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| author | Justin Gassner <justin.gassner@mailbox.org> | 2023-09-12 07:36:33 +0200 |
|---|---|---|
| committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-01-13 20:41:27 +0100 |
| commit | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (patch) | |
| tree | dc42d2ae9b4a8e7ee467f59e25c9e122e63f2e04 | |
| download | site-777f9d3fd8caf56e6bc6999a4b05379307d0733f.tar.zst | |
Initial commit
68 files changed, 2327 insertions, 0 deletions
diff --git a/.cspell.yaml b/.cspell.yaml new file mode 100644 index 0000000..896a692 --- /dev/null +++ b/.cspell.yaml @@ -0,0 +1,29 @@ +version: "0.2" +language: "en_US" +minWordLength: 3 +dictionaryDefinitions: + - name: names + addWords: true + path: "./.cspell/my-cspell-dicts/names.txt" +ignorePaths: + - "./.cspell/" +dictionaries: + - latex + - names +words: + - abs + - gfm + - hilb + - innerp + - katex + - norm + - qed + - thms + - jxir + - srv + - mathbb + - enspace + - callouts + - injective + - Closedness + - vcs diff --git a/.cspell/my-cspell-dicts b/.cspell/my-cspell-dicts new file mode 160000 +Subproject 5f3c438665f20f7fa0ad17c35b87946426a0cc4 diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..2ca8682 --- /dev/null +++ b/.gitignore @@ -0,0 +1,4 @@ +_site/ +.sass-cache/ +.jekyll-cache/ +.jekyll-metadata diff --git a/.gitmodules b/.gitmodules new file mode 100644 index 0000000..7f77fb6 --- /dev/null +++ b/.gitmodules @@ -0,0 +1,3 @@ +[submodule ".cspell/my-cspell-dicts"] + path = .cspell/my-cspell-dicts + url = git.jxir.de:my-cspell-dicts diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml new file mode 100644 index 0000000..c518df3 --- /dev/null +++ b/.pre-commit-config.yaml @@ -0,0 +1,46 @@ +repos: + - repo: https://github.com/pre-commit/pre-commit-hooks + rev: v4.4.0 + hooks: + - id: file-contents-sorter + files: \.(txt)$ + - id: check-added-large-files + - id: check-case-conflict + - id: check-executables-have-shebangs + - id: check-merge-conflict + - id: check-shebang-scripts-are-executable + - id: check-symlinks + - id: check-vcs-permalinks + - id: check-yaml + - id: destroyed-symlinks + - id: end-of-file-fixer + - id: fix-byte-order-marker + - id: mixed-line-ending + - id: trailing-whitespace + - repo: https://github.com/pre-commit/mirrors-prettier + rev: v3.0.3 + hooks: + - id: prettier + types_or: [ruby, yaml] + - repo: https://github.com/streetsidesoftware/cspell-cli + rev: v7.3.0 + hooks: + - id: cspell + - repo: https://github.com/klieret/jekyll-relative-url-check + rev: v2.0.2 + hooks: + - id: jekyll-relative-url-check-markdown + - id: jekyll-relative-url-check-html + - repo: https://github.com/adrienverge/yamllint.git + rev: v1.29.0 + hooks: + - id: yamllint + args: [--strict] + - repo: https://github.com/Calinou/pre-commit-luacheck + rev: v1.0.0 + hooks: + - id: luacheck + - repo: https://github.com/JohnnyMorganz/StyLua + rev: v0.18.2 + hooks: + - id: stylua-system @@ -0,0 +1,10 @@ +# frozen_string_literal: true + +source "https://rubygems.org" + +gem "jekyll" +gem "just-the-docs" +gem "kramdown-math-katex" +gem "mini_racer" +gem 'jekyll-scholar' +gem 'jekyll-include-cache' diff --git a/_bibliography/functional-analysis-basics.bib b/_bibliography/functional-analysis-basics.bib new file mode 100644 index 0000000..45bcd37 --- /dev/null +++ b/_bibliography/functional-analysis-basics.bib @@ -0,0 +1,6 @@ +@book{kreyszig, + title = {Introductory Functional Analysis With Applications}, + author = {Erwin Kreyszig}, + publisher = {John Wiley & Sons}, + year = {1978}, +} diff --git a/_bibliography/general-topology.bib b/_bibliography/general-topology.bib new file mode 100644 index 0000000..7118fd3 --- /dev/null +++ b/_bibliography/general-topology.bib @@ -0,0 +1,8 @@ +@book{munkres, + title = {Topology}, + author = {James Munkres}, + publisher = {Pearson Education Limited}, + isbn = {978-1-292-02362-5}, + year = {2014}, + edition = {2}, +} diff --git a/_config.yaml b/_config.yaml new file mode 100644 index 0000000..0bcb235 --- /dev/null +++ b/_config.yaml @@ -0,0 +1,78 @@ +title: jxir's math pages +description: Desc + +source: . +destination: /srv/http/ +url: https://jxir.de +# baseurl: /masters-thesis +permalink: /:basename:output_ext + +theme: just-the-docs +plugins: + - jekyll-include-cache + - jekyll-scholar + +defaults: + - scope: + path: "pages" + values: + layout: default + +search: + tokenizer_separator: /[\s\-\u2013/]+/ + +#color_scheme: dark + +kramdown: + input: GFMKatex + parse_block_html: true + math_engine: katex + math_engine_opts: + { + macros: + { + "\\NN": "\\mathbb{N}", + "\\ZZ": "\\mathbb{Z}", + "\\QQ": "\\mathbb{Q}", + "\\RR": "\\mathbb{R}", + "\\CC": "\\mathbb{C}", + "\\KK": "\\mathbb{K}", + "\\hilb": "\\mathcal{#1}", + "\\abs": "\\lvert #1 \\rvert", + "\\norm": "\\lVert #1 \\rVert", + "\\braces": "\\lbrace #1 \\rbrace", + "\\innerp": "\\langle #1, #2 \\rangle", + "\\dom": "D(#1)", + "\\ran": "R(#1)", + "\\graph": "G(#1)", + "\\pspec": "\\sigma_p(#1)", + "\\cspec": "\\sigma_c(#1)", + "\\rspec": "\\sigma_r(#1)", + "\\Res": "\\operatorname{Res}", + }, + } + +qed: '<span style="float:right;">$\square\enspace$</span>' + +callouts: + definition: + title: Definition + color: grey-dk + theorem: + title: Theorem + color: green + proposition: + title: Proposition + color: green + lemma: + title: Lemma + color: green + corollary: + title: Corollary + color: green + axiom: + title: Axiom + color: yellow + +scholar: + style: chicago-author-date diff --git a/_includes/footer_custom.html b/_includes/footer_custom.html new file mode 100644 index 0000000..a29ea6e --- /dev/null +++ b/_includes/footer_custom.html @@ -0,0 +1,4 @@ +<p class="text-small text-grey-dk-100 mb-0"> + Written by Justin Gassner. + <a href="https://git.jxir.de/site/tree/{{ page.path }}">View Source</a> +</p> diff --git a/_includes/head_custom.html b/_includes/head_custom.html new file mode 100644 index 0000000..365d5f5 --- /dev/null +++ b/_includes/head_custom.html @@ -0,0 +1,33 @@ +<link + rel="stylesheet" + type="text/css" + href="{{ '/assets/katex/katex.min.css' | relative_url }}" +/> +<link + rel="preload" + href="{{ '/assets/katex/fonts/KaTeX_Main-Regular.woff2' | relative_url }}" + as="font" + type="font/woff2" + crossorigin +/> +<link + rel="preload" + href="{{ '/assets/katex/fonts/KaTeX_Math-Italic.woff2' | relative_url }}" + as="font" + type="font/woff2" + crossorigin +/> +<link + rel="preload" + href="{{ '/assets/katex/fonts/KaTeX_AMS-Regular.woff2' | relative_url }}" + as="font" + type="font/woff2" + crossorigin +/> +<link + rel="preload" + href="{{ '/assets/katex/fonts/KaTeX_Caligraphic-Regular.woff2' | relative_url }}" + as="font" + type="font/woff2" + crossorigin +/> diff --git a/_includes/nav_footer_custom.html b/_includes/nav_footer_custom.html new file mode 100644 index 0000000..46b6c5d --- /dev/null +++ b/_includes/nav_footer_custom.html @@ -0,0 +1,2 @@ +<footer class="site-footer"> +</footer> diff --git a/_includes/navigation.html b/_includes/navigation.html new file mode 100644 index 0000000..ab853aa --- /dev/null +++ b/_includes/navigation.html @@ -0,0 +1,5 @@ +<nav> + {% for item in site.data.navigation %} + <a href="{{ item.link }}" {% if page.url == item.link %}class="current"{% endif %}>{{ item.name }}</a> + {% endfor %} +</nav> diff --git a/_plugins/enunciation.rb b/_plugins/enunciation.rb new file mode 100644 index 0000000..75c777a --- /dev/null +++ b/_plugins/enunciation.rb @@ -0,0 +1,32 @@ +module Jekyll + class EnunciationTagBlock < Liquid::Block + + def initialize(tag_name, arg, parse_context) + super + @arg = arg.strip + end + + def render(context) + text = super + # If the enunciation ends with a KaTeX displayed equation, + # then we remove the bottom margin. + if text[-3,3] == "$$\n" + text += '{: .katex-display .mb-0 }' + end + # Check if a description was given. + if @arg.length < 1 + "{: .#{block_name} }\n #{text.gsub!(/^/,'> ')}" + else + "{: .#{block_name}-title }\n> #{block_name.capitalize} (#{@arg})\n>\n#{text.gsub!(/^/,'> ')}" + end + end + + end +end + +Liquid::Template.register_tag('definition', Jekyll::EnunciationTagBlock) +Liquid::Template.register_tag('theorem', Jekyll::EnunciationTagBlock) +Liquid::Template.register_tag('proposition', Jekyll::EnunciationTagBlock) +Liquid::Template.register_tag('lemma', Jekyll::EnunciationTagBlock) +Liquid::Template.register_tag('corollary', Jekyll::EnunciationTagBlock) +Liquid::Template.register_tag('axiom', Jekyll::EnunciationTagBlock) diff --git a/_plugins/proof.rb b/_plugins/proof.rb new file mode 100644 index 0000000..ab824e5 --- /dev/null +++ b/_plugins/proof.rb @@ -0,0 +1,12 @@ +module Jekyll + class ProofTagBlock < Liquid::Block + + def render(context) + text = super + "<span style=\"text-transform: uppercase; font-weight: bold; font-size: .75em;\">Proof</span> #{text} <span style=\"float:right;\">$\\square\\enspace$</span>" + end + + end +end + +Liquid::Template.register_tag('proof', Jekyll::ProofTagBlock) diff --git a/_plugins/singledollar.rb b/_plugins/singledollar.rb new file mode 100644 index 0000000..fe665ab --- /dev/null +++ b/_plugins/singledollar.rb @@ -0,0 +1,19 @@ +# https://gist.github.com/sp301415/db9f7bee03bda483aa2ca7c0ca92fbfa + +require 'kramdown/parser/kramdown' +require 'kramdown-parser-gfm' + +class Kramdown::Parser::GFMKatex < Kramdown::Parser::GFM + # Override inline math parser + @@parsers.delete(:inline_math) + + INLINE_MATH_START = /(\$+)([^\$]+)(\$+)/m + + def parse_inline_math + start_line_number = @src.current_line_number + @src.pos += @src.matched_size + @tree.children << Element.new(:math, @src.matched[1..