Added Gantmacher's theorem.

src/fa/lc/tensor.tex

@@ -53,6 +53,32 @@
     In constructing the \hyperref[projective tensor product]{definition:projective-tensor-product}, it may be more natural to obtain its topology as a projective topology using its universal property. However, doing so requires taking a least upper bound across \textit{all continuous linear maps defined on} $E \times F$, a collection too big to be a set. As such, constructing it as a projective topology is logically dubious, or at the very least beyond my abilities. 
 \end{remark}
 
+\begin{proposition}
+\label{proposition:projective-tensor-product-dual}
+    Let $E, F$ be locally convex space over $K \in \RC$, then
+    \[
+    (E \wh \otimes_\pi F)^* = (E \otimes_\pi F)^* \iso L^2(E, F; K) \iso L(E; F^*) \iso L(F; E^*)
+    \]
+
+    where:
+    \begin{enumerate}
+        \item The dual pairing between $E \otimes_\pi F$ and $L^2(E, F; K)$ is given by
+        \[
+        \angles{\sum_{k = 1}^n x_k \otimes y_k, \lambda}_{E \otimes_\pi F} = \sum_{k = 1}^n \lambda(x_k, y_k)
+        \]
+        \item The dual pairing between $E \otimes_\pi F$ and $L(E; F^*)$ is given by
+        \[
+        \angles{\sum_{k = 1}^n x_k \otimes y_k, T}_{E \otimes_\pi F} = \sum_{k = 1}^n \dpn{y_k, Tx_k}{F}
+        \]
+        \item The dual pairing between $E \otimes_\pi F$ and $L(F; E^*)$ is given by
+        \[
+        \angles{\sum_{k = 1}^n x_k \otimes y_k, T}_{E \otimes_\pi F} = \sum_{k = 1}^n \dpn{x_k, Ty_k}{E}
+        \]
+    \end{enumerate}
+    
+\end{proposition}
+
+
 
 
 \begin{definition}[Cross Seminorm]

src/fa/lp/seq.tex

@@ -1,6 +1,22 @@
-\section{$l^p$ Direct Sums}
+\section{Sequence Spaces}
 \label{section:lp-direct-sum}
 
+\begin{definition}[$c_0$-Direct Sum]
+\label{definition:c0-direct-sum}
+    Let $\seqi{X}$ be normed vector spaces over $K \in \RC$. For any $x \in \prod_{i \in I}X_i$, $x$ \textbf{vanishes at infinity} if for each $\eps > 0$, $\bracs{i \in I| \norm{x_i}_{X_i} \ge \eps}$ is finite. The space
+    \[
+    [c_0(I); X_i] = \bracs{x \in \prod_{i \in I}X_i \bigg | x \text{ vanishes at infinity}}
+    \]
+
+    equipped with the uniform norm
+    \[
+    \norm{x}_{[c_0(I); X_i]} = \sup_{i \in I}\norm{x_i}_{X_i}
+    \]
+
+    is the \textbf{$c_0$-direct sum} of $\seqi{X}$.
+\end{definition}
+
+
 \begin{definition}[$l^p$-Direct Sum]
 \label{definition:lp-direct-sum}
     Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$, then the \textbf{$l^p$-direct sum} of $\seqi{X}$ is the space
@@ -63,6 +79,42 @@
     \]
 \end{proof}
 
