Added the compact null sequence lemma.

src/fa/lc/compact.tex

@@ -1,6 +1,27 @@
 \section{Compact Convex Sets}
 \label{section:compact-convex}
 
+
+\begin{theorem}[Mazur]
+\label{theorem:convex-hull-complete}
+    Let $E$ be a locally convex space over $K \in \RC$ and $A \subset E$ be compact, then
+    \begin{enumerate}
+        \item $\conv(A)$ is totally bounded in $E$. 
+        \item If $E$ is complete, then $\conv(A)$ is relatively compact. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    (1): Let $U \in \cn_E(0)$ be convex and circled, then since $A$ is compact, there exists $B \subset A$ finite such that $A \subset B + U$. As such, $\conv(A) \subset \conv(B + U) = \conv(B) + U$. Since $B$ is finite, $\conv(B)$ is compact. Thus there exists $C \subset \conv(B)$ finite such that
+    \[
+    \conv(A) \subset \conv(B + U) = \conv(B) + U \subset C + U
+    \]
+
+    which yields a finite covering of $\conv(A)$ using $U$. 
+
+    (2): By \autoref{proposition:compact-uniform}. 
+\end{proof}
+
+
 \begin{definition}[Extreme Point]
 \label{definition:extreme-point}
     Let $E$ be a vector space over $\real$, $K \subset E$, and $x \in K$, then $x$ is \textbf{extremal} if there exists no $y, z \in K$ such that $x \in (y, z) \subset K$. 
@@ -83,3 +104,36 @@
     
 \end{proof}
 
+\begin{lemma}
+\label{lemma:compact-null-auxiliary}
+    Let $E$ be a Banach space over $K \in \RC$ and $A \subset E$ be compact, then:
+    \begin{enumerate}
+        \item There exists a null sequence $\seq{x_n} \subset E$ such that $A \subset \ol{\conv}(\seq{x_n})$.
+        \item There exists a compact, convex, and circled set $B \subset E$ such that $A$ is compact in $E_B$. 
+    \end{enumerate}
+\end{lemma}
+\begin{proof}[Proof, {{\cite[Lemma III.9.1]{SchaeferWolff}}}. ]
+    (1): Assume without loss of generality that $A \ne \emptyset$. Let $\bracsn{\lambda_n}_0^\infty \subset (0, \infty)$ such that $\sum_{n \in \natz}\lambda_n = 1$. For each $n \in \natz$, let $r_n = \lambda_{n+1}^2$ and $A_n \subset A$ be finite such that $A \subset \bigcup_{x \in A_n}B_E(x, r_n)$. 
+
+    Define $B_0 = \lambda_0^{-1}A_0$. For each $n \in \natp$, write $A_n = \bracsn{x_j}_1^k$. By definition of $A_{n-1}$, there exists $\bracsn{y_j}_1^k \subset A_{n-1}$ such that $d(x_j, y_j) < r_{n-1}$ for all $1 \le j \le k$. For every $1 \le j \le k$, let $z_j = (x_j - y_j)/\lambda_n$, and define $B_n = \bracsn{z_j}_1^k$. 
+
+    By the above construction, $A_n \subset \sum_{j = 0}^n \lambda_j B_j$ for all $n \in \natz$. As $\sum_{n \in \natz}\lambda_n = 1$, 
+    \[
+    A \subset \ol{\conv}\braks{\bigcup_{n \in \natz}A_n} \subset \ol{\conv}\braks{\bracs{0} \cup \bigcup_{n \in \natz}B_n}
+    \]
+
+    Finally, for each $n \in \natp$, $B_n$ is finite with $\norm{z}_E \le r_{n-1}/\lambda_{n} = \lambda_n$ for all $z \in B_n$. As $B_0$ is finite as well, any enumeration of $\bracs{0} \cup \bigcup_{n \in \natz}B_n$ yields a null sequence. 
+
+    (2): Using (1) and \hyperref[Mazur's Theorem]{theorem:convex-hull-complete}, assume without loss of generality that there exists a null sequence $\seq{x_n} \subset E$ such that $A$ is the closed convex hull of $\seq{x_n}$. 
+
+    Since $\seq{x_n} \subset E$ is a null sequence, there exists $\seq{\lambda_n} \subset [1, \infty)$ such that:
+    \begin{enumerate}[label=(\roman*)]
+        \item $\lambda_n \to \infty$ as $n \to \infty$. 
+        \item $\lambda_n x_n \to 0$ as $n \to \infty$. 
+    \end{enumerate}
+
+    Let $B = \ol{\aconv}(\seq{\lambda_n x_n})$, then by \hyperref[Mazur's Theorem]{theorem:convex-hull-complete}, $B$ is a compact, convex, and circled subset of $E$ with $\seq{x_n} \subset B$. In addition, $\seq{x_n} \subset E_B$ with $\norm{x_n}_{E_B} \le \lambda_n^{-1}$ for all $n \in \natp$. Thus $\seq{x_n}$ is a null sequence in $E_B$ as well. 
+
+    Now, let $A'$ be the closed convex hull of $\seq{x_n}$ with respect to $E_B$. Since the inclusion $E_B \to E$ is continuous, $A'$ is a compact convex set in $E$ by \autoref{proposition:compact-extensions}. As such, $A' = A$ by \autoref{proposition:closure-of-image}. Therefore $A$ is a compact subset of $E_B$. 
+\end{proof}
+

src/fa/lc/tensor.tex

@@ -28,7 +28,7 @@
     
     The space $E \otimes_\pi F$ is the \textbf{projective tensor product} of $E$ and $F$, and the mapping $\iota \in L^2(E, F; E \otimes_\pi F)$ is the \textbf{canonical embedding}.
 
-    The space $E \widetilde{\otimes}_\pi F$ denotes the Hausdorff completion of $E \otimes_\pi F$.
+    The space $E \wh{\otimes}_\pi F$ denotes the Hausdorff completion of $E \otimes_\pi F$.
 \end{definition}
 \begin{proof}
     Let $E \otimes_\pi F = E \otimes F$ be the \hyperref[tensor product]{definition:tensor-product} of $E$ and $F$ as vector spaces. Let $\mathscr{T} \subset 2^{2^X}$ be the collection of all locally convex topologies satisfying (1) and (2), and let $\mathcal{S}$ be the projective topology on $E \otimes_\pi F$ generated by $\mathscr{T}$. 

src/fa/notation.tex

@@ -26,7 +26,7 @@
     $[\cdot]_A$ & Gauge of a radial set $A$. & \autoref{definition:gauge} \\
     $\rho_M$ & Quotient of seminorm $\rho$ by subspace $M$. & \autoref{definition:quotient-norm} \\
     $E \otimes_\pi F$ & Projective tensor product of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\
-    $E \,\widetilde{\otimes}_\pi F$ & Projective completion of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\
+    $E \,\wh{\otimes}_\pi F$ & Projective completion of $E$ and $F$. & \autoref{definition:projective-tensor-product} \\
     $p \otimes q$ & Cross seminorm of $p$ and $q$. & \autoref{definition:cross-seminorm} \\
     $N(E; F)$ & Nuclear mappings from $E$ to $F$. & \autoref{definition:nuclear-operator-normed} \\ 
     % ---- Order Structures ----