Added the c_0 sequence space and its duality result.

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