-2], nil, category: :span, location: start_line_number) + end + + define_parser(:inline_math, INLINE_MATH_START, '\$') +end diff --git a/_sass/custom/custom.scss b/_sass/custom/custom.scss new file mode 100644 index 0000000..43a40c8 --- /dev/null +++ b/_sass/custom/custom.scss @@ -0,0 +1,3 @@ +blockquote.theorem { + border-left: none; +} diff --git a/_scripts/new.lua b/_scripts/new.lua new file mode 100755 index 0000000..e57100d --- /dev/null +++ b/_scripts/new.lua @@ -0,0 +1,112 @@ +#!/usr/bin/lua + +local lfs = require("lfs") + +local subcommand = arg[1] +if not subcommand then + print("No arguments given.") + goto exit +end + +if not (subcommand == "page" or subcommand == "dir") then + print("Unknown subcommand '" .. subcommand .. "'.") + goto exit +end + +local title = table.concat(arg, " ", 2) +if not title or title == "" then + print("No title given.") + goto exit +end + +local function slugify(str) + local strlow = string.lower(str) + local slug = string.gsub(strlow, "%W", "-") + return slug +end + +local titleslug = slugify(title) +print("Title: " .. title) +print(" Slug: " .. titleslug) + +local parent, grand_parent +local file = io.open("index.md", "r") +if file then + print("Found index.md.") + io.input(file) + for line in io.lines() do + local match = string.match(line, "title: (.*)") + if match then + parent = match + print("--> Parent: " .. parent) + end + match = string.match(line, "parent: (.*)") + if match then + grand_parent = match + print("--> Grand parent: " .. grand_parent) + end + end + io.close(file) +else + print("No index.md found.") +end + +if arg[1] == "page" then + file = io.open(titleslug .. ".md", "w") + if file then + io.output(file) + print("Writing file: " .. titleslug .. ".md") + io.write("---\ntitle: " .. title .. "\n") + if parent then + io.write("parent: " .. parent .. "\n") + end + if grand_parent then + io.write("grand_parent: " .. grand_parent .. "\n") + end + io.write([[ +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> ... + +{% proof %} +{% endproof %} +]]) + io.close(file) + end +elseif arg[1] == "dir" then + if grand_parent then + print("Cannot have deeper directories.") + goto exit + end + print("Making directory: " .. titleslug .. "/") + lfs.mkdir(titleslug) + file = io.open(titleslug .. "/index.md", "w") + if file then + io.output(file) + print("Writing file: " .. titleslug .. "/index.md") + io.write("---\ntitle: " .. title .. "\n") + if parent then + io.write("parent: " .. parent .. "\n") + end + io.write([[ +nav_order: 1 +has_children: true +has_toc: false +# cspell:words +--- + +# {{ page.title }} +]]) + io.close(file) + end +end + +::exit:: diff --git a/about.md b/about.md new file mode 100644 index 0000000..0d7d816 --- /dev/null +++ b/about.md @@ -0,0 +1,11 @@ +--- +title: About +nav_order: 1 +description: "Just the Docs is a responsive Jekyll theme with built-in search that is easily customizable and hosted on GitHub Pages." +layout: default +permalink: / +--- + +# About page + +This page tells you a little bit about me. diff --git a/favicon.ico b/favicon.ico Binary files differnew file mode 100644 index 0000000..cc016c3 --- /dev/null +++ b/favicon.ico diff --git a/pages/complex-analysis/index.md b/pages/complex-analysis/index.md new file mode 100644 index 0000000..d07109e --- /dev/null +++ b/pages/complex-analysis/index.md @@ -0,0 +1,8 @@ +--- +title: Complex Analysis +nav_order: 2 +has_children: true +# cspell:words +--- + +# {{ page.title }} diff --git a/pages/complex-analysis/one-complex-variable/basics.md b/pages/complex-analysis/one-complex-variable/basics.md new file mode 100644 index 0000000..b30d18c --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/basics.md @@ -0,0 +1,23 @@ +--- +title: Basics +parent: One Complex Variable +grand_parent: Complex Analysis +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem } +> {: #holomorphic-function-is-constant-if-derivative-vanishes } +> +> If the derivative of a holomorphic function vanishes +> throughout a connected open subset of the complex plane, +> then it must be constant on that set. +> +> More generally, if the derivative of a holomorphic function vanishes +> throughout an open subset of the complex plane, +> then it must be constant on any connected component of that set. + +{% proof %} +{% endproof %} diff --git a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md new file mode 100644 index 0000000..ccdd0ea --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md @@ -0,0 +1,102 @@ +--- +title: Cauchy's Integral Formula +parent: One Complex Variable +grand_parent: Complex Analysis +nav_order: 3 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> +> Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$. +> Let $\gamma$ be a contour in $G$ such that the interior of $\gamma$ is contained in $G$. +> Then for any point $a$ in the interior of $\gamma$, +> +> $$ +> f(a) = \frac{1}{2 \pi i} \int_{\gamma} \frac{f(z)}{z-a} \, dz. +> $$ +> {: .katex-display .mb-0 } + +{% proof %} +{% endproof %} + +{: .theorem-title } +> {{ page.title }} (Generalization) +> {: #cauchys-integral-formula-generalized } +> +> Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$. +> Then the $n$th derivative $f^{(n)}$ exists for every $n \in \NN$. +> If $\gamma$ is a contour in $G$ such that the interior of $\gamma$ is contained in $G$, +> then for any point $a$ in the interior of $\gamma$, +> +> $$ +> f^{(n)} (a) = \frac{n!}{2 \pi i} \int_{\gamma} \frac{f(z)}{(z-a)^{n+1}} \, dz. +> $$ +> {: .katex-display .mb-0 } + +{% proof %} +{% endproof %} + +The last formula may be rewritten as + +$$ +\int_{\gamma} \frac{f(z)}{(z-a)^n} \, dz = \frac{2 \pi i}{(n-1)!} f^{(n-1)}(a) +$$ + +and is often used to compute the integral. + +## Many Consequences + +{: .theorem-title } +> Cauchy's Estimate +> {: #cauchys-estimate } +> +> Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$. +> Then +> +> $$ +> \norm{f^{(n)}(a)} \le \frac{n!}{r^n} \sup_{\abs{z-a} = r} \norm{f(z)} \qquad \forall n \in \NN. +> $$ +> {: .katex-display .mb-0 } + +{% proof %} +From [{{ page.title }}](#cauchys-integral-formula-generalized) +for the circular contour around $a$ with radius $r$ we obtain + +$$ +\begin{aligned} +\norm{f^{(n)}(a)} &\le \frac{n!}{2\pi} \sup_{\abs{z-a} = r} \norm{f(z)} \, \int_{\abs{z-a} = r} \frac{dz}{\abs{z-a}^{n+1}}. +\end{aligned} +$$ + +Note that the supremum is finite (and is attained), +because $f$ is continuous and the circle is compact. +Clearly, the integral evaluates to $2 \pi r / r^{n+1}$ +and the right hand side of the inequality reduces to the desired expression. +{% endproof %} + +--- + +Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane. + +{: .theorem-title } +> Liouville's Theorem +> {: #liouvilles-theorem } +> +> Every bounded entire function is constant. + +{% proof %} +Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$. +Since $f$ is holomorphic on the whole plane, we may make +[Cauchy's Estimate](#cauchys-estimate) +for all disks centered at any point $a \in \CC$ and with any radius $r>0$. +For the first derivative, we have $\norm{f'(a)} \le M/r$, which tends to $0$ for $r \to \infty$. +Hence $f' = 0$ in the whole plane. This +[implies](/pages/complex-analysis/one-complex-variable/basics.html#holomorphic-function-is-constant-if-derivative-vanishes) +that $f$ is constant. +{% endproof %} + +--- diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md new file mode 100644 index 0000000..15412bc --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md @@ -0,0 +1,39 @@ +--- +title: Cauchy's Theorem +parent: One Complex Variable +grand_parent: Complex Analysis +nav_order: 2 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} (Homotopy Version) +> +> Let $G$ be a connected open subset of the complex plane. +> Let $f : G \to \CC$ be a holomorphic function. +> If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then +> +> $$ +> \int_{\gamma_0} \! f(z) \, dz = +> \int_{\gamma_1} \! f(z) \, dz +> $$ +> +> If $\gamma$ is a null-homotopic closed curve in $G$, then +> +> $$ +> \int_{\gamma} f(z) \, dz = 0 +> $$ + +{% proof %} +{% endproof %} + +{{ page.title }} has a converse: + +{: .theorem-title } +> Morera's Theorem +> +> Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function. +> If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$, +> then $f$ is holomorphic in $G$. diff --git a/pages/complex-analysis/one-complex-variable/index.md b/pages/complex-analysis/one-complex-variable/index.md new file mode 100644 index 0000000..4942ff8 --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/index.md @@ -0,0 +1,9 @@ +--- +title: One Complex Variable +parent: Complex Analysis +nav_order: 1 +has_children: true +# cspell:words +--- + +# {{ page.title }} diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md new file mode 100644 index 0000000..0147f31 --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/power-series.md @@ -0,0 +1,62 @@ +--- +title: Power Series +parent: One Complex Variable +grand_parent: Complex Analysis +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +{: .definition-title } +> Definition ({{ page.title }}) +> +> Let $X$ be a complex Banach space. +> A *power series* (with values in $X$) is an infinite series of the form +> +> +> $$ +> \sum_{n=0}^{\infty} x_n (z - a)^n = x_0 + x_1 (z-a) + x_2 (z-a)^2 + \cdots, +> $$ +> +> where $x_n \in X$ is the *$n$th coefficient*, +> $z$ is a complex variable and +> $a$ is the *center* of the series. + +{: .lemma } +> Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$. +> Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$. + +{% proof %} +TODO +{% endproof %} + +{: .theorem } +> Suppose $\sum_{n=0}^{\infty} x_n (z - a)^n$ is a Banach space valued power series. +> Then either +> +> - the series converges only for $z=a$ (formally $R=0$), or +> - there exists a number $0<R<\infty$ such that +> the series converges absolutely whenever $\abs{z-a} < R$ +> and diverges whenever $\abs{z-a} > R$, or +> - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$). +> +> The number $R \in [0,\infty]$ is called the *radius of convergence* of the power series. + +{% proof %} +TODO +{% endproof %} + +{: .theorem-title } +> Cauchy–Hadamard Formula +> +> Let $\sum_{n=0}^{\infty} x_n (z - a)^n$ be a Banach space valued power series +> with radius of convergence $R$. Then +> +> $$ +> \frac{1}{R} = \limsup_{n \to \infty} \norm{x_n}^{1/n}. +> $$ + +{% proof %} +TODO +{% endproof %} diff --git a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md new file mode 100644 index 0000000..b49cdf4 --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md @@ -0,0 +1,60 @@ +--- +title: The Calculus of Residues +parent: One Complex Variable +grand_parent: Complex Analysis +nav_order: 4 +# cspell:words +#published: false +--- + +# {{ page.title }} + +{: .definition-title } +> Definition (Residue) +> +> TODO + +Calculation of Residues + +If $f$ has a simple pole at $c$, then +$\Res(f,c) = \lim_{z \to c} (z-c) f(z)$. + +If $f$ has a pole of order $k$ at $c$, then + +$$ +\Res(f,c) = \frac{1}{(k-1)!} g^{(k-1)}(c), \quad \text{where} \ g(z) = (z-c)^k f(z). +$$ + +If $g$ and $h$ are holomorphic near $c$ and $h$ has a simple zero at $c$, +then $f = g/h$ has a simple pole at $c$ and + +$$ +\Res(f,c) = \frac{g(c)}{h'(c)} +$$ + + + + +{: .theorem-title } +> Residue Theorem (Basic Version) +> {: #residue-theorem-basic-version } +> +> Let $f$ be a function meromorphic in an open subset $G \subset \CC$. +> Let $\gamma$ be a contour in $G$ such that +> the interior of $\gamma$ is contained in $G$ +> and contains finitely many poles $c_1, \ldots, c_n$ of $f$. +> Then +> +> +> $$ +> \int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k) +> $$ +> {: .katex-display .mb-0 } + +{% proof %} +{% endproof %} + +TODO +- argument principle +- Rouché's theorem +- winding number diff --git a/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md new file mode 100644 index 0000000..5adc3f6 --- /dev/null +++ b/pages/complex-analysis/several-complex-variables/edge-of-the-wedge.md @@ -0,0 +1,18 @@ +--- +title: Edge of the Wedge +parent: Several Complex Variables +grand_parent: Complex Analysis +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> ... + +{% proof %} +{% endproof %} diff --git a/pages/complex-analysis/several-complex-variables/index.md b/pages/complex-analysis/several-complex-variables/index.md new file mode 100644 index 0000000..49763d5 --- /dev/null +++ b/pages/complex-analysis/several-complex-variables/index.md @@ -0,0 +1,12 @@ +--- +title: Several Complex Variables +parent: Complex Analysis +nav_order: 2 +has_children: true +# cspell:words +--- + +# {{ page.title }} + +TODO +- Edge of the Wedge Theorem diff --git a/pages/complex-analysis/weak-and-strong-analyticity.md b/pages/complex-analysis/weak-and-strong-analyticity.md new file mode 100644 index 0000000..7db1dbf --- /dev/null +++ b/pages/complex-analysis/weak-and-strong-analyticity.md @@ -0,0 +1,18 @@ +--- +title: Weak and Strong Analyticity +parent: Complex Analysis +nav_order: 3 +published: false +# cspell:words +--- + +# {{ page.title }} + +{: .definition-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> ... + +{% proof %} +{% endproof %} diff --git a/pages/distribution-theory/definitions.md b/pages/distribution-theory/definitions.md new file mode 100644 index 0000000..a800e03 --- /dev/null +++ b/pages/distribution-theory/definitions.md @@ -0,0 +1,9 @@ +--- +title: Definitions +parent: Distribution Theory +nav_order: 10 +# cspell:words +published: false +--- + +# {{ page.title }} diff --git a/pages/distribution-theory/index.md b/pages/distribution-theory/index.md new file mode 100644 index 0000000..b4b50a8 --- /dev/null +++ b/pages/distribution-theory/index.md @@ -0,0 +1,26 @@ +--- +title: Distribution Theory +nav_order: 3 +has_children: true +has_toc: false +published: true +--- + +# {{ page.title }} + +As usual, let $\mathcal{S}$ denote the space of Schwartz test functions on $\RR^n$. + +{: .definition-title } +> Definition (Operator Valued Distribution) +> +> Let $\hilb{H}$ be a Hilbert space. +> An *operator valued tempered distribution* $\Phi$ (on $\RR^n$) +> is a mapping that associates to each test function $f \in \mathcal{S}$ +> an unbounded linear operator $\Phi(f)$ in $\hilb{H}$ such that +> {: .mb-0 } +> +> {: .my-0 } +> - there is a dense linear subspace $\mathcal{D}$ of $\hilb{H}$ that +> 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. diff --git a/pages/distribution-theory/sobolev-theory.md b/pages/distribution-theory/sobolev-theory.md new file mode 100644 index 0000000..931731f --- /dev/null +++ b/pages/distribution-theory/sobolev-theory.md @@ -0,0 +1,9 @@ +--- +title: Sobolev Theory +parent: Distribution Theory +nav_order: 10 +# cspell:words +published: false +--- + +# {{ page.title }} diff --git a/pages/functional-analysis-basics/banach-alaoglu-theorem.md b/pages/functional-analysis-basics/banach-alaoglu-theorem.md new file mode 100644 index 0000000..59e4a92 --- /dev/null +++ b/pages/functional-analysis-basics/banach-alaoglu-theorem.md @@ -0,0 +1,19 @@ +--- +title: Banach–Alaoglu Theorem +parent: Functional Analysis Basics +nav_order: 3 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> The closed unit ball in the dual of a normed space is weak\* compact. + +{% proof %} +{% endproof %} + +## Generalization: Alaoglu–Bourbaki diff --git a/pages/functional-analysis-basics/compact-operators.md b/pages/functional-analysis-basics/compact-operators.md new file mode 100644 index 0000000..b114c24 --- /dev/null +++ b/pages/functional-analysis-basics/compact-operators.md @@ -0,0 +1,44 @@ +--- +title: Compact Operators +parent: Functional Analysis Basics +nav_order: 4 +published: false +# cspell:words +--- + +# {{ page.title }} + +{: .definition-title } +> Definition (Compact Linear Operator) +> {: #compact-operator } +> +> A linear operator $T : X \to Y$, +> where $X$ and $Y$ are normed spaces, +> is said to be a *compact linear operator*, +> if for every bounded subset $M \subset X$ +> the image $TM$ is relatively compact in $Y$. + +{: .proposition-title } +> Proposition (Characterisation of Compactness) +> +> Let $X$ and $Y$ be normed spaces. +> A linear operator $T : X \to Y$ is compact if and only if +> for every bounded sequence $(x_n)$ in $X$ +> the image sequence $(Tx_n)$ in $Y$ has a convergent subseqence. + +{: .proposition-title } +> Every compact linear operator is bounded. + +{: .proposition-title } +> Proposition (Compactness of Zero and Identity) +> +> The zero operator on any normed space is compact. +> The indentity operator on a normed space $X$ is compact if and only if $X$ has finite dimension. + +{: .proposition-title } +> Proposition (The Space of Compact Linear Operators) +> +> The set $C(X,Y)$ of compact linear operator from a normed space $X$ into a normed space $Y$ +> form a linear subspace of the space $B(X,Y)$ of bounded linear operators from $X$ into $Y$. +> If $Y$ is a Banach space, then $C(X,Y)$ is a closed linear subspace of the Banach space +> $B(X,Y)$ and hence itself a Banach space. diff --git a/pages/functional-analysis-basics/index.md b/pages/functional-analysis-basics/index.md new file mode 100644 index 0000000..1b7fd69 --- /dev/null +++ b/pages/functional-analysis-basics/index.md @@ -0,0 +1,11 @@ +--- +title: Functional Analysis Basics +nav_order: 3 +has_children: true +--- + +# {{ page.title }} + +## Recommended Textbooks + +{% bibliography --file functional-analysis-basics %} diff --git a/pages/functional-analysis-basics/reflexive-spaces.md b/pages/functional-analysis-basics/reflexive-spaces.md new file mode 100644 index 0000000..dee0e55 --- /dev/null +++ b/pages/functional-analysis-basics/reflexive-spaces.md @@ -0,0 +1,123 @@ +--- +title: Reflexive Spaces +parent: Functional Analysis Basics +nav_order: 2 +# cspell:words +--- + +# {{ page.title }} + +{: .definition-title } +> Definition (Canonical Embedding) +> +> Let $X$ be a normed space. +> The mapping +> +> $$ +> C : X \longrightarrow X'', \quad x \mapsto g_x, +> $$ +> +> where the functional $g_x$ on $X'$ is defined by +> +> $$ +> g_x(f) = f(x) \quad \text{for $f \in X'$,} +> $$ +> +> is called the *canonical embedding* of $X$ into its bidual $X''$. + +{: .lemma } +> The canonical embedding $C : X \to X''$ of a normed space into its bidual +> is well-defined and an embedding of normed spaces. + +{% proof %} +{% endproof %} + +In particular, $C$ is isometric, hence injective. + +{: .definition-title } +> Definition (Reflexivity) +> +> A normed space is said to be *reflexive* +> if the canonical embedding into its bidual +> is surjective. + +If a normed space $X$ is reflexive, +then $X$ is isomorphic with $X''$, its bidual. +James gives a counterexample for the converse statement. + +{: .theorem } +> If a normed space is reflexive, +> then it is complete; hence a Banach space. + +{% proof %} +{% endproof %} + +{: .theorem } +> If a normed space $X$ is reflexive, +> then the weak and weak$^*$ topologies on $X'$ agree. + +{% proof %} +By definition, the weak and weak$^*$ topologies on $X'$ +are the initial topologies induced by the sets of functionals +$X''$ and $C(X)$, respectively. +Since $X$ is reflexive, those sets are equal. +{% endproof %} + +The converse is true as well. Proof: TODO + +{: .theorem } +> If a normed space $X$ is reflexive, +> then its dual $X'$ is reflexive. + +{% proof %} +Since $X$ is reflexive, +the canonical embedding + +$$ +C : X \longrightarrow X'', \quad C(x)(f) = f(x), \quad x \in X, f \in X', +$$ + +is an isomorphism. +Therefore, the the dual map + +$$ +C' : X''' \longrightarrow X', \quad C'(h)(x) = h(C(x)), \quad x \in X, h \in X''', +$$ + +is an isomorphism as well. +A priori, it is not clear how $C'$ is related to +the canonical embedding + +$$ +D : X' \longrightarrow X''', \quad D(f)(g) = g(f), \quad f \in X', g \in X''. +$$ + +To show that $D$ is surjective, +consider any element $h$ in $X'''$. +We claim that $h=D(f)$ with $f=C'(h)$. +Let $g$ be any element of $X''$. +It is of the form $g=C(x)$ with $x \in X$ unique, because $X$ is reflexive. +We have + +$$ +h(g) = h(C(x)) = C'(h)(x) = f(x) +$$ + +by the definition of $C'$. +On the other hand, + +$$ +D(f)(g) = g(f) = C(x)(f) = f(x) +$$ + +by the definitions of $D$ and $C$. +This shows that $D$ is surjective, hence $X'$ is reflexive. +In fact, we have shown more: $D = (C')^{-1}$. +{% endproof %} + +{: .theorem } +> Every finite-dimensional normed space is reflexive. +> + +{: .theorem } +> Every Hilbert space is reflexive. 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 new file mode 100644 index 0000000..f8b8254 --- /dev/null +++ b/pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md @@ -0,0 +1,31 @@ +--- +title: Closed Graph Theorem +parent: The Fundamental Four +grand_parent: Functional Analysis Basics +nav_order: 4 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> An (everywhere-defined) linear operator between Banach spaces is bounded +> iff its graph is closed. + +We prove a slightly more general version: + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }}-variant } +> +> Let $X$ and $Y$ be Banach spaces +> and $T : \dom{T} \to Y$ a linear operator +> with domain $\dom{T}$ closed in $X$. +> Then $T$ is bounded if and only if +> its graph $\graph{T}$ is closed. + +{% proof %} +{% 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 new file mode 100644 index 0000000..9d21d41 --- /dev/null +++ b/pages/functional-analysis-basics/the-fundamental-four/hahn-banach-theorem.md @@ -0,0 +1,147 @@ +--- +title: Hahn–Banach Theorem +parent: The Fundamental Four +grand_parent: Functional Analysis Basics +nav_order: 1 +--- + +# {{ page.title }} + +In fact, there are multiple theorems and corollaries +which bear the name Hahn–Banach. +All have in common that +they guarantee the existence of linear functionals +with various additional properties. + +{: .definition-title } +> Definition (Sublinear Functional) +> +> A functional $p$ on a real vector space $X$ +> is called *sublinear* if it is +> {: .mb-0 } +> +> {: .mt-0 .mb-0 } +> - *positive-homogenous*, that is +> {: .mt-0 .mb-0 } +> +> $$ +> p(\alpha x) = \alpha \, p(x) \qquad \forall \alpha \ge 0, \ \forall x \in X, +> $$ +> +> - and satisfies the *triangle inequality* +> {: .mt-0 .mb-0 } +> +> $$ +> p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X. +> $$ +> {: .katex-display .mb-0 } + +If $p$ is a sublinear functional, +then $p(0)=0$ and $p(-x) \ge -p(x)$ for all $x$. + +Every norm on a real vector space is a sublinear functional. + +{: .theorem-title } +> {{ page.title }} (Basic Version) +> +> Let $p$ be a sublinear functional on a real vector space $X$. +> Then there exists a linear functional $f$ on $X$ satisfying +> $f(x) \le p(x)$ for all $x \in X$. + +## Extension Theorems + +{: .theorem-title } +> {{ page.title }} (Extension, Real Vector Spaces) +> +> Let $p$ be a sublinear functional on a real vector space $X$. +> Let $f$ be a linear functional +> which is defined on a linear subspace $Z$ of $X$ +> and satisfies +> +> $$ +> f(x) \le p(x) \qquad \forall x \in Z. +> $$ +> +> Then $f$ has a linear extension $\tilde{f}$ to $X$ such that +> +> $$ +> \tilde{f}(x) \le p(x) \qquad \forall x \in X. +> $$ + +{% proof %} +{% endproof %} + +{: .definition-title } +> Definition (Semi-Norm) +> +> We call a real-valued functional $p$ on a real or complex vector space $X$ +> a *semi-norm* if it is +> {: .mb-0 } +> +> {: .mt-0 .mb-0 } +> - *absolutely homogenous*, that is +> {: .mt-0 .mb-0 } +> +> $$ +> p(\alpha x) = \abs{\alpha} \, p(x) \qquad \forall \alpha \in \KK \ \forall x \in X, +> $$ +> - and satisfies the *triangle inequality* +> {: .mt-0 .mb-0 } +> +> $$ +> p(x+y) \le p(x) + p(y) \qquad \forall x,y \in X. +> $$ +> {: .katex-display .mb-0 } + +{: .theorem-title } +> {{ page.title }} (Extension, Real and Complex Vector Spaces) +> +> Let $p$ be a semi-norm on a real or complex vector space $X$. +> Let $f$ be a linear functional +> which is defined on a linear subspace $Z$ of $X$ +> and satisfies +> +> $$ +> \abs{f(x)} \le p(x) \qquad \forall x \in Z. +> $$ +> +> Then $f$ has a linear extension $\tilde{f}$ to $X$ such that +> +> $$ +> \abs{\tilde{f}(x)} \le p(x) \qquad \forall x \in X. +> $$ + +{: .theorem-title } +> {{ page.title }} (Extension, Normed Spaces) +> +> Let $X$ be a real or complex normed space +> and let $f$ be a bounded linear functional +> defined on a linear subspace $Z$ of $X$. +> Then $f$ has a bounded linear extension $\tilde{f}$ to $X$ such that $\norm{\tilde{f}} = \norm{f}$. + +{% proof %} +We apply the preceding theorem with $p(x) = \norm{f} \norm{x}$ +and obtain a linear extension $\tilde{f}$ of $f$ to $X$ +satisfying $\abs{\tilde{f}(x)} \le \norm{f} \norm{x}$ for all $x \in X$. +This implies that $\tilde{f}$ is bounded and $\norm{\tilde{f}} \le \norm{f}$. +We have $\norm{\tilde{f}} \ge \norm{f}$, because $\tilde{f}$ extends $f$. +{% endproof %} + +Corollaries + +Important consequence: canonical embedding into bidual + +## Separation Theorems + +{: .theorem-title } +> {{ page.title }} (Separation, Point and Closed Subspace) +> +> Suppose $Z$ is a closed subspace +> of a normed space $X$ and $x$ lies in $X \setminus Z$. +> Then there exists a bounded linear functional on $X$ +> which vanishes on $Z$ but has a nonzero value at $x$. + +{: .theorem-title } +> {{ page.title }} (Separation, Convex Sets) +> +> TODO diff --git a/pages/functional-analysis-basics/the-fundamental-four/index.md b/pages/functional-analysis-basics/the-fundamental-four/index.md new file mode 100644 index 0000000..e814571 --- /dev/null +++ b/pages/functional-analysis-basics/the-fundamental-four/index.md @@ -0,0 +1,8 @@ +--- +title: The Fundamental Four +parent: Functional Analysis Basics +nav_order: 2 +has_children: true +--- + +# {{ page.title }} 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 new file mode 100644 index 0000000..53da008 --- /dev/null +++ b/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.md @@ -0,0 +1,109 @@ +--- +title: Open Mapping Theorem +parent: The Fundamental Four +grand_parent: Functional Analysis Basics +nav_order: 3 +# cspell:words surjective bijective +--- + +# {{ page.title }} + +Recall that a mapping $T : X \to Y$, +where $X$ and $Y$ are topological spaces, +is called *open* if the image under $T$ of each open set of $X$ +is open in $Y$. + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> A bounded linear operator between Banach spaces is open +> if and only if it is surjective. + +{% proof %} +Let $X$ and $Y$ be Banach spaces +and let $T : X \to Y$ be a bounded linear operator. +Let $B_X$ and $B_Y$ denote the open unit balls in $X$ and $Y$, respectively. + +First, suppose that $T$ is surjective. +The balls $m B_X$, $m \in \NN$, cover $X$. +Since $T$ is surjective, +their images $mTB_X$ cover $Y$. +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 +[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$ +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 +the translation by a vector and the multiplication with a nonzero scalar +are homeomorphisms and thus compatible with taking the closure. +We conclude $\alpha B_Y \subset \overline{T(mB_X-q)}$. +Since $mB_X-q$ is a bounded set, +it is contained in a ball $\beta B_X$ for some $\beta > 0$. +Thus, $\alpha B_Y \subset \overline{T \beta B_X} = \beta \overline{TB_X}$. +With $\gamma := \alpha / \beta > 0$ we obtain $\gamma B_Y \subset \overline{TB_X}$. + +Clearly, every $y \in \gamma B_Y$ is the limit of a sequence $(Tx_n)$, +where $x_n \in B_X$. +However, the sequence $(x_n)$ *may not converge*! +We show that it is possible to find a *convergent* sequence $(s_n)$ in $4B_X$ +such that $Ts_n \to y$. +To construct $(s_n)$, we recursively define a sequence $(y_k)$ +with $y_k \in 2^{-k} \gamma B_Y$ for $k \in \NN_0$. +The sequence starts with $y_0 := y \in 2^0 \gamma B_Y$. +Given $y_k \in 2^{-k} \gamma B_Y$, one has $y_k \in \overline{T 2^{-k} B_X}$. +By the definition of closure, there exists a $x_k \in 2^{-k} B_X$ +such that $Tx_k$ lies in the open $2^{-(k+1)} \gamma$-ball about $y_k$. +This means that $y_{k+1} := y_k - Tx_k \in 2^{-(k+1)}\gamma B_Y$. +Now define $s_n$ as the $n$-th partial sum of the series $\sum_{k=0}^{\infty} x_k$. +The series converges, +because it converges absolutely (Here we use the completeness of $X$). +The latter is true because $\sum \norm{x_k} \le \sum 2^{-k} = 3$. +This also shows that each $s_n$ and the limit $x := \lim s_n$ lie in $4B_X$. +The auxiliary sequence $(y_n)$ converges to $0$ by construction. +Therefore, in the limit $n \to \infty$ + +$$ +Ts_n = \sum_{k=0}^{n} Tx_k = \sum_{k=0}^{n} y_k - y_{k+1} += y_0 - y_{n+1} \to y_0 = y, +$$ + +as desired. +It follows from the continuity of $T$ that $Ts_n \to Tx$, thus $Tx = y$. + +In the preceding paragraph it was proven that $\gamma B_Y \subset 4TB_X$. +Hence, $\delta B_Y \subset TB_X$ where $\delta := \gamma/4$. +To show that $T$ is open, consider any open set $U \subset X$. +If $y$ lies in $TU$, there exists a $x \in U$ such that $Tx=y$. +Since $U$ is open, there is an $\epsilon > 0$ such that $x+\epsilon B_X \subset U$. +Applying $T$, we find $y + \epsilon TB_X \subset TU$. +Combine with $\delta B_Y \subset TB_X$ to see $y + \epsilon \delta B_X \subset TU$. +Hence, $TU$ is open. +This shows that $T$ is indeed an open mapping. + +Conversely, suppose that $T$ is open. TODO +{% endproof %} + +--- + +XXX injective +For a bijective mapping between topological spaces, to say that it is open, +is equivalent to saying that its inverse is continuous. +The inverse of a bijective linear map between normed spaces is automatically linear +and thus continuous if and only if it is bounded. +As a corollary to the {{ page.title }} we obtain the following: + +{: .corollary-title } +> Bounded Inverse Theorem +> {: #bounded-inverse-theorem } +> +> If a bounded linear operator between Banach spaces is bijective, +> then its inverse is bounded. +XXX relax to injective + +Also known as *Inverse Mapping 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 new file mode 100644 index 0000000..13460da --- /dev/null +++ b/pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md @@ -0,0 +1,76 @@ +--- +title: Uniform Boundedness Theorem +parent: The Fundamental Four +grand_parent: Functional Analysis Basics +nav_order: 2 +description: > + The +# spellchecker:words preimages pointwise +--- + +# {{ page.title }} + +Also known as *Uniform Boundedness Principle* and *Banach–Steinhaus Theorem*. + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> If $\mathcal{T}$ is a set of bounded linear operators +> from a Banach space $X$ into a normed space $Y$ such that +> $\braces{\norm{Tx} : T \in \mathcal{T}}$ +> is a bounded set for every $x \in X$, then +> $\braces{\norm{T} : T \in \mathcal{T}}$ +> is a bounded set. + +{% proof %} +For each $n \in \NN$ the set + +$$ +A_n = \bigcap_{T \in \mathcal{T}} \braces{x \in X : \norm{Tx} \le n} +$$ + +is closed, since it is the intersection +of the preimages of the closed interval $[0,n]$ +under the continuous maps $x \mapsto \norm{Tx}$. +Given any $x \in X$, +the set $\braces{\norm{Tx} : T \in \mathcal{T}}$ is bounded by assumption. +This means that there exists a $n \in \NN$ +such that $\norm{Tx} \le n$ for all $T \in \mathcal{T}$. +In other words, $x \in A_n$. +Thus we have show that $\bigcup A_n = X$. +XXX Apart from the trivial case $X = \emptyset$, +the union $\bigcup A_n$ has nonempty interior. +Now, utilizing the completeness of $X$, the +[Baire Category Theorem]({% link pages/general-topology/baire-spaces.md %}) +implies that there exists a $m \in \NN$ such that $A_m$ has nonempty interior. +It follows that $A_m$ contains an open ball $B(y,\epsilon)$. + +To show that $\braces{\norm{T} : T \in \mathcal{T}}$ is bounded, +let $z \in X$ with $\norm{z} \le 1$. +Then $y+\epsilon z \in B(y,\epsilon)$. +Using the reverse triangle inequality and the linearity of $T$, we find + +$$ +\epsilon \norm{Tz} \le \norm{Ty} + \norm{T(y + \epsilon z)} \le 2m. +$$ + +This proves $\norm{T} \le 2m/\epsilon$ for all $T \in \mathcal{T}$. +{% endproof %} + +--- + +In particular, for a sequence of operators $(T_n)$, +if there are pointwise bounds $c_x$ such that + +$$ +\norm{T_n x} \le c_x \quad \forall n \in \NN, \forall x \in X, +$$ + +the theorem implies the existence of bound $c$ such that + +$$ +\norm{T_n} \le c \quad \forall n \in \NN. +$$ + +If $X$ is not complete, this may be false. diff --git a/pages/general-topology/baire-spaces.md b/pages/general-topology/baire-spaces.md new file mode 100644 index 0000000..6bd7d9f --- /dev/null +++ b/pages/general-topology/baire-spaces.md @@ -0,0 +1,83 @@ +--- +title: Baire Spaces +parent: General Topology +nav_order: 1 +description: > + A Baire space is a topological space with the property that the intersection + of countably many dense open subsets is still dense. One version of the Baire + Category Theorem states that complete metric spaces are Baire spaces. We give + a self-contained proof of Baire's Category Theorem by contradiction. +# spellchecker:words +--- + +# {{ page.title }} + +{: .definition } +> A topological space is said to be a *Baire space*, +> if any of the following equivalent conditions holds: +> {: .mb-0 } +> +> - The intersection of countably many dense open subsets is still dense. +> - The union of countably many closed subsets with empty interior has empty interior. +> {: .mt-0 .mb-0 } + +Note that +a set is dense in a topological space +if and only if +its complement has empty interior. + +Any sufficient condition +for a topological space to be a Baire space +constitutes a *Baire Category Theorem*, +of which there are several. +Here we give one +that is commonly used in functional analysis. + +{: .theorem-title } +> Baire Category Theorem #1 +> {: #baire-category-theorem } +> +> Every complete metric space is a Baire space. + +{% proof %} +Let $X$ be a metric space +with complete metric $d$. +Suppose that $X$ is not a Baire space. +Then there is a countable collection $\braces{U_n}$ of dense open subsets of $X$ +such that the intersection $U := \bigcap U_n$ is not dense in $X$. + +In a metric space, any nonempty open set contains an open ball. +It is also true, that any nonempty open set contains a closed ball, +since $\overline{B(y,\delta_1)} \subset B(y,\delta_2)$ if $\delta_1 < \delta_2$. + +We construct a sequence $(B_n)$ of open balls $B_n := B(x_n,\epsilon_n)$ satisfying + +$$ +\overline{B_{n+1}} \subset B_n \cap U_n \quad \epsilon_n < \tfrac{1}{n} \quad \forall n \in \NN, +$$ + +as follows: By hypothesis, +the interior of $X \setminus U$ is not empty (otherwise $U$ would be dense in $X$), +so we may choose an open ball $B_1$ with $\epsilon_1 < 1$ +such that $\overline{B_1} \subset X \setminus U$. +Given $B_n$, +the set $B_n \cap U_n$ is nonempty, because $U_n$ is dense in $X$, +and it is open, because $B_n$ and $U_n$ are open. +This allows us to choose an open ball $B_{n+1}$ as desired. + +Note that by construction $B_m \subset B_n$ for $m \ge n$, +thus $d(x_m,x_n) < \epsilon_n < \tfrac{1}{n}$. +Therefore, the sequence $(x_n)$ is Cauchy +and has a limit point $x$ by completeness. +In the limit $m \to \infty$, we obtain $d(x,x_n) \le \epsilon_n$ (strictness is lost), +hence $x \in \overline{B_n}$ for all $n$. +This shows that $x \in U_n$ for all $n$, that is $x \in U$. +On the other hand, $x \in \overline{B_1} \subset X \setminus U$, +in contradiction to the preceding statement. +{% endproof %} + +{: .theorem-title } +> Baire Category Theorem #2 +> {: #baire-category-theorem } +> +> Every compact Hausdorff space is a Baire space. diff --git a/pages/general-topology/baire-spaces.md.txt b/pages/general-topology/baire-spaces.md.txt new file mode 100644 index 0000000..eabe792 --- /dev/null +++ b/pages/general-topology/baire-spaces.md.txt @@ -0,0 +1,73 @@ +--- +title: Baire Spaces +parent: General Topology +nav_order: 1 +description: > + A Baire space is a topological space with the property that the intersection + of countably many dense open subsets is still dense. One version of the Baire + Category Theorem states that complete metric spaces are Baire spaces. We give + a self-contained proof of Baire's Category Theorem by contradiction. +# spellchecker:words +--- + +# + + +A topological space is said to be a *Baire space*, +if any of the following equivalent conditions holds: +> +- The intersection of countably many dense open subsets is still dense. +- The union of countably many closed subsets with empty interior has empty interior. + + +Note that +a set is dense in a topological space +if and only if +its complement has empty interior. + +Any sufficient condition +for a topological space to be a Baire space +constitutes a *Baire Category Theorem*, +of which there are several. +Here we give one +that is commonly used in functional analysis. + + +Baire Category Theorem + +> +Complete metric spaces are Baire spaces. + +**Proof:** +Let C-C-C be a metric space +with complete metric D-D-D. +Suppose that F-F-F is not a Baire space. +Then there is a countable collection G-G-G of dense open subsets of B-B-B +such that the intersection C-C-C is not dense in D-D-D. + +In a metric space, any nonempty open set contains an open ball. +It is also true, that any nonempty open set contains a closed ball, +since F-F-F if G-G-G. + +We construct a sequence B-B-B of open balls C-C-C satisfying + +V-V-V +as follows: By hypothesis, +the interior of D-D-D is not empty (otherwise F-F-F would be dense in G-G-G), +so we may choose an open ball B-B-B with C-C-C +such that D-D-D. +Given F-F-F, +the set G-G-G is nonempty, because B-B-B is dense in C-C-C, +and it is open, because D-D-D and F-F-F are open. +This allows us to choose an open ball G-G-G as desired. + +Note that by construction B-B-B for C-C-C, +thus D-D-D. +Therefore, the sequence F-F-F is Cauchy +and has a limit point G-G-G by completeness. +In the limit B-B-B, we obtain C-C-C (strictness is lost), +hence D-D-D for all F-F-F. +This shows that G-G-G for all B-B-B, that is C-C-C. +On the other hand, D-D-D, +in contradiction to the preceding statement. + diff --git a/pages/general-topology/compactness/basics.md b/pages/general-topology/compactness/basics.md new file mode 100644 index 0000000..a1dded7 --- /dev/null +++ b/pages/general-topology/compactness/basics.md @@ -0,0 +1,43 @@ +--- +title: Basics +parent: Compactness +grand_parent: General Topology +nav_order: 1 +published: false +# cspell:words +--- + +# {{ page.title }} of Compact Spaces + +*Compact space* is short for compact topological space. + +{: .definition } +> Suppose $X$ is a topological space. +> A *covering* of $X$ is a collection $\mathcal{A}$ +> of subsets of $X$ such that +> $\bigcup \mathcal{A} = X$. +> A covering $\mathcal{A}$ of $X$ is called *open* +> if each member of the collection $\mathcal{A}$ +> is open in $X$. +> A covering $\mathcal{A}$ is called *finite* +> the collection $\mathcal{A}$ is finite. +> A *subcovering* of a covering $\mathcal{A}$ of $X$ +> is a subcollection $\mathcal{B}$ of $\mathcal{A}$ +> such that $\mathcal{B}$ is a covering of $X$. + +{: .definition } +> A topological space $X$ is called *compact* +> if every open covering of $X$ +> has a finite subcovering. + +{: .theorem } +> Every closed subspace of a compact space is compact. + +{% proof %} +{% endproof %} + +{: .theorem } +> Every compact subspace of a Hausdorff space is closed. + +{% proof %} +{% endproof %} diff --git a/pages/general-topology/compactness/index.md b/pages/general-topology/compactness/index.md new file mode 100644 index 0000000..60c29a0 --- /dev/null +++ b/pages/general-topology/compactness/index.md @@ -0,0 +1,9 @@ +--- +title: Compactness +parent: General Topology +nav_order: 1 +has_children: true +# cspell:words +--- + +# {{ page.title }} diff --git a/pages/general-topology/compactness/tychonoff-product-theorem.md b/pages/general-topology/compactness/tychonoff-product-theorem.md new file mode 100644 index 0000000..2ae78e4 --- /dev/null +++ b/pages/general-topology/compactness/tychonoff-product-theorem.md @@ -0,0 +1,19 @@ +--- +title: Tychonoff Product Theorem +parent: Compactness +grand_parent: General Topology +nav_order: 2 +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> The product of (an arbitrary family of) compact spaces is compact. + +{% proof %} +TODO +{% endproof %} diff --git a/pages/general-topology/index.md b/pages/general-topology/index.md new file mode 100644 index 0000000..507c29a --- /dev/null +++ b/pages/general-topology/index.md @@ -0,0 +1,11 @@ +--- +title: General Topology +nav_order: 1 +has_children: true +--- + +# {{ page.title }} + +## Recommended Textbooks + +{% bibliography --file general-topology %} diff --git a/pages/general-topology/jordan-curve-theorem.md b/pages/general-topology/jordan-curve-theorem.md new file mode 100644 index 0000000..9da141e --- /dev/null +++ b/pages/general-topology/jordan-curve-theorem.md @@ -0,0 +1,18 @@ +--- +title: Jordan Curve Theorem +parent: General Topology +nav_order: 1 +published: false +# cspell:words +--- + +# {{ page.title }} + +{: .theorem-title } +> {{ page.title }} +> {: #{{ page.title | slugify }} } +> +> ... + +{% proof %} +{% endproof %} diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md new file mode 100644 index 0000000..2fb8f03 --- /dev/null +++ b/pages/operator-algebras/banach-algebras/index.md @@ -0,0 +1,259 @@ +--- +title: Banach Algebras +parent: Operator Algebras +nav_order: 1 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +{% definition Banach Algebra %} +A *Banach algebra* $\mathcal{A}$ is a complex Banach space +endowed with a binary operation $(x,y) \mapsto xy$, called *product*, +that makes the underlying vector space into an associative algebra, +and that satisfies + +$$ +\norm{xy} \le \norm{x} \norm{y} \quad \forall x,y \in \mathcal{A}. +$$ +{% enddefinition %} + +The algebraic properties required of the product are explicitly: + +$$ +\begin{align*} +x(y+y') &= xy + xy' &\quad +(\lambda x)y &= \lambda (xy) &\quad +(xy)z &= x(yz) \\ +(x+x')y &= xy + x'y & +x(\lambda y) &= \lambda (xy) +\end{align*} +$$ + +The topological property is sometimes described by saying +that the norm is *submultiplicative*. + +{% definition Commutative Banach Algebra %} +A Banach algebra $\mathcal{A}$ is said to be *commutative* (or *abelian*) if +$xy = yx$ holds for all $x,y \in \mathcal{A}$. +{% enddefinition %} + +{% definition Unital Banach Algebra %} +An element $e$ of a Banach algebra $\mathcal{A}$ is called a *unit* (or an *identity*), +if $\norm{e} = 1$ and $ex=x=xe$ for all $x \in \mathcal{A}$. +We say that $\mathcal{A}$ is an *unital* Banach algebra, if $\mathcal{A}$ contains a unit. +{% enddefinition %} + +It is easy to see that a Banach algebra has at most one unit. + +{: .proposition-title #neumann-series } +> Proposition (Neumann Series) +> +> Let $\mathcal{A}$ be a unital Banach algebra +> and let $x \in \mathcal{A}$ satisfy $\norm{x} < 1$. +> Then $\mathbf{1}-x$ is invertible +> and the inverse is given by the series +> +> $$ +> (\mathbf{1}-x)^{-1} = \sum_{n=0}^{\infty} x^n, +> $$ +> +> which converges absolutely in norm. +> Moreover, we have the estimate +> +> $$ +> \norm{(\mathbf{1}-x)^{-1}} \le \frac{1}{1 - \norm{x}}. +> $$ +> {: .katex-display .mb-0 } + +{% proof %} +Since the Banach algebra norm is submultiplicative, +we have $\norm{x^n} \le \norm{x}^n$ for all $n \in \NN$. +This implies that the series $\sum \norm{x^n}$ +is majorized by the geometric series $\sum \norm{x}^n$, +which is known to be convergent for $\norm{x} < 1$. +It follows that the series $\sum x^n$ is absolutely convergent. +Denote its limit by $s = \lim_{n \to \infty} s_n = \sum_{n=0}^{\infty} x$, +where $s_n = \mathbf{1} + x + \cdots + x^n$ is the $n$th partial sum. +Clearly, + +$$ +(\mathbf{1}-x) s_n = s_n (\mathbf{1}-x) = \mathbf{1} - x^{n+1}. +$$ + +In the limit $n \to \infty$ we obtain $(\mathbf{1}-x) s = s (\mathbf{1}-x) = \mathbf{1}$, +because multiplication in a Banach algebra is continuous, and because $y^n \to 0$ when $\norm{y} < 