nav_order: 1 --- # {{ page.title }} -{: .no_toc } In fact, there are multiple theorems and corollaries which bear the name Hahn–Banach. @@ -14,15 +13,6 @@ All have in common that they guarantee the existence of linear functionals with various additional properties. -<details open markdown="block"> - <summary> - Table of contents - </summary> - {: .text-delta } -- TOC -{:toc} -</details> - {% definition Sublinear Functional %} A functional $p$ on a real vector space $X$ is called *sublinear* if it is @@ -146,7 +136,7 @@ we define a functional $f_0$ by $f_0(\alpha x) = \alpha \norm{x}$ for $\alpha \i It is easy to check that $f_0$ is linear and bounded with norm $\norm{f_0} = 1$. By the Hahn–Banach Extension Theorem for Normed Spaces, there exists a bounded linear functional $f$ on $X$ extending $f_0$ with identical norm. -Hence we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$. +Hence, we have $f(x) = f_0(x) = \norm{x}$ and $\norm{f} = \norm{f_0} = 1$. {% endproof %} Recall that for a normed space $X$ we denote its (topological) dual space by $X'$. @@ -155,7 +145,7 @@ Recall that for a normed space $X$ we denote its (topological) dual space by $X' For every element $x$ of a real or complex normed space $X$ one has $$ -\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} +\norm{x} = \sup_{f \in X' \setminus \braces{0}} \frac{\abs{f(x)}}{\norm{f}} $$ and the supremum is attained. diff --git a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md index b191bb2..e7f2b70 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md @@ -30,10 +30,10 @@ This remains true, if we take closures: $\bigcup \overline{mTB_X} = Y$. Hence, we have written the space $Y$, which is assumed to have a complete norm, -as the union of countably many closed sets. It follows form the +as the union of countably many closed sets. It follows from the [Baire Category Theorem]({% link pages/general-topology/baire-spaces.md %}) that $\overline{mTB_X}$ has nonempty interior for some $m$. -Thus there are $q \in Y$ and $\alpha > 0$ +Thus, there are $q \in Y$ and $\alpha > 0$ such that $q + \alpha B_Y \subset \overline{mTB_X}$. Choose a $p \in X$ with $Tp=q$. It is a well known fact, that in a normed space diff --git a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md index 47ddd3f..1140e45 100644 --- a/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md +++ b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md @@ -14,11 +14,13 @@ Let $X$, $Y$ be normed spaces. We say that a collection $\mathcal{T}$ of bounded linear operators from $X$ to $Y$ is {: .mb-0 } -- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded for every $x \in X$, +- *pointwise bounded* if the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded + for every $x \in X$, - *uniformly bounded* if the set $\braces{\norm{T} : T \in \mathcal{T}}$ is bounded. {% enddefinition %} -Clearly, every uniformly bounded collection of operators is pointwise bounded since $\norm{Tx} \le \norm{T} \norm{x}$. +Clearly, every uniformly bounded collection of operators is pointwise bounded +since $\norm{Tx} \le \norm{T} \norm{x}$. The converse is true, if $X$ is complete: {% theorem * Uniform Boundedness Theorem %} diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md index 37e9b4d..6c2e274 100644 --- a/pages/general-topology/compactness/index.md +++ b/pages/general-topology/compactness/index.md @@ -26,7 +26,8 @@ if and only if it has the following property: then there exists a finite subcollection of $\mathcal{O}$ that covers $X$. If $\mathcal{A}$ is a collection of subsets of $X$, -let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote the collection of the complements of its members. +let $\mathcal{A}^c = \braces{ X \setminus A : A \in \mathcal{A}}$ denote +the collection of the complements of its members. Clearly, $\mathcal{B}$ is a subcollection of $\mathcal{A}$ if and only if $\mathcal{B}^c$ is a subcollection of $\mathcal{A}^c$. Moreover, note that $\mathcal{B}$ covers $X$ if and only if @@ -43,4 +44,3 @@ if and only if $\mathcal{A}^c$ consists of closed subsets of $X$. {% definition Finite Intersection Property%} TODO {% enddefinition %} - diff --git a/pages/general-topology/continuity-and-convergence.md b/pages/general-topology/continuity-and-convergence.md index 7ae4534..57e5ca9 100644 --- a/pages/general-topology/continuity-and-convergence.md +++ b/pages/general-topology/continuity-and-convergence.md @@ -1,5 +1,5 @@ --- -title: Continuity & Convergence +title: Continuity & Convergence parent: General Topology nav_order: 2 --- diff --git a/pages/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md index c0dc45a..52b2b4c 100644 --- a/pages/general-topology/metric-spaces/index.md +++ b/pages/general-topology/metric-spaces/index.md @@ -46,13 +46,13 @@ Clearly, a metric subspace of a metric space is itself a metric space. {% proposition %} Let $(X,d)$ be a (semi-)metric space. - For all $x,y,z \in X$ we have the *inverse triangle inequality* - + $$ \abs{d(x,y) - d(y,z)} \le d(x,z). $$ - For all $v,w,x,y \in X$ we have the *quadrilateral inequality* - + $$ \abs{d(v,w) - d(x,y)} \le d(v,x) + d(w,y) $$ @@ -141,27 +141,31 @@ every sequence in $X$ has at most one limit. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A mapping $f: X \to Y$ is called - *continuous at a point $x \in X$* if - + $$ - \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : + \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen $$ - *continuous* if it is continuous at every point of $X$, that is - + $$ - \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + \forall x \in X \ \ \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x' \in X : + \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen $$ - *uniformly continuous* if - + $$ - \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen + \forall\epsilon>0 \ \ \exists\delta>0 \ \ \forall x,x' \in X : + \big\lparen d_X(x,x') \le \delta \implies d_Y(f(x),f(x')) < \epsilon \big\rparen $$ - *Lipschitz continuous* if - + $$ - \exists L \ge 0 \ \ \forall x,x' \in X : d_Y(f(x),f(x')) \le L \, d_X(x,x') + \exists L \ge 0 \ \ \forall x,x' \in X : + d_Y(f(x),f(x')) \le L \, d_X(x,x') $$ {% enddefinition %} diff --git a/pages/general-topology/topological-spaces.md b/pages/general-topology/topological-spaces.md index b0b1834..cb0c30b 100644 --- a/pages/general-topology/topological-spaces.md +++ b/pages/general-topology/topological-spaces.md @@ -75,7 +75,8 @@ is the smallest topology on $X$ containing $\mathcal{A}$. {% definition Basis for a Topology %} A *basis for a topology* on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ -such that for every point $x \in X$ +such that for every point $x \in X$ + - there exists $B \in \mathcal{B}$ such that $x \in B$, - if $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$, then there exists a $B_3 \in \mathcal{B}$ @@ -85,6 +86,7 @@ such that for every point $x \in X$ {% theorem Topology Generated by a Basis %} If $X$ is set and $\mathcal{B}$ is a basis for a topology on $X$, then the topology generated by $\mathcal{B}$ equals + - the collection of all subsets $S \subset X$ with the property that for every $x \in S$ there exists a basis element $B \in \mathcal{B}$ such that $x \in B$ and $B \subset S$; @@ -125,7 +127,7 @@ then the topology generated by $\mathcal{S}$ equals Suppose $(X,\mathcal{T})$ is a topological space. A subset $S$ of $X$ is called *open* with respect to $\mathcal{T}$ -when it belongs to $\mathcal{T}$ +when it belongs to $\mathcal{T}$, and it is called *closed* with respect to $\mathcal{T}$ when its complement $X \setminus S$ belongs to $\mathcal{T}$. {% enddefinition %} @@ -137,10 +139,8 @@ if and only if its complement is closed. Let $\mathcal{C}$ be the collection of closed subsets of a topological space. Then {: .mb-0 } - $X$ and $\varnothing$ belong to $\mathcal{C}$, -- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$, -- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$. +- the