+
+\begin{theorem}
+\label{theorem:c0-sum-dual}
+    Let $\seqi{X}$ be normed vector spaces over $K \in \RC$. For each $y \in [l^1(I); X_i^*]$, let
+    \[
+    \phi_y: [c_0(I); X_i] \to K \quad x \mapsto \sum_{i \in I}\dpn{x_i, y_i}{X_i}
+    \]
+
+    then the mapping
+    \[
+    [l^1(I); X_i^*] \to [c_0(I); X_i]^* \quad y \mapsto \phi_y
+    \]
+
+    is an isometric isomorphism. 
+\end{theorem}
+\begin{proof}
+    By \hyperref[Hölder's Inequality]{proposition:lp-direct-sum-gymnastics}, for each $y \in [l^1(I); X_i^*]$, $\norm{\phi_y}_{[c_0(I); X_i]^*} \le \norm{y}_{[l^1(I); X_i]}$. 
+
+    Let $\phi \in [c_0(I); X_i]^*$, then there exists $y \in [l^\infty(I); X_i^*]$ such that for each $i \in I$ and $x \in X_i$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]} = \dpn{x_i, y_i}{X_i}$. 
+
+    Let $J \subset I$ be finite and $\alpha \in (0, 1)$, then there exists $x \in [c_0(I); X_i]$ such that
+    \begin{enumerate}
+        \item $\{i \in I|x_i \ne 0\} \subset J$. 
+        \item $\norm{x}_{[c_0(I); X_i]} \le 1$. 
+        \item For each $j \in J$, $\dpn{x_j, y_j}{X_j} \ge \alpha\norm{y_j}_{X_j^*}$. 
+    \end{enumerate}
+
+    Thus
+    \[
+    \alpha\sum_{j \in J}\norm{y_j}_{X_j^*} \le \sum_{j \in J}\dpn{x_j, y_j}{X_j} = \dpn{x, y}_{[c_0(I); X_i]} \le \norm{\phi}_{[c_0(I); X_i]^*}
+    \]
+
+    As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^1(I); X_i]$ with $\norm{y}_{[l^1(I); X_i^*]} \le \norm{\phi_y}_{[c_0(I); X_i]^*}$. By the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $\dpn{x, \phi_y}{[c_0(I); X_i]} = \dpn{x, \phi}{[c_0(I); X_i]}$ for all $x \in [c_0(I); X_i]$. Hence the map is an isometric isomorphism.  
+\end{proof}
+
+
 \begin{theorem}
 \label{theorem:lp-sum-dual}
     Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$ and $q \in (1, \infty]$ be Hölder conjugates. For each $y \in [l^q(I); X_i^*]$, let

src/fa/norm/compact.tex

@@ -26,3 +26,34 @@
     ($\Leftarrow$): By \autoref{proposition:operator-space-completeness}, $E^*$ is complete. The preceding case then implies that $T^{**} \in L(E^{**}; F^{**})$ is compact. As such, its restriction to $E$, being identified with $T$, is also compact. 
 \end{proof}
 
+\begin{theorem}[Gantmacher]
+\label{theorem:weakly-compact-biadjoint}
+    Let $E, F$ be Banach spaces over $K \in \RC$, and $T \in L(E; F)$, then the following are equivalent:
+    \begin{enumerate}
+        \item $T(B_E(0, 1))$ is relatively $\sigma(F, F^*)$-compact. 
+        \item $T^{**}(E^{**}) \subset F \subset F^{**}$. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}
+    Let $B_E$ be the closed unit ball of $E$, and $B_{E^{**}}$ be the closed unit ball of $E^{**}$. By \hyperref[Goldstine's Theorem]{theorem:goldstine-weak}, $B_E$ is $\sigma(E^{**}, E^*)$-dense in $B_{E^{**}}$. Since $T^{**}$ is a $\sigma(E^{**}, E^*)$-$\sigma(F^{**}, F^*)$-continuous extension of $T$, 
+    \[
+    T^{**}(B_{E^{**}}) = T^{**}\paren{\ol{B_E}^{\sigma(E^{**}, E^*)}} \subset \ol{T(B_E)}^{\sigma(F^{**}, F^*)}
+    \]
+
+    by \autoref{proposition:closure-of-image}. On the other hand, $B_{E^{**}}$ is $\sigma(E^{**}, E^*)$-compact by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}. Hence \autoref{proposition:compact-extensions} implies that $T^{**}(B_{E^{**}})$ is $\sigma(F^{**}, F^*)$-closed, so
+    \[
+    T^{**}(B_{E^{**}}) = T^{**}\paren{\ol{B_E}^{\sigma(E^{**}, E^*)}} = \ol{T(B_E)}^{\sigma(F^{**}, F^*)}
+    \]
+
+    The above equality shows that the following five statements are equivalent:
+    \begin{enumerate}[label=(\roman*)]
+        \item $T(B_E)$ is relatively $\sigma(F, F^*)$-compact. 
+        \item $\ol{T(B_E)}^{\sigma(F^{**}, F^*)} = \ol{T(B_E)}^{\sigma(F, F^*)}$. 
+        \item $\ol{T(B_E)}^{\sigma(F^{**}, F^*)} \subset F$. 
+        \item $T^{**}(B_{E^{**}}) \subset F$. 
+        \item $T^{**}(E^{**}) \subset F$. 
+    \end{enumerate}
+
+    where (i) is equivalent to (1), and (v) is equivalent to (2). 
+\end{proof}
+