From 668679063bb93da2516001ed3937c78486a19727 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 3 Jun 2026 16:56:40 -0400 Subject: [PATCH] Added some example facts. --- .vscode/project.code-snippets | 5 +++++ src/op/example/bc.tex | 38 ++++++++++++++++++++++++++++++++ src/op/example/convolution.tex | 29 +++++++++++++++++++++++- src/op/example/disk.tex | 11 +++++++-- src/op/example/hardy.tex | 2 +- src/topology/main/compact.tex | 9 ++++++++ src/topology/main/compactify.tex | 2 +- 7 files changed, 91 insertions(+), 5 deletions(-) create mode 100644 src/op/example/bc.tex diff --git a/.vscode/project.code-snippets b/.vscode/project.code-snippets index 6500efa..f915781 100644 --- a/.vscode/project.code-snippets +++ b/.vscode/project.code-snippets @@ -171,5 +171,10 @@ "scope": "latex", "prefix": "rank", "body": ["\\text{rk}"] + }, + "Cite Proof": { + "scope": "latex", + "prefix": "cproof", + "body": ["{Proof, {{\\cite[$1]{$2}}}. }$0"] } } diff --git a/src/op/example/bc.tex b/src/op/example/bc.tex new file mode 100644 index 0000000..aa37293 --- /dev/null +++ b/src/op/example/bc.tex @@ -0,0 +1,38 @@ +\section{$BC(X)$} +\label{section:bounded-continuous-functions-algebra} + +\begin{definition}[Algebra of Bounded Continuous Functions] +\label{definition:bc-algebra} + Let $X$ be a topological space, then $BC(X; \complex)$ is the \textbf{space of bounded continuous functions} on $X$. + + Equipped with pointwise operations and the uniform norm, $BC(X; \complex)$ forms a commutative Banach algebra. +\end{definition} + + + +\begin{theorem} +\label{theorem:multiplicative-functional-bc} + Let $X$ be a completely regular space and $\beta X$ be its \hyperref[Stone-Čech compactification]{definition:stone-cech}. For each $f \in BC(X; \complex)$, let $\beta f \in BC(\beta X; \complex)$ be its unique extension to $\beta X$, then the mapping + \[ + E: \beta X \to \Omega(BC(X; \complex)) \quad E(x)(f) = (\beta f)(x) + \] + + is a homeomorphism. +\end{theorem} +\begin{proof} + Let $\phi \in BC(X; \complex)^* \setminus \ol{E(X)}$, then there exists $\seqf{f_k} \subset BC(X; \complex)$ and $\eps > 0$ such that for every $x \in X$, + \[ + f(x) = \sum_{k = 1}^n |f_k(x) - \dpn{f_k, \phi}{BC(X; \complex)}|^2 \ge \eps^2 + \] + + In which case, if $\phi \in \Omega(BC(X; \complex))$, then + \[ + \dpn{f, \phi}{BC(X; \complex)} = \sum_{k = 1}^n \phi(f_k - \phi(f_k))\phi(\ol{f_k - \phi(f_k)}) = 0 + \] + + Since $f(x) > 0$ for all $x \in X$, $f \in G(BC(X; \complex))$. As $\phi \in \Omega(BC(X; \complex))$, the above contradicts (3) of \autoref{proposition:multiplicative-unit}. Thus $\Omega(BC(X; \complex)) \setminus \ol{E(X)}$ is empty, and $E(X)$ is dense in $\Omega(BC(X; \complex))$. + + By (U) of the \hyperref[Stone-Čech compactification]{definition:stone-cech}, $E|_X$ extends uniquely to a surjective continuous map. Since $BC(\beta X; \complex)$ and $BC(X; \complex)$ are in bijection via the restriction map, $E$ is injective. Therefore $E$ is bijective, and $E$ is a homeomorphism by \autoref{proposition:compact-hausdorff-homeomorphism}. +\end{proof} + + diff --git a/src/op/example/convolution.tex