1$. +This proves that $s$ is the inverse of $\mathbf{1}-x$. + +The estimate follows from $\norm{s} \le \sum \norm{x}^n = 1 / (1 - \norm{x})$. +{% endproof %} + +## The Spectrum + +{: .definition-title } +> Definition (Spectrum, Resolvent Set) +> +> Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. +> {: .mb-0 } +> +> {: .my-0 } +> - The *spectrum* of $x$ is the set $\sigma(x) = \lbrace\lambda \in \CC : x - \lambda$ is not invertible in $\mathcal{A}\rbrace$. \ +> The elements of $\sigma(x)$ are called *spectral values* of $x$. +> - The *resolvent set* of $x$ is the set $\rho (x) = \CC \setminus \sigma(x)$. \ +> For $\lambda \in \rho(x)$ the *resolvent* of $x$ is the algebra element $R_{\lambda} = (\lambda - x)^{-1}$. \ +> The mapping $R : \rho(x) \to \mathcal{A}$, $\lambda \mapsto R_{\lambda}$, is called *resolvent map*. + +{% theorem %} +Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. +If $\lambda$ lies in the resolvent set of $x$, +then so do all complex numbers $\mu$ with the property that + +$$ +\abs{\lambda - \mu} < \frac{1}{\norm{(\lambda - x)^{-1}}}. \tag{$*$} +$$ + +For such $\mu$ the resolvent of $x$ is represented by the absolutely convergent power series + +$$ +(\mu - x)^{-1} = \sum_{n=0}^{\infty} (\mu - \lambda)^n (\lambda - x)^{-(n+1)}. +$$ +{% endtheorem %} + +{% proof %} +Let $\lambda$ be in the resolvent set of $x$. +Then $\lambda - x$ is invertible and we have for all $\mu \in \CC$ + +$$ +\mu - x = \bigl(\mathbf{1} - (\lambda - \mu) (\lambda - x)^{-1}\bigr) (\lambda - x). +$$ + +If $\mu$ satisfies condition ($*$), the first factor is invertible +and the inverse is given by a [Neumann series](#neumann-series): + +$$ +\bigl(\mathbf{1} - (\lambda - \mu) (\lambda - x)^{-1}\bigr)^{-1} += \sum_{n=0}^{\infty} (\lambda - \mu)^n (\lambda - x)^{-n}. +$$ + +As a product of invertible algebra elements, $\mu - x$ must itself be invertible; +the claimed formula for its inverse follows by an application of +the rule $(ab)^{-1} = b^{-1} a^{-1}$ for invertible $a,b \in \mathcal{A}$. +{% endproof %} + +{: .corollary #resolvent-set-is-open #spectrum-is-closed } +> The resolvent set $\rho(x)$ is open and the spectrum $\sigma(x)$ is closed. + +{: .corollary #resolvent-map-is-analytic } +> Suppose $x$ is an element of a unital Banach algebra $\mathcal{A}$. +> The resolvent map +> +> $$ +> R : \rho(x) \longrightarrow \mathcal{A}, \quad \lambda \longmapsto R_{\lambda} = (\lambda - x)^{-1}, +> $$ +> +> is (strongly) analytic. + + --- + +{: .proposition #spectrum-is-not-empty } +> Suppose $x$ is an element of a unital Banach algebra. +> Then its spectrum $\sigma(x)$ is not empty. + +{% proof %} +We assume that $\sigma(x)$ is empty +and derive a contradiction. +Observe that the resolvent map $R$ is defined on the whole complex plane. +By [this corollary](#resolvent-map-is-analytic), $R$ is analytic, hence entire. +Analytic functions are countinuous; +therefore $R$ is bounded on the compact disk $\abs{\lambda} \le 2 \norm{x}$. +For $\abs{\lambda} > 2 \norm{x}$ we may expand $R_{\lambda}$ into a [Neumann series](#neumann-series), + +$$ +R_{\lambda} += (\lambda - x)^{-1} += \lambda^{-1} (\mathbf{1} - \lambda^{-1} x)^{-1} += \lambda^{-1} \sum_{n=0}^{\infty} (\lambda^{-1} x)^n, +$$ + +and make the estimate + +$$ +\norm{R_{\lambda}} +\le \abs{\lambda}^{-1} (1 - \norm{\lambda^{-1} x})^{-1} += (\abs{\lambda} - \norm{x})^{-1} +< \norm{x}^{-1}. +$$ + +This shows that $R$ is a bounded entire function. Now +[Liouville's Theorem](/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.html#liouvilles-theorem) +(for vector-valued functions) implies that $R$ is constant. +This is contradictiory because XXX +{% endproof %} + +{: .theorem-title } +> Gelfand–Mazur Theorem +> +> Every Banach algebra in which all nonzero elements are invertible is isometrically isomorphic to $\CC$. + +{% proof %} +For any Banach algebra $A$, +the mapping $\varphi : \CC \to A$, $\lambda \mapsto \lambda \mathbf{1}$, +is linear, multiplicative and isometric, hence injective. +Let $x$ be any element of $A$. +Since its +[spectrum is not empty](/pages/operator-algebras/banach-algebras/index.html#spectrum-is-not-empty), +there must exist a complex number $\lambda$ +such that $x - \lambda \mathbf{1}$ is not invertible. +Now suppose that all nonzero elements of $A$ are invertible. +Then necessarily $x - \lambda \mathbf{1} = 0$, or $x = \lambda \mathbf{1}$. +This proves that the mapping $\varphi$ is also surjective +and thus an isometric isomorphism. +{% endproof %} + +Other ways of stating that +all nonzero elements of a Banach algebra $\mathcal{A}$ are invertible +include: +{: .mb-0 } + +{: .mt-0 } +- $\mathcal{A}$ is a division algebra. +- The underlying ring of $\mathcal{A}$ is a field. + +{: .theorem-title } +> Spectral Radius Formula +> +> For every Banach algebra element $x$ the spectral radius is given by +> +> $$ +> r(x) = \lim_{n \to \infty} \norm{x^n}^{1/n}. +> $$ +> {: .katex-display .mb-0 } + +## Gelfand’s Theory + +Proposition +Let $\mathcal{A}$ be a unital commutative Banach algebra. +If $\phi$ is a nonzero multiplicative linear functional on $\mathcal{A}$, +then its kernel $\ker \phi$ is a maximal ideal in $\mathcal{A}$. +Every maximal ideal $\mathcal{I}$ in $\mathcal{A}$ is of the form +$I = \ker \phi$ for some nonzero multiplicative linear functional $\phi$ on $\mathcal{A}$. + +In other words, the mapping $\phi \mapsto \ker \phi$ is gives a bijection +between the sets of nonzero multiplicative linear functionals and maximal ideals. + + +Definition +The *maximal ideal space* $\mathcal{M}_{\mathcal{A}}$ of a unital commutative Banach algebra $\mathcal{A}$ +is the set of maximal ideals of $\mathcal{A}$; its topology is inherited from +the weak* topology on the dual of $\mathcal{A}$ via the correspondece described above. + +Proposition +The *maximal ideal space* of a unital commutative Banach algebra is a compact Hausdorff space. + +{% definition bla, blubb %} +a +b +{% enddefinition %} + + diff --git a/pages/operator-algebras/c-star-algebras/index.md b/pages/operator-algebras/c-star-algebras/index.md new file mode 100644 index 0000000..adc1981 --- /dev/null +++ b/pages/operator-algebras/c-star-algebras/index.md @@ -0,0 +1,8 @@ +--- +title: C*-Algebras +parent: Operator Algebras +nav_order: 2 +has_children: true +--- + +# {{ page.title }} diff --git a/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md new file mode 100644 index 0000000..05b1d4f --- /dev/null +++ b/pages/operator-algebras/c-star-algebras/positive-linear-functionals.md @@ -0,0 +1,39 @@ +--- +title: Positive Linear Functionals +parent: C*-Algebras +grand_parent: Operator Algebras +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +all algebra are assumed to be unital + +{: .definition-title } +> Hermitian Functional, Positive Functional, State +> +> A linear functional on a $C^*$-algebra $\mathcal{A}$ is said to be +> +> - *Hermitian* if $\phi(x*) = \overline{\phi(x)}$ for all $x \in \mathcal{A}$. +> - *positive* if $\phi(x) \ge 0$ for all $x \ge 0$. +> - a *state* if $\phi$ is positive and $\phi(\mathbf{1}) = 1$. +> + +{: .definition-title } +> State +> +> A norm-one positive linear functional on a $C^*$-algebra is called a *state*. + +{: .definition-title } +> 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}$. + +Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$. + +{: .proposition } +> The state space of a $C^*$-algebra is convex and weak* compact. + +{% proof %} +{% endproof %} diff --git a/pages/operator-algebras/c-star-algebras/states.md b/pages/operator-algebras/c-star-algebras/states.md new file mode 100644 index 0000000..619bc9a --- /dev/null +++ b/pages/operator-algebras/c-star-algebras/states.md @@ -0,0 +1,43 @@ +--- +title: States +parent: C*-Algebras +grand_parent: Operator Algebras +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +{: .definition-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*.