From 8b9bb9346c217874670b0f1798ab6f1cb28fdb83 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Tue, 20 Feb 2024 12:01:07 +0100 Subject: Update --- .../the-calculus-of-residues.md | 1 + .../several-complex-variables/index.md | 1 + pages/distribution-theory/index.md | 2 +- .../banach-alaoglu-theorem.md | 2 +- .../compact-operators.md | 2 +- pages/functional-analysis-basics/hilbert-spaces.md | 3 +- .../normed-spaces/index.md | 9 ++++ .../functional-analysis-basics/reflexive-spaces.md | 30 ++++++++++-- .../uniform-boundedness-theorem.md | 6 +++ .../functional-analysis-basics/weak-convergence.md | 13 +++++ pages/general-topology/connectedness.md | 1 + pages/general-topology/separation/index.md | 1 + .../bochner-integral/index.md | 1 + .../lebesgue-integral/convergence-theorems.md | 4 ++ .../lebesgue-integral/fubini-theorem.md | 1 + .../lebesgue-integral/transformation-formula.md | 1 + .../fixed-point-theorems/index.md | 1 + pages/operator-algebras/banach-algebras/index.md | 7 +-- pages/spectral-theory/test/basic.md | 55 +++++++++++----------- pages/unbounded-operators/adjoint-operators.md | 6 +-- pages/unbounded-operators/graph-and-closedness.md | 6 +-- .../hellinger-toeplitz-theorem.md | 2 +- pages/unbounded-operators/quadratic-forms.md | 10 ---- 23 files changed, 105 insertions(+), 60 deletions(-) create mode 100644 pages/functional-analysis-basics/normed-spaces/index.md create mode 100644 pages/functional-analysis-basics/weak-convergence.md (limited to 'pages') diff --git a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md index a2fa53d..e915097 100644 --- a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md +++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md @@ -3,6 +3,7 @@ title: The Calculus of Residues parent: One Complex Variable grand_parent: Complex Analysis nav_order: 4 +published: false --- # {{ page.title }} diff --git a/pages/complex-analysis/several-complex-variables/index.md b/pages/complex-analysis/several-complex-variables/index.md index 803eea4..c630e80 100644 --- a/pages/complex-analysis/several-complex-variables/index.md +++ b/pages/complex-analysis/several-complex-variables/index.md @@ -3,6 +3,7 @@ title: Several Complex Variables parent: Complex Analysis nav_order: 2 has_children: true +published: false --- # {{ page.title }} diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md index 08a3ab2..5d1b7bf 100644 --- a/pages/distribution-theory/index.md +++ b/pages/distribution-theory/index.md @@ -3,7 +3,7 @@ title: Distribution Theory nav_order: 5 has_children: true has_toc: false -published: true +published: false --- # {{ page.title }} diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md index 0913776..1b6ff81 100644 --- a/pages/functional-analysis-basics/banach-alaoglu-theorem.md +++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md @@ -1,7 +1,7 @@ --- title: Banach–Alaoglu Theorem parent: Functional Analysis Basics -nav_order: 3 +nav_order: 5 --- # {{ page.title }} diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md index 92e94ba..dc20bad 100644 --- a/pages/functional-analysis-basics/compact-operators.md +++ b/pages/functional-analysis-basics/compact-operators.md @@ -1,7 +1,7 @@ --- title: Compact Operators parent: Functional Analysis Basics -nav_order: 4 +nav_order: 6 published: false --- diff --git a/pages/functional-analysis-basics/hilbert-spaces.md b/pages/functional-analysis-basics/hilbert-spaces.md index b3ef52b..a77e5c7 100644 --- a/pages/functional-analysis-basics/hilbert-spaces.md +++ b/pages/functional-analysis-basics/hilbert-spaces.md @@ -1,7 +1,8 @@ --- title: Hilbert Spaces parent: Functional Analysis Basics -nav_order: 1 +nav_order: 7 +published: false --- # {{ page.title }} diff --git a/pages/functional-analysis-basics/normed-spaces/index.md b/pages/functional-analysis-basics/normed-spaces/index.md new file mode 100644 index 0000000..c92d8c1 --- /dev/null +++ b/pages/functional-analysis-basics/normed-spaces/index.md @@ -0,0 +1,9 @@ +--- +title: Normed Spaces +parent: Functional Analysis Basics +nav_order: 1 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md index 58ca1d3..69fefc4 100644 --- a/pages/functional-analysis-basics/reflexive-spaces.md +++ b/pages/functional-analysis-basics/reflexive-spaces.md @@ -1,7 +1,9 @@ --- title: Reflexive Spaces parent: Functional Analysis Basics -nav_order: 2 +nav_order: 4 +description: > + A normed space is said to be reflexive if the canonical embedding into its bidual is surjective. --- # {{ page.title }} @@ -28,10 +30,30 @@ The canonical embedding $C : X \to X''$ of a normed space into its bidual is well-defined and an embedding of normed spaces. {% endlemma %} +In particular, $C$ is isometric, hence injective. + {% proof %} -{% endproof %} +We have to show that, for any given $x \in X$, +$g_x$ is a bounded linear functional on $X'$. +Linearity follows from the fact that +the vector space structure on $X'$ is given by pointwise operations. +To see that $g_x$ is bounded, observe that -In