b/src/op/example/convolution.tex index 7827676..d1784fd 100644 --- a/src/op/example/convolution.tex +++ b/src/op/example/convolution.tex @@ -17,7 +17,7 @@ The space $\ell^1(\integer)$ is the \textbf{convolution algebra} on $\integer$. \end{definition} \begin{proof} - For each $f, g \in \ell^1(\integer)$, + For each $f, g \in \ell^1(\integer)$, by \hyperref[Fubini's Theorem]{theorem:fubini-tonelli}, \begin{align*} \normn{f * g}_{\ell^1(\integer)} &= \sum_{n \in \integer} \abs{\sum_{k \in \integer}f(k)g(n - k)} \\ &\le \sum_{n, k \in \integer}|f(k)| \cdot |g(n-k)| \le \sum_{k \in \integer}|f(k)| \cdot \sum_{n \in \integer}|g(n - k)| \\ @@ -45,4 +45,31 @@ so $\normn{f^2}_{\ell^1(\integer)} = 7 < \normn{f}_{\ell^1(\integer)}^2$. By \autoref{proposition:gelfand-isometric}, the Gelfand transform is not isometric. \end{proof} +\begin{theorem} +\label{theorem:convolution-integer-gelfand} + Let $\ell^1(\integer)$ be the convolution algebra on $\integer$, then the mapping + \[ + F: \mathbf{S}^1 = \partial B_\complex(0, 1) \to \Omega(\ell^1(\integer)) \quad + (F(z))(f) = \sum_{n \in \integer}f(n)z^n + \] + + is a homeomorphism. +\end{theorem} +\begin{proof}{Proof, {{\cite[Theorem 6.3]{Zhu}}}. } + Let $z \in \mathbf{S}^1$ and $f, g \in \ell^1(\integer)$, then by \hyperref[Fubini's Theorem]{theorem:fubini-tonelli}, + \begin{align*} + F(z)(f * g) &= \sum_{n \in \integer}z^n \sum_{k \in \integer}f(k)g(n-k) \\ + &= \sum_{k \in \natp}f(k)z^k \sum_{n \in \integer}z^{n-k}g(n-k) \\ + &= F(z)(f) \cdot F(z)(g) + \end{align*} + + so $F(z)$ is a multiplicative functional for all $z \in \mathbf{S}^1$. + + For each $n \in \integer$, let $\delta_n = \one_{\bracs{k = n}}$, then for any $m, n \in \integer$, $\delta_m * \delta_n = \delta_{m + n}$. Let $\phi \in \Omega(\ell^1(\integer))$ and $z = \phi(\delta_1)$, then $\phi(\delta_n) = z^n$ for all $n \in \integer$. Since the span of $\bracsn{\delta_n|n \in \integer}$ is dense in $\ell^1(\integer)$, $\phi = F(z)$ by the \hyperref[Dominated Convergence Theorem]{theorem:dct}. + + Finally, by the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $F: \textbf{S}^1 \to \Omega(\ell^1(\integer))$ is continuous. Since $\textbf{S}^1$ is compact, $\Omega(\ell^1(\integer))$ is Hausdorff, and $F$ is a bijection, $F$ is a homeomorphism by \autoref{proposition:compact-hausdorff-homeomorphism}. +\end{proof} + + + diff --git a/src/op/example/disk.tex b/src/op/example/disk.tex index ffbfbf5..1db6809 100644 --- a/src/op/example/disk.tex +++ b/src/op/example/disk.tex @@ -13,10 +13,17 @@ \begin{proposition} \label{proposition:disk-algebra-multiplicative-functional} - Let $\phi \in A(D)^*$ be a multiplicative linear functional, then there exists $z_0 \in \ol D$ such that for every $f \in A(D)$, $\phi(f) = f(z_0)$. + Let $D = B_\complex(0, 1)$ and $A(D)$ be the disk algebra, then the mapping + \[ + E: \ol D \to \Omega(D) \quad E(z_0)(f) = f(z_0) + \] + + is a homeomorphism. \end{proposition} \begin{proof} - By \autoref{proposition:multiplicative-unit}, $\norm{\phi}_{A(D)^*} = 1$. Let $p$ be the identity polynomial, and $z_0 = \phi(p)$, then for