Added second proof for von Neumann.

src/measure/measurable-maps/product.tex

@@ -37,21 +37,26 @@
 \end{enumerate}
 \end{proof}
 
-\begin{proposition}[{{\cite[Proposition 1.5]{Folland}}}]
+\begin{proposition}
 \label{proposition:product-sigma-algebra-metric}
-    Let $\seqf{X_j}$ be topological spaces, $X = \prod_{j = 1}^n X_j$, then:
+    Let $\seq{X_j}$ be topological spaces, $X = \prod_{n \in \natp} X_n$, then:
     \begin{enumerate}
-        \item $\bigotimes_{j = 1}^n \cb_{X_j} \subset \cb_{X}$.
+        \item $\bigotimes_{n \in \natp} \cb_{X_n} \subset \cb_{X}$.
-        \item If $\seqf{X_j}$ are separable, then $\bigotimes_{j = 1}^n \cb_{X_j} = \cb_{X}$
+        \item If $\seq{X_n}$ are separable and metrisable, then $\bigotimes_{n \in \natp} \cb_{X_n} = \cb_{X}$. 
     \end{enumerate}
 \end{proposition}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Proposition 1.5]{Folland}}} and {{\cite[Proposition 8.1.7]{CohnMeasure}}}. ]
-    (2): By \autoref{proposition:separable-product}, $X$ is also separable. Let $U \subset X$, then there exists $\seq{x_j} \subset X$ such that $\overline{\bracs{x_j|j \in \natp}} \supset U$. For each $j \in \nat$, let $\bracs{U_{j, k}}$ such that $U_{j, k} \in \cn_{X_k}(\pi_k(x_j))$ for each $1 \le k \le n$ and $x_j \in \bigcap_{k = 1}^n \pi_k^{-1}(U_{j, k}) \subset U$.
+    (2): For each $J \subset \natp$ finite, $x \in X$, and $\eps > 0$, define
-
-    Since $\bigcap_{k = 1}^n \pi_k^{-1}(U_{j, k}) \in \bigotimes_{j = 1}^n \cb_{X_j}$ and
     \[
-    U = \bigcup_{j \in \natp}\bigcap_{k = 1}^n \pi_k^{-1}(U_{j, k})
+    B_J(x, \eps) = \bigcap_{j \in J}\pi_j^{-1}(B_{X_j}(x_j, \eps))
     \]
 
-    The open set $U$ is in $\bigotimes_{j = 1}^n \cb_{X_j}$.
+    By \autoref{proposition:separable-product}, $X$ is also separable. Let $\seq{x_n} \subset X$ be a countable dense subset, and
+    \[
+    \fB = \bracsn{B_J(x_n, 1/k)|n, k \in \natp, J \subset \natp \text{ finite}}
+    \]
+
+    For each $x \in X$ and $U \in \cn_X(x)$, there exists $J \subset X$ finite and $k \in \natp$ such that $x \in B_J(x, 1/k) \subset U$. By density of $\seq{x_n}$, there exists $n \in \natp$ such that $x_n \in B_J(x, 1/(2k))$. In which case, $x \in B_J(x_n, 1/(2k)) \subset B_J(x, 1/k) \subset U$. Thus $\fB$ is a countable base for the product topology, consisting of cylinder sets in $\bigotimes_{n \in \natp} \cb_{X_n}$. 
+
+    As such, every open set in $X$ is a countable union of elements of $\fB$. Therefore $\bigotimes_{n \in \natp} \cb_{X_n} \supset \cb_{X}$.
 \end{proof}