particular, $C$ is isometric, hence injective. +$$ +\abs{g_x(f)} = \abs{f(x)} \le \norm{f} \norm{x} +$$ + +holds for all $f \in X'$. +Moreover, this implies that $\norm{g_x} \le \norm{x}$. +Thanks to +[Hahn–Banach](/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.html#hahn-banach-theorem-existence-of-functionals), +we know that there exists a bounded linear functional +$f \in X'$ with $\norm{f} = 1$ such that $f(x) = \norm{x}$; +hence, $\norm{g_x} = \norm{x}$. +This means that the mapping $x \mapsto g_x$ is isometric. +Clearly, this mapping is also linear, and thus an embedding +of normed spaces. +{% endproof %} {% definition Reflexivity %} A normed space is said to be *reflexive* @@ -40,7 +62,7 @@ is surjective. {% enddefinition %} If a normed space $X$ is reflexive, -then $X$ is isomorphic with $X''$, its bidual. +then $X$ is isometrically isomorphic with $X''$, its bidual. James gives a counterexample for the converse statement. {% theorem %} 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 1140e45..c88608b 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 @@ -3,6 +3,10 @@ title: Uniform Boundedness Theorem parent: The Fundamental Four grand_parent: Functional Analysis Basics nav_order: 2 +description: > + The Uniform Boundedness Theorem states that a pointwise bounded collection of + bounded linear operators from a Banach space into a normed space must be + uniformly bounded. We give a proof based on Baire’s Category Theorem. --- # {{ page.title }} @@ -84,7 +88,9 @@ $$ If $X$ is not complete, this may be false. +{% comment %} TODO: - strong operator convergence - Kreyszig 4.9-5 - Haase 15.6 +{% endcomment %} diff --git a/pages/functional-analysis-basics/weak-convergence.md b/pages/functional-analysis-basics/weak-convergence.md new file mode 100644 index 0000000..6e7f417 --- /dev/null +++ b/pages/functional-analysis-basics/weak-convergence.md @@ -0,0 +1,13 @@ +--- +title: Weak Convergence +parent: Functional Analysis Basics +nav_order: 3 +--- + +# {{ page.title }} + +{% theorem %} +{% endtheorem %} + +{% proof %} +{% endproof %} diff --git a/pages/general-topology/connectedness.md b/pages/general-topology/connectedness.md index b422775..c57dfdc 100644 --- a/pages/general-topology/connectedness.md +++ b/pages/general-topology/connectedness.md @@ -2,6 +2,7 @@ title: Connectedness parent: General Topology nav_order: 5 +published: false --- # {{ page.title }} diff --git a/pages/general-topology/separation/index.md b/pages/general-topology/separation/index.md index b4916f5..d5fdd5b 100644 --- a/pages/general-topology/separation/index.md +++ b/pages/general-topology/separation/index.md @@ -4,6 +4,7 @@ parent: General Topology nav_order: 4 has_children: true has_toc: false +published: false --- # {{ page.title }} diff --git a/pages/measure-and-integration/bochner-integral/index.md b/pages/measure-and-integration/bochner-integral/index.md index 5517934..ff9e442 100644 --- a/pages/measure-and-integration/bochner-integral/index.md +++ b/pages/measure-and-integration/bochner-integral/index.md @@ -4,6 +4,7 @@ parent: Measure and Integration nav_order: 3 has_children: true has_toc: false +published: false --- # {{ page.title }} diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md index f9ebc4a..1a34820 100644 --- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -3,6 +3,10 @@ title: Convergence Theorems parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 2 +description: > +We state and prove the most important convergence theorems of Lebesgue +integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the +Dominated Convergence Theorem. --- # {{ page.title }} diff --git a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md index 6e5179c..c87607a 100644 --- a/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md +++ b/pages/measure-and-integration/lebesgue-integral/fubini-theorem.md @@ -3,6 +3,7 @@ title: Fubini Theorem parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 2 +published: false --- # {{ page.title }} diff --git a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md index 6f02bc8..00441c3 100644 --- a/pages/measure-and-integration/lebesgue-integral/transformation-formula.md +++ b/pages/measure-and-integration/lebesgue-integral/transformation-formula.md @@ -3,6 +3,7 @@ title: Transformation Formula parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 3 +published: false --- # {{ page.title }} diff --git a/pages/more-functional-analysis/fixed-point-theorems/index.md b/pages/more-functional-analysis/fixed-point-theorems/index.md index b90bf00..1065c70 100644 --- a/pages/more-functional-analysis/fixed-point-theorems/index.md +++ b/pages/more-functional-analysis/fixed-point-theorems/index.md @@ -4,6 +4,7 @@ parent: More Functional Analysis nav_order: 1 has_children: true has_toc: false +published: false --- # {{ page.title }} diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md index 9d70df8..3a427a7 100644 --- a/pages/operator-algebras/banach-algebras/index.md +++ b/pages/operator-algebras/banach-algebras/index.md @@ -157,9 +157,10 @@ is (strongly) analytic. --- -{: .proposition #spectrum-is-not-empty } -> Suppose $x$ is an element of a unital Banach algebra. -> Then its spectrum $\sigma(x)$ is not empty. +{% proposition %} +Suppose $x$ is an element of a unital Banach algebra. +Then its spectrum $\sigma(x)$ is not empty. +{% endproposition %} {% proof %} We assume that $\sigma(x)$ is empty diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md index b1015d1..9fa409b 100644 --- a/pages/spectral-theory/test/basic.md +++ b/pages/spectral-theory/test/basic.md @@ -7,34 +7,33 @@ nav_order: 2 # {{ page.title }} -{: .definition-title } -> Definition (resolvent operator, regular value, resolvent set, spectrum, spectral value) -> -> Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$. -> We write -> -> $$ -> T_{\lambda} = T - \lambda = T - \lambda I, -> $$ -> -> where $\lambda$ is a complex number and -> $I$ is the identical operator on the domain of $T$. -> If the operator $T_{\lambda}$ is injective, -> that is, it has an inverse $T_{\lambda}^{-1}$ -> (with domain $\ran{T_{\lambda}}$), -> then we call -> -> $$ -> R_{\lambda}(T) = T_{\lambda}^{-1} = (T - \lambda)^{-1} = (T - \lambda I)^{-1} -> $$ -> -> the *resolvent operator* of $T$ for $\lambda$. -> A *regular value* of $T$ is a complex number $\lambda$ for which the resolvent $R_{\lambda}(T)$ exists, -> has dense domain and is bounded. -> The set of all regular values of $T$ is called the *resolvent set* of $T$ and denoted $\rho(T)$. -> The complement of the resolvent set in the complex plane -> is called the *spectrum* of $T$ and denoted $\sigma(T)$. -> The elements of the spectrum of $T$ are called the *spectral values* of $T$. +{% definition %} +Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$. +We write + +$$ +T_{\lambda} = T - \lambda = T - \lambda I, +$$ + +where $\lambda$ is a complex number and +$I$ is the identical operator on the domain of $T$. +If the operator $T_{\lambda}$ is injective, +that is, it has an inverse $T_{\lambda}^{-1}$ +(with domain $\ran{T_{\lambda}}$), +then we call + +$$ +R_{\lambda}(T) = T_{\lambda}^{-1} = (T - \lambda)^{-1} = (T - \lambda I)^{-1} +$$ + +the *resolvent operator* of $T$ for $\lambda$. +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 elements of the spectrum of $T$ are called the *spectral values* of $T$. +{% enddefinition %} {% definition Point Spectrum, Residual Spectrum, Continuous Spectrum %} Let $T : \dom{T} \to X$ be an operator in a complex normed space $X$. diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md index 96933db..b1c9207 100644 --- a/pages/unbounded-operators/adjoint-operators.md +++ b/pages/unbounded-operators/adjoint-operators.md @@ -1,12 +1,8 @@ --- title: Adjoint Operators parent: Unbounded Operators -nav_order: 1 +nav_order: 3 published: false -description: > - The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric - operator on a Hilbert space is bounded. We give a proof using the Uniform - Boundedness Theorem. We give another proof using the Closed Graph Theorem. --- # {{ page.title }} diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md index 04a6789..73b00a0 100644 --- a/pages/unbounded-operators/graph-and-closedness.md +++ b/pages/unbounded-operators/graph-and-closedness.md @@ -1,11 +1,7 @@ --- title: Graph and Closedness parent: Unbounded Operators -nav_order: 1 -description: > - The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric - operator on a Hilbert space is bounded. We give a proof using the Uniform - Boundedness Theorem. We give another proof using the Closed Graph Theorem. +nav_order: 2 --- # {{ page.title }} diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md index 09046f4..67cdeea 100644 --- a/pages/unbounded-operators/hellinger-toeplitz-theorem.md +++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md @@ -1,7 +1,7 @@ --- title: Hellinger–Toeplitz Theorem parent: Unbounded Operators -nav_order: 10 +nav_order: 1 description: > The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric operator on a Hilbert space is bounded. We give a proof using the Uniform diff --git a/pages/unbounded-operators/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md index cb1c44a..e2183ff 100644 --- a/pages/unbounded-operators/quadratic-forms.md +++ b/pages/unbounded-operators/quadratic-forms.md @@ -3,16 +3,6 @@ title: Quadratic Forms parent: Unbounded Operators nav_order: 5 published: false -description: > - The Hellinger–Toeplitz Theorem states that an everywhere-defined symmetric - operator on a Hilbert space is bounded. We give a proof using the Uniform - Boundedness Theorem. We give another proof using the Closed Graph Theorem. --- # {{ page.title }} - -{% definition Graph of an Operator %} -The *graph* of an operator $T$ in a Hilbert space $\hilb{H}$ -is the set of all pairs $(x,y) \in \hilb{H}\times\hilb{H}$ -where $x$ lies in the domain of $T$ and $y=Tx$. -{% enddefinition %} -- cgit v1.2.3-54-g00ecf