intersection of any subcollection of $\mathcal{C}$ belongs to $\mathcal{C}$, +- the union of any finite subcollection $\mathcal{C}$ belongs to $\mathcal{C}$. {% endproposition %} ## The Subspace Topology - - diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md index 67f0996..f9ebc4a 100644 --- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -32,10 +32,10 @@ $$ In the following proof we omit $X$ and $d\mu$ for visual clarity. {% proof %} -By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, where $g_n = \inf_{k \ge n} f_k$. +By definition, we have $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} g_n$, +where $g_n = \inf_{k \ge n} f_k$. Now $(g_n)$ is a monotonic sequence of nonnegative measurable functions. -By the -[Monotone Convergence Theorem](#monotone-convergence-theorem) +By the [Monotone Convergence Theorem](#monotone-convergence-theorem) $$ \int \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int g_n. diff --git a/pages/measure-and-integration/lebesgue-integral/index.md b/pages/measure-and-integration/lebesgue-integral/index.md index a857d95..3418e10 100644 --- a/pages/measure-and-integration/lebesgue-integral/index.md +++ b/pages/measure-and-integration/lebesgue-integral/index.md @@ -106,7 +106,7 @@ For any measurable subset $A \subset X$ we define the *integral on $A$* of a (quasi-)integrable function $f : X \to \overline{\RR}$ (or $\CC$) by $$ -\int_A f \, d\mu = +\int_A f \, d\mu = \int_X \chi_A f \, d\mu. $$ {% enddefinition %} diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md index 023c253..8482e87 100644 --- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -1,5 +1,5 @@ --- -title: The L<sup>p</sup> Spaces +title: The L<sup>p</sup> Spaces parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 4 diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md index 637ab0c..c843881 100644 --- a/pages/measure-and-integration/measure-theory/measures.md +++ b/pages/measure-and-integration/measure-theory/measures.md @@ -14,7 +14,7 @@ is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that - $\mu(\varnothing) = 0$, - for every sequence $(A_n)_{n \in \NN}$ of pairwise disjoint sets $A_n \in \mathcal{A}$ - + $$ \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen = \sum_{n=0}^{\infty} \mu(A_n). diff --git a/pages/measure-and-integration/measure-theory/sigma-algebras.md b/pages/measure-and-integration/measure-theory/sigma-algebras.md index 5d22f6b..8f58f09 100644 --- a/pages/measure-and-integration/measure-theory/sigma-algebras.md +++ b/pages/measure-and-integration/measure-theory/sigma-algebras.md @@ -47,4 +47,3 @@ defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A} By the previous proposition, $\sigma(\mathcal{E})$ is in fact a σ-algebra on $X$. ## Products of {{ page.title }} - diff --git a/pages/measure-and-integration/measure-theory/signed-measures.md b/pages/measure-and-integration/measure-theory/signed-measures.md index 77b2416..657a28f 100644 --- a/pages/measure-and-integration/measure-theory/signed-measures.md +++ b/pages/measure-and-integration/measure-theory/signed-measures.md @@ -18,7 +18,7 @@ is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that - for every sequence $(A_n)_{n \in \NN}$ of pairwise disjoint sets $A_n \in \mathcal{A}$ {: .my-0 } - + $$ \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen = \sum_{n=0}^{\infty} \mu(A_n). diff --git a/pages/more-functional-analysis/topological-vector-spaces/index.md b/pages/more-functional-analysis/topological-vector-spaces/index.md index 745d53b..a3e6220 100644 --- a/pages/more-functional-analysis/topological-vector-spaces/index.md +++ b/pages/more-functional-analysis/topological-vector-spaces/index.md @@ -49,7 +49,7 @@ A subset $A \subset X$ is said to be {% theorem %} These properties of subsets of $X$ are stable under arbitrary intersections. -Thus we obtain the notions of +Thus, we obtain the notions of *convex hull* $\co A$, *balanced hull* $\bal A$, and *absolutely convex