src/op/system/dilation.tex

@@ -120,3 +120,74 @@
 
     Thus for any $p \in \complex[x]$, $p(T) = P_Hp(U)|_H$. As $U$ is unitary, the \hyperref[continuous functional calculus]{definition:continuous-functional-calculus} shows that $\norm{p(U)}_{B(K)} \le \sup_{z \in B_\complex(0, 1)}|p(z)|$. Therefore $\norm{p(T)}_{B(H)} \le \sup_{z \in B_\complex(0, 1)}|p(z)|$. 
 \end{proof}
+\begin{proof}[Proof, {{\cite[Exercise 2.15]{Paulsen}}}. ]
+    Assume without loss of generality that $\norm{T}_{B(H)} < 1$. Let
+    \[
+    P(T; t) = \frac{1}{1 - e^{-it}T} + \frac{1}{1 - e^{it}T^*} - 1
+    \]
+    
+    
+    Let $p, q \in \complex[z]$ and $\gamma: [0, 2\pi] \to \complex$ be defined by $\gamma(t) = e^{it}$, then by the \hyperref[holomorphic functional calculus]{definition:holomorphic-functional-calculus},
+    \[
+        p(T) = \frac{1}{2\pi i}\int_\gamma \frac{p(\lambda)}{\lambda - T} d\lambda
+    \]
+
+    For each $n \in \natz$, since $\gamma([0, 2\pi]) \subset \partial B_\complex(0, 1)$, $\ol{\lambda^n} = \lambda^{-n}$. After passing through a \hyperref[Neumann series]{lemma:neumann-series}, 
+    \begin{align*}
+    \int_\gamma \frac{\ol{\lambda^n}}{\lambda - T} d\lambda &= \int_\gamma \frac{\lambda^{-n}}{\lambda - T} d\lambda = \int_\gamma \frac{\lambda^{-{(n+1)}}}{1 - T/\lambda} d\lambda \\
+    &= \sum_{k = 0}^\infty T^k\int_{\gamma} \lambda^{-(n+k+1)}d\lambda = \begin{cases}
+        1 & n = 0 \\
+        0 & n > 0
+    \end{cases}
+    \end{align*}
+
+    By linearity and \hyperref[Cauchy's integral formula]{theorem:cauchy-formula}, 
+    \[
+    \frac{1}{2\pi i}\int_\gamma\frac{\ol{q(\lambda)}}{\lambda - T} d\lambda = \ol{q(0)} = \frac{1}{2\pi i}\int_\gamma \frac{\ol{q(\lambda)}}{\lambda} d\lambda
+    \]
+
+    so
+    \begin{align*}
+        p(T) + \overline{q(0)} &= \frac{1}{2\pi i} \int_\gamma \frac{p(\lambda) + \ol{q(\lambda)}}{\lambda - T} d\lambda \\
+        p(T) + \overline{q(0)} &= \frac{1}{2\pi i} \int_0^{2\pi}\frac{ie^{it}[p(e^{it}) + \ol{q(e^{it})}]}{e^{it} - T} dt \\
+        &= \frac{1}{2\pi}\int_0^{2\pi}\frac{p(e^{it}) + \ol{q(e^{it})}}{1 - e^{-it}T}dt
+    \end{align*}
+    
+    Similarly, 
+    \[
+    q(T)^* + p(0) = \frac{1}{2\pi}\int_0^{2\pi}\frac{p(e^{it}) + \ol{q(e^{it})}}{1 - e^{it}T^*}dt
+    \]
+
+    Therefore
+    \begin{align*}
+        p(T) + q(T)^* + p(0) + \ol{q(0)} &= \frac{1}{2\pi}\int_0^{2\pi}[p(e^{it}) + \ol{q(e^{it})}](P(T; t) - 1)dt \\
+        p(T) + q(T)^* &= \frac{1}{2\pi}\int_0^{2\pi}[p(e^{it}) + \ol{q(e^{it})}]P(T; t)dt
+    \end{align*}
+    
+    Now, for each $t \in [0, 2\pi]$, 
+    \[
+    (1 - e^{-it}T)(1 - e^{it}T^*) = 1 - e^{-it}T - e^{it}T^* + TT^*
+    \]
+    
+    so by the resolvent identity, 
+    \begin{align*}
+        P(T; t) &= \frac{1}{1 - e^{-it}T} + \frac{1}{1 - e^{it}T^*} - 1 \\
+        &= (1 - e^{-it}T)^{-1}\braks{1 - TT^*}(1 - e^{it}T^*)^{-1} \ge 0
+    \end{align*}
+
+    For each $\xi, \eta \in H$, let $\dpn{\xi, \eta}{t} = \dpn{P(T; t)\xi, \eta}{H}$, then $\dpn{\cdot, \cdot}{t}$ is a pseudo inner product on $H$. Hence by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, 
+    \begin{align*}
+        \dpn{p(T)\xi, \eta}{H} &= \frac{1}{2\pi}\int_0^{2\pi} p(T)\dpn{P(T; t)\xi, \eta}{H}dt = \frac{1}{2\pi}\int_0^{2\pi} p(T)\dpn{\xi, \eta}{t}dt \\ 
+        |\dpn{p(T)\xi, \eta}{H}| &\le \frac{1}{2\pi}\int_0^{2\pi} \norm{p}_{H^\infty(D)}\dpn{\xi, \xi}{t}^{1/2}\dpn{\eta, \eta}t^{1/2}dt\\
+        &\le \norm{p}_{H^\infty(D)}\braks{\frac{1}{2\pi}\int_0^{2\pi}\dpn{\xi, \xi}{t} dt}^{1/2}\braks{\frac{1}{2\pi}\int_0^{2\pi}\dpn{\eta, \eta}{t} dt}^{1/2}
+    \end{align*}
+
+    Since $I_H(\xi) = \frac{1}{2\pi}\int_0^{2\pi} P(T; t)dt$, 
+    \[
+    \frac{1}{2\pi}\int_0^{2\pi}\dpn{\xi, \xi}{t} dt = \frac{1}{2\pi}\int_0^{2\pi} \dpn{P(T; t)\xi, \xi}{H}dt = \norm{\xi}_H^2
+    \]
+
+    Therefore $\dpn{p(T)\xi, \eta}{H} \le \norm{p}_{H^\infty(D)}\norm{\xi}_H\norm{\eta}_H$, and $\norm{p(T)}_H \le \norm{p}_{H^\infty(D)}$.
+    
+    
+\end{proof}

src/topology/main/interiorclosureboundary.tex

@@ -91,10 +91,16 @@
 
 \begin{proposition}
 \label{proposition:separable-product}
-    Let $\seqf{X_j}$ be separable topological spaces, then $\prod_{j = 1}^n X_j$ is also separable.
+    Let $\seq{X_n}$ be separable topological spaces, then $\prod_{n \in \natp}X_n$ is separable. 
 \end{proposition}
 \begin{proof}
-    Let $\seq{A_j}$ such that $A_j \subset X_j$ is countable and dense for each $1 \le j \le n$. By \autoref{proposition:dense-product}, $\prod_{j = 1}^n A_j$ is countable and dense in $\prod_{j = 1}^n X_j$.
+    For each $n \in \nat$, let $A_n \subset X_n$ be a countable dense subset. For each $N \in \nat$, let $B_N \subset X$ such that:
+    \begin{enumerate}
+        \item $B_N$ is countable.
+        \item For each $\bracsn{a_n}_1^N \subset \prod_{n = 1}^N A_n$, there exists $b \in B_N$ such that $a_n = b_n$ for all $1 \le j \le N$. 
+    \end{enumerate}
+
+    Then, $\bigcup_{N \in \natp}B_N$ is a countable dense subset of $\prod_{n \in \natp}X_n$.
 \end{proof}