\ +> The *state space* $S(\mathcal{A})$ of a C\*-algebra $\mathcal{A}$ is the set of all its states. + +Note that $S(\mathcal{A})$ is a subset of the unit ball in the dual space of $\mathcal{A}$. + +{: .corollary } +> A linear functional $\omega$ on a C\*-algebra is a state +> if and only if $\omega(\mathbf{1}) = 1 = \norm{\omega}$. + +{: .proposition } +> The state space of a C\*-algebra is convex and weak\* compact. + +{% proof %} +First, we show convexity. +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. + +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 %}), +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}$. +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. +This shows that the state space is weak* closed, completing the proof. +{% endproof %} diff --git a/pages/operator-algebras/index.md b/pages/operator-algebras/index.md new file mode 100644 index 0000000..2024202 --- /dev/null +++ b/pages/operator-algebras/index.md @@ -0,0 +1,7 @@ +--- +title: Operator Algebras +nav_order: 4 +has_children: true +--- + +# {{ page.title }} diff --git a/pages/quantum-field-theory/index.md b/pages/quantum-field-theory/index.md new file mode 100644 index 0000000..0099c72 --- /dev/null +++ b/pages/quantum-field-theory/index.md @@ -0,0 +1,8 @@ +--- +title: Quantum Field Theory +nav_order: 15 +has_children: true +published: false +--- + +# {{ page.title }} diff --git a/pages/quantum-field-theory/wightman-axioms/index.md b/pages/quantum-field-theory/wightman-axioms/index.md new file mode 100644 index 0000000..ecc204e --- /dev/null +++ b/pages/quantum-field-theory/wightman-axioms/index.md @@ -0,0 +1,13 @@ +--- +title: Wightman Axioms +parent: Quantum Field Theory +nav_order: 1 +has_children: true +# cspell:words +--- + +# {{ page.title }} + +TODO: +- Haag’s Theorem +- Källén–Lehmann representation diff --git a/pages/quantum-field-theory/wightman-axioms/scalar-field.md b/pages/quantum-field-theory/wightman-axioms/scalar-field.md new file mode 100644 index 0000000..d37bcd0 --- /dev/null +++ b/pages/quantum-field-theory/wightman-axioms/scalar-field.md @@ -0,0 +1,19 @@ +--- +title: Scalar Field +parent: Wightman Axioms +grand_parent: Quantum Field Theory +nav_order: 1 +# cspell:words +--- + +# Wightman Axioms + +Also known as *Gårding–Wightman axioms*. + +## Wightman Axioms for a Hermitian Scalar Field + +{: .axiom-title } +> Axiom 1 +> +> j + diff --git a/pages/spectral-theory/index.md b/pages/spectral-theory/index.md new file mode 100644 index 0000000..d88fd6d --- /dev/null +++ b/pages/spectral-theory/index.md @@ -0,0 +1,8 @@ +--- +title: Spectral Theory +nav_order: 3 +has_children: true +published: false +--- + +# {{ page.title }} diff --git a/pages/spectral-theory/of-unbounded-operators/index.md b/pages/spectral-theory/of-unbounded-operators/index.md new file mode 100644 index 0000000..0b2a6f9 --- /dev/null +++ b/pages/spectral-theory/of-unbounded-operators/index.md @@ -0,0 +1,9 @@ +--- +title: of Unbounded Operators +parent: Spectral Theory +nav_order: 1 +has_children: true +# cspell:words +--- + +# {{ page.title }} diff --git a/pages/spectral-theory/test/basic.md b/pages/spectral-theory/test/basic.md new file mode 100644 index 0000000..8c42f6d --- /dev/null +++ b/pages/spectral-theory/test/basic.md @@ -0,0 +1,59 @@ +--- +title: Test +parent: Test +grand_parent: Spectral Theory +nav_order: 2 +description: > + The +# spellchecker:words Steinhaus preimages Baire pointwise +--- + +# {{ 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-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)$ | + +By definition, the sets $\pspec{T}$, $\rspec{T}$, $\cspec{T}$ and $\rho(T)$ form a partition of the complex plane. diff --git a/pages/spectral-theory/test/index.md b/pages/spectral-theory/test/index.md new file mode 100644 index 0000000..690750d --- /dev/null +++ b/pages/spectral-theory/test/index.md @@ -0,0 +1,8 @@ +--- +title: Test +parent: Spectral Theory +nav_order: 1 +has_children: true +--- + +# {{ page.title }} diff --git a/pages/tomita-takesaki-theory/index.md b/pages/tomita-takesaki-theory/index.md new file mode 100644 index 0000000..dd2a312 --- /dev/null +++ b/pages/tomita-takesaki-theory/index.md @@ -0,0 +1,9 @@ +--- +title: Tomita Takesaki Theory +nav_order: 10 +has_children: true +published: false +# cspell:words +--- + +# {{ page.title }} diff --git a/pages/tomita-takesaki-theory/standard-subspaces.md b/pages/tomita-takesaki-theory/standard-subspaces.md new file mode 100644 index 0000000..970c51a --- /dev/null +++ b/pages/tomita-takesaki-theory/standard-subspaces.md @@ -0,0 +1,19 @@ +--- +title: Standard Subspaces +parent: Tomita Takesaki Theory +nav_order: 1 +# cspell:words +--- + +# {{ page.title }} + +{: .definition-title } +> Definition (Cyclic, Separating, Standard Subspace) +> +> A closed real linear subspace $H$ of a complex Hilbert space $\hilb{H}$ is called +> * *cyclic*, if $H+iH$ is dense in $\hilb{H}$, +> * *separating*, if $H \cap iH = \braces{0}$, and +> * *standard*, if $H$ is cyclic and separating. + +**Proof:** +{{ site.qed }} diff --git a/pages/unbounded-operators/adjoint-operators.md b/pages/unbounded-operators/adjoint-operators.md new file mode 100644 index 0000000..a93e6d4 --- /dev/null +++ b/pages/unbounded-operators/adjoint-operators.md @@ -0,0 +1,15 @@ +--- +title: Adjoint Operators +parent: Unbounded Operators +nav_order: 1 +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. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + diff --git a/pages/unbounded-operators/graph-and-closedness.md b/pages/unbounded-operators/graph-and-closedness.md new file mode 100644 index 0000000..a9bf738 --- /dev/null +++ b/pages/unbounded-operators/graph-and-closedness.md @@ -0,0 +1,22 @@ +--- +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. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + + +{: .definition-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$. diff --git a/pages/unbounded-operators/hellinger-toeplitz-theorem.md b/pages/unbounded-operators/hellinger-toeplitz-theorem.md new file mode 100644 index 0000000..07d6b81 --- /dev/null +++ b/pages/unbounded-operators/hellinger-toeplitz-theorem.md @@ -0,0 +1,116 @@ +--- +title: Hellinger–Toeplitz Theorem +parent: Unbounded Operators +nav_order: 10 +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. +# cspell:words Hellinger Toeplitz Schwarz Riesz functionals +--- + +# {{ page.title }} + +Conventions: +{: .mb-0 } + +- Hilbert spaces are complex. +- The inner product is anti-linear in its first argument. +- 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 + +$$ +\innerp{Tx}{y} = \innerp{x}{Ty} \quad \forall x,y \in D(T). +$$ + +{: .theorem-title } +> Hellinger–Toeplitz theorem +> +> An everywhere-defined symmetric operator on a Hilbert space is bounded. + +Consequently, a symmetric Hilbert space operator +that is (truly) unbounded +cannot be defined everywhere. + +--- + +## Proof using the Uniform Boundedness Theorem + +Assume that $T$ is not bounded. +Then there exists a sequence $(x_n)$ of unit vectors in $\hilb{H}$ +such that $\norm{Tx_n} \to \infty$. +Consider the sequence $(f_n)$ of linear functionals on $\hilb{H}$, +defined by + +$$ +f_n(y) = \innerp{Tx_n}{y} = \innerp{x_n}{Ty} \quad y \in \hilb{H}. +$$ + +The second identity is due to the symmetry of $T$. +Apply Cauchy-Schwarz to both expressions to obtain the inequalities + +$$ +\abs{f_n(y)} \le \norm{Tx_n} \norm{y} +\quad \text{and} \quad +\abs{f_n(y)} \le \norm{x_n} \norm{Ty} +$$ + +for each $n \in \NN$ and $y \in \hilb{H}$. +The first inequality shows that the functionals $f_n$ are bounded. +The second one shows that, for fixed $y$, +the sequence $(\abs{f_n(y)})$ is bounded by $\norm{Ty}$, +since $\norm{x_n} = 1$ for all $n$. +By the [Uniform Boundedness Theorem]({% link +pages/functional-analysis-basics/the-fundamental-four/uniform-boundedness-theorem.md +%}), $(\norm{f_n})$ is a bounded sequence. +One has + +$$ +\norm{Tx_n}^2 = \abs{f_n(Tx_n)} \le \norm{f_n} \norm{Tx_n} \quad n \in \NN. +$$ + +Divide by $\norm{Tx_n}$ (if nonzero) +to obtain $\norm{Tx_n} \le \norm{f_n}$ for all but finitely many $n$. +Thus $(\norm{Tx_n})$ is a bounded sequence, +contradicting $\norm{Tx_n} \to \infty$. +{{ site.qed }} + +--- + +## Proof using the Closed Graph Theorem + +By the [Closed Graph Theorem]({% link +pages/functional-analysis-basics/the-fundamental-four/closed-graph-theorem.md %}), +it is sufficient to show that the graph of $T$ is closed. +Let $(x_n)$ be a convergent sequence of vectors in $\hilb{H}$ +such that the image sequence $(Tx_n)$ converges as well. +Naming the limits $x$ and $z$, respectively, we have + +$$ +x_n \to x +\quad \text{and} \quad +Tx_n \to z. +$$ + +Continuity of the inner product implies + +$$ +\innerp{x_n}{Ty} \to \innerp{x}{Ty} +\quad \text{and} \quad +\innerp{Tx_n}{y} \to \innerp{z}{y} +$$ + +for all $y \in \hilb{H}$. +Since $T$ is symmetric, +the first assertion can be rewritten as + +$$ +\innerp{Tx_n}{y} \to \innerp{Tx}{y}. +$$ + +A sequence of complex numbers has at most one limit, +hence $\innerp{Tx}{y} = \innerp{z}{y}$ for all $y$. +By the Riesz representation theorem, $Tx=z$. +{{ site.qed }} diff --git a/pages/unbounded-operators/index.md b/pages/unbounded-operators/index.md new file mode 100644 index 0000000..54ad701 --- /dev/null +++ b/pages/unbounded-operators/index.md @@ -0,0 +1,7 @@ +--- +title: Unbounded Operators +nav_order: 4 +has_children: true +--- + +# {{ page.title }} diff --git a/pages/unbounded-operators/quadratic-forms.md b/pages/unbounded-operators/quadratic-forms.md new file mode 100644 index 0000000..5831b88 --- /dev/null +++ b/pages/unbounded-operators/quadratic-forms.md @@ -0,0 +1,23 @@ +--- +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. +# spellchecker:dictionaries latex +# spellchecker:words Hellinger Toeplitz innerp hilb Schwarz functionals enspace Riesz +--- + +# {{ page.title }} + + +{: .definition-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 $y=Tx$. |