hull* $\aco A$. diff --git a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md index 277ecd3..ea0b22b 100644 --- a/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md +++ b/pages/more-functional-analysis/topological-vector-spaces/polar-topologies.md @@ -29,7 +29,7 @@ We say that the bilinear form $b : V \times W \to \KK$ is *nondegenerate*, if it $$ \begin{gather*} \forall v \in V : \qquad ( \forall w \in W : \angles{v,w} = 0 ) \implies v = 0 \\ -\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 +\forall w \in W : \qquad ( \forall v \in V : \angles{v,w} = 0 ) \implies w = 0 \end{gather*} $$ @@ -45,7 +45,6 @@ $$ Then $b$ is nondegenerate if and only if both $c$ and $\tilde{c}$ are injective. - {% definition Dual Pair %} A *dual pair* (or *dual system* or *duality*) $\angles{V,W}$ over a field $\KK$ is constituted by two vector spaces $V$ and $W$ over $\KK$ @@ -56,9 +55,9 @@ and a nondegenerate bilinear form $\angles{\cdot,\cdot} : V \times W \to \KK$. {% definition Weak Topology %} Suppose $\angles{X,Y}$ is a dual pair of vector spaces over a field $\KK$. -We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as -the [initial topology](/pages/general-topology/universal-constructions.html#initial-topology) induced by the maps -$\angles{\cdot,y} : X \to \KK$, where $y \in Y$. +We define the *weak topology on $X$*, denoted by $\sigma(X,Y)$, as the +[initial topology](/pages/general-topology/universal-constructions.html#initial-topology) +induced by the maps $\angles{\cdot,y} : X \to \KK$, where $y \in Y$. Similarly, the *weak topology on $Y$*, denoted by $\sigma(Y,X)$, is the initial topology induced by the maps $\angles{x,\cdot} : Y \to \KK$, where $x \in X$. diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md index 3335d78..9d70df8 100644 --- a/pages/operator-algebras/banach-algebras/index.md +++ b/pages/operator-algebras/banach-algebras/index.md @@ -80,7 +80,7 @@ $$ (\mathbf{1}-x) s_n = s_n (\mathbf{1}-x) = \mathbf{1} - x^{n+1}. $$ -In the limit $n \to \infty$ we obtain $(\mathbf{1}-x) s = s (\mathbf{1}-x) = \mathbf{1}$, +In the limit $n \to \infty$ we obtain $(\mathbf{1}-x) s = s (\mathbf{1}-x) = \mathbf{1}$, because multiplication in a Banach algebra is continuous, and because $y^n \to 0$ when $\norm{y} < 1$. This proves that $s$ is the inverse of $\mathbf{1}-x$. @@ -94,10 +94,12 @@ Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. {: .mb-0 } {: .my-0 } -- The *spectrum* of $x$ is the set $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \ +- The *spectrum* of $x$ is the set + $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \ The elements of $\sigma(x)$ are called *spectral values* of $x$. - The *resolvent set* of $x$ is the set $\rho (x) = \CC \setminus \sigma(x)$. \ - For $\lambda \in \rho(x)$ the *resolvent* of $x$ is the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \ + For $\lambda \in \rho(x)$ the *resolvent* of $x$ is + the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \ The mapping $R : \rho(x) \to \mathcal{A}$, $\lambda \mapsto R_{\lambda}$, is called *resolvent map*. {% enddefinition %} @@ -119,7 +121,7 @@ $$ {% proof %} Let $\lambda$ be in the resolvent set of $x$. -Then $\lambda - x$ is invertible, and we have for all $\mu \in \CC$ +Then $\lambda - x$ is invertible, and we have for all $\mu \in \CC$ $$ \mu - x = \bigl(\mathbf{1} - (\lambda - \mu) (\lambda - x)^{-1}\bigr) (\lambda - x). @@ -138,20 +140,22 @@ the claimed formula for its inverse follows by an application of the rule $(ab)^{-1} = b^{-1} a^{-1}$ for invertible $a,b \in \mathcal{A}$. {% endproof %} -{: .corollary #resolvent-set-is-open #spectrum-is-closed } -> The resolvent set $\rho(x)$ is open and the spectrum $\sigma(x)$ is closed. +{% corollary %} +The resolvent set $\rho(x)$ is open and the spectrum $\sigma(x)$ is closed. +{% endcorollary %} -{: .corollary #resolvent-map-is-analytic } -> Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. -> The resolvent map -> -> $$ -> R : \rho(x) \longrightarrow \mathcal{A}, \quad \lambda \longmapsto R_{\lambda} = (\lambda - x)^{-1}, -> $$ -> -> is (strongly) analytic. +{% corollary %} +Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. +The resolvent map + +$$ +R : \rho(x) \longrightarrow \mathcal{A}, \quad \lambda \longmapsto R_{\lambda} = (\lambda - x)^{-1}, +$$ + +is (strongly) analytic. +{% endcorollary %} - --- +--- {: .proposition #spectrum-is-not-empty } > Suppose $x$ is an element of a unital Banach algebra. @@ -169,7 +173,7 @@ For $\abs{\lambda} > 2 \norm{x}$ we may expand $R_{\lambda}$ into a [Neumann ser $$ R_{\lambda} = (\lambda - x)^{-1} -= \lambda^{-1} (\mathbf{1} - \lambda^{-1} x)^{-1} += \lambda^{-1} (\mathbf{1} - \lambda^{-1} x)^{-1} = \lambda^{-1} \sum_{n=0}^{\infty} (\lambda^{-1} x)^n, $$ @@ -177,7 +181,7 @@ and make the estimate $$ \norm{R_{\lambda}} -\le \abs{\lambda}^{-1} (1 - \norm{\lambda^{-1} x})^{-1} +\le \abs{\lambda}^{-1} (1 - \norm{\lambda^{-1} x})^{-1} = (\abs{\lambda} - \norm{x})^{-1} < \norm{x}^{-1}. $$ @@ -197,7 +201,7 @@ For any Banach algebra $A$, the mapping $\varphi : \CC \to A$, $\lambda \mapsto \lambda \mathbf{1}$, is linear, multiplicative and isometric, hence injective. Let $x$ be any element of $A$. -Since its +Since its [spectrum is not empty](/pages/operator-algebras/banach-algebras/index.html#spectrum-is-not-empty), there must exist a complex number $\lambda$ such that $x - \lambda \mathbf{1}$ is not invertible. diff --git a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md index ea15f87..bdf7b3d 100644 --- a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md +++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md @@ -22,7 +22,9 @@ A norm-one positive linear functional on a $C^*$-algebra is called a *state*. {% enddefinition %} {% definition State Space %} -The *state space* of a $C^*$-algebra $\mathcal{A}$, denoted by $S(\mathcal{A})$, is the set of all states of $\mathcal{A}$. +The *state space* of a $C^*$-algebra $\mathcal{A}$, +denoted by $S(\mathcal{A})$, +is the set of all states of $\mathcal{A}$. {% enddefinition %} Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$. diff --git a/pages/operator-algebras/c-star-algebras/states.md b/pages/operator-algebras/c-star-algebras/states.md index 29cf5f5..a483915 100644 --- a/pages/operator-algebras/c-star-algebras/states.md +++ b/pages/operator-algebras/c-star-algebras/states.md @@ -8,7 +8,9 @@ nav_order: 1 # {{ page.title }} {% definition State, State Space %} -A norm-one [positive linear functional]( {% link pages/operator-algebras/c-star-algebras/positive-linear-functionals.md %} ) on a C\*-algebra is called a *state*.\ +A norm-one +[positive linear functional](/pages/operator-algebras/c-star-algebras/positive-linear-functionals.html) +on a C\*-algebra is called a *state*.\ The *state space* $S(\mathcal{A})$ of a C\*-algebra $\mathcal{A}$ is the set of all its states. {% enddefinition %} @@ -30,20 +32,23 @@ Let $\omega_0, \omega_1$ be states on $\mathcal{A}$ and let $t \in (0,1)$. Consider the convex combination $\omega = (1-t)\omega_0 + t\omega_1$. Clearly, $\omega$ is linear and $\omega(\mathbf{1}) = 1$. By the triangle inequality, $\norm{\omega} \le 1$. -It follows from the lemma above that $\omega$ lies in $S(\mathcal{A})$. This proves that $S(\mathcal{A})$ is convex. +It follows from the lemma above that $\omega$ lies in $S(\mathcal{A})$. +This proves that $S(\mathcal{A})$ is convex. Next we show weak\* compactness. Since $S(\mathcal{A})$ is contained in the closed unit ball in the dual of $\mathcal{A}$, which is weak\* compact by the -[Banach–Alaoglu Theorem]({% link pages/functional-analysis-basics/banach-alaoglu-theorem.md %}), +[Banach–Alaoglu Theorem](/pages/functional-analysis-basics/banach-alaoglu-theorem.html), it will suffice to show that $S(\mathcal{A})$ is weak\* closed. -Let $(\omega_i)$ be a net of states that weak\* converges to some bounded linear functional $\omega$ on $\mathcal{A}$. +Let $(\omega_i)$ be a net of states +that