every $q \in \complex[x]$, $\phi(q) = q(x_0)$. By density of polynomials in $A(D)$, $\phi(f) = f(z_0)$ for all $f \in A(D)$. + Let $\phi \in \Omega(D)$. By \autoref{proposition:multiplicative-unit}, $\norm{\phi}_{A(D)^*} = 1$. Let $p$ be the identity polynomial, and $z_0 = \phi(p)$, then for every $q \in \complex[x]$, $\phi(q) = q(x_0)$. By density of polynomials in $A(D)$, $\phi(f) = f(z_0)$ for all $f \in A(D)$. Therefore $\ol D$ is in bijection with $\Omega(D)$. + + Since $A(D) \subset C(\ol D)$, $E$ is continuous. As $\ol D$ is compact and $\Omega(D)$ is Hausdorff, $E$ is a homeomorphism by \autoref{proposition:compact-hausdorff-homeomorphism}. \end{proof} diff --git a/src/op/example/hardy.tex b/src/op/example/hardy.tex index f4ba611..e3a8034 100644 --- a/src/op/example/hardy.tex +++ b/src/op/example/hardy.tex @@ -3,7 +3,7 @@ \begin{definition}[Hardy Space] \label{definition:hardy-space} - Let $D = B_\complex(0, 1)$, then the \textbf{Hardy space} $H^\infty(D)$ is the space of all bounded holomorphic functions on $D$, equipped with the uniform norm. + Let $D = B_\complex(0, 1)$, then the \textbf{Hardy space} $H^\infty(D)$ is the algebra of all bounded holomorphic functions on $D$, equipped with the uniform norm. \end{definition} \begin{proposition} diff --git a/src/topology/main/compact.tex b/src/topology/main/compact.tex index 5a049c4..e97bde3 100644 --- a/src/topology/main/compact.tex +++ b/src/topology/main/compact.tex @@ -70,6 +70,15 @@ Let $x \in E^c$. For each $y \in E$, there exists $U_y \in \cn(y)$ and $V_y \in \cn(x)$ such that $U_y \cap V_y = \emptyset$. By compactness, there exists $E_0 \subset E$ finite such that $\bigcup_{y \in E_0}U_y \supset E$. In which case, $E \cap \bigcap_{y \in E_0}V_y = \emptyset$ and $\bigcap_{y \in E_0}V_y \in \cn(x)$. Thus $E^c$ is open. \end{proof} +\begin{proposition} +\label{proposition:compact-hausdorff-homeomorphism} + Let $X$ be a compact topological space, $Y$ be a Hausdorff space, and $f \in C(X; Y)$ be a bijection, then $f$ is a homeomorphism. +\end{proposition} +\begin{proof} + For each $K \subset X$ closed, $K$ and $f(K)$ are compact by \autoref{proposition:compact-extensions}. By \autoref{proposition:compact-closed}, $f(K)$ is closed. Thus $f^{-1}$ maps closed sets to closed sets, and hence open sets to open sets. +\end{proof} + + \begin{proposition} \label{proposition:compact-hausdorff-normal} Let $X$ be a Hausdorff space, $A, B \subset X$ be compact with $A \cap B = \emptyset$, then there exists $U \in \cn(A)$ and $V \in \cn(B)$ such that $U \cap V = \emptyset$. diff --git a/src/topology/main/compactify.tex b/src/topology/main/compactify.tex index 4ebd4c7..79ffbc5 100644 --- a/src/topology/main/compactify.tex +++ b/src/topology/main/compactify.tex @@ -25,7 +25,7 @@ \] \end{enumerate} - Moreover, if $(\beta X, e)$ is \textit{any} pair that satisfies (1) and (U1), then + Moreover, if $(\beta X, e)$ is \textit{any} pair that satisfies (U1), then \begin{enumerate} \item[(U2)] For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\beta \varphi \in C(\beta X; Y)$ such that the following diagram commutes: \[