weak\* converges to some bounded linear functional $\omega$ on $\mathcal{A}$. This means that $\omega_i(x) \to \omega(x)$ for every $x \in \mathcal{A}$. -For all $i$ we have $\omega_i(x) \ge 0$ for $x \ge 0$ and $\omega_i(\mathbf{1}) = 1$; hence $\omega(x) \ge 0$ for $x \ge 0$ and $\omega(\mathbf{1}) = 1$. Thus $\omega$ is again a state. +For all $i$ we have $\omega_i(x) \ge 0$ for $x \ge 0$ and $\omega_i(\mathbf{1}) = 1$; +hence $\omega(x) \ge 0$ for $x \ge 0$ and $\omega(\mathbf{1}) = 1$. +Thus, $\omega$ is again a state. This shows that the state space is weak* closed, completing the proof. {% endproof %} TODO: state space is nonempty TODO: pure states - diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md index 05405f4..b1015d1 100644 --- a/pages/spectral-theory/test/basic.md +++ b/pages/spectral-theory/test/basic.md @@ -32,25 +32,19 @@ nav_order: 2 > A *regular value* of $T$ is a complex number $\lambda$ for which the resolvent $R_{\lambda}(T)$ exists, > has dense domain and is bounded. > The set of all regular values of $T$ is called the *resolvent set* of $T$ and denoted $\rho(T)$. -> The complement of the resolvent set in the complex plane is called the *spectrum* of $T$ and denoted $\sigma(T)$. +> The complement of the resolvent set in the complex plane +> is called the *spectrum* of $T$ and denoted $\sigma(T)$. > The elements of the spectrum of $T$ are called the *spectral values* of $T$. -{: .definition-title } -> Definition (point spectrum, residual spectrum, continuous spectrum) -> -> Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$. -> The *point spectrum* $\pspec{T}$ is the set of all $\lambda \in \CC$ -> for which the resolvent $R_\lambda(T)$ does not exist. -> The *residual spectrum* $\rspec{T}$ is the set of all $\lambda \in \CC$ -> for which the resolvent $R_\lambda(T)$ exists, but is not densely defined. -> The *continuous spectrum* $\cspec{T}$ is the set of all $\lambda \in \CC$ -> for which the resolvent $R_\lambda(T)$ exists and is densely defined, but unbounded. - -| If $R_\lambda(T)$ exists, | is densely defined | and is bounded ... | ... then $\lambda$ belongs to the | -|:-------------------------:|:------------------:|:------------------:|-----------------------------------| -| ✗ | - | - | point spectrum $\pspec{T}$ | -| ✓ | ✗ | ? | residual spectrum $\rspec{T}$ | -| ✓ | ✓ | ✗ | continuous spectrum $\cspec{T}$ | -| ✓ | ✓ | ✓ | resolvent set $\rho(T)$ | +{% definition Point Spectrum, Residual Spectrum, Continuous Spectrum %} +Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$. +The *point spectrum* $\pspec{T}$ is the set of all $\lambda \in \CC$ +for which the resolvent $R_\lambda(T)$ does not exist. +The *residual spectrum* $\rspec{T}$ is the set of all $\lambda \in \CC$ +for which the resolvent $R_\lambda(T)$ exists, but is not densely defined. +The *continuous spectrum* $\cspec{T}$ is the set of all $\lambda \in \CC$ +for which the resolvent $R_\lambda(T)$ exists and is densely defined, but unbounded. +{% enddefinition %} -By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ form a partition of the complex plane. +By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ +form a partition of the complex plane. diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md index 1b9fb0c..96933db 100644 --- a/pages/unbounded-operators/adjoint-operators.md +++ b/pages/unbounded-operators/adjoint-operators.md @@ -10,4 +10,3 @@ description: > --- # {{ page.title }} - diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md index fea54be..09046f4 100644 --- a/pages/unbounded-operators/hellinger-toeplitz-theorem.md +++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md @@ -18,7 +18,7 @@ Conventions: - Operators are linear and possibly unbounded. Recall that an operator $T : D(T) \to \hilb{H}$ in a Hilbert space $\hilb{H}$ -is called *symmetric*, if is has the property +is called *symmetric*, if it has the property $$ \innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T). |