Added duality for the sequence spaces.

src/fa/lp/seq.tex

@@ -63,5 +63,46 @@
     \]
 \end{proof}
 
+\begin{theorem}
+\label{theorem:lp-sum-dual}
+    Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p, q \in [1, \infty]$ be Hölder conjugates. For each $y \in [l^q(I); X_i^*]$, let
+    \[
+    \phi_y: [l^p(I); X_i] \to K \quad x \mapsto \sum_{i \in I}\dpn{x_i, y_i}{X_i}
+    \]
 
+    then the mapping
+    \[
+    [l^q(I); X_i^*] \to [l^p(I); X_i]^* \quad y \mapsto \phi_y
+    \]
+
+    is an isometric isomorphism. 
+\end{theorem}
+\begin{proof}
+    Let $\phi \in [l^p(I); X_i^*]$, then there exists $y \in [l^\infty(I); X_i^*]$ such that for each $i \in I$ and $x \in X_i$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]} = \dpn{x_i, y_i}{X_i}$. 
+
+    Since the $q = \infty$ case has been ruled out, assume that $q \in [1, \infty)$. For each $\alpha \in (0, 1)$, there exists $x \in [l^\infty(I); X_i]$ with $\norm{x_i}_{X_i} \le 1$ and $\dpn{x_i, y_i}{X_i} \ge \alpha \norm{y_i}_{X_i}$. If $q = 1$, then
+    \begin{align*}
+    \alpha \norm{y}_{[l^1(I); X_i^*]} &\le \alpha \sum_{i \in I} \norm{y_i}_{X_i^*} \le 
+    \sum_{i \in I}\dpn{x_i, y_i}{X_i} \\
+    &= 
+    \dpn{x, \phi_y}{[l^\infty(I); X_i]} \le \norm{\phi}_{[l^\infty(I); X_i]^*}
+    \end{align*}
+    
+    Otherwise, for each $J \subset I$ finite and $i \in I$, let $F_J(i) = \one_{J} \cdot \norm{y_i}_{X_i^*}^{q - 1}$, then by \autoref{lemma:holder-conjugate-gymnastics},
+    \[
+    \norm{F_J x}_{[l^p(I); X_i]}^p = \sum_{j \in J}\norm{y_j}_{X_i^*}^{p(q - 1)}= \sum_{j \in J}\norm{y_j}_{X_i^*}^{q}
+    \]
+
+    so 
+    \begin{align*}
+    \alpha \sum_{j \in J} \norm{y_i}_{X_i^*}^q & \le 
+    \sum_{i \in I}F_J(i)\dpn{x_i, y_i}{X_i} = 
+    \dpn{F_J x, \phi_y}{[l^p(I); X_i]} \\
+    &\le \norm{\phi}_{[l^p(I); X_i]^*} \cdot \braks{\sum_{j \in J}\norm{y_j}_{X_i^*}^{q}}^{1/p} \\
+    \alpha \braks{\sum_{j \in J}\norm{y_i}_{X_i^*}^q}^{1/q} &\le \norm{\phi}_{[l^p(I); X_i]^*}
+    \end{align*}
+
+    As the above holds for all $\alpha \in (0, 1)$, $y \in [l^p(I); X_i]^*$ with $\norm{y}_{[l^q(I); X_i^*]} = \norm{\phi_y}_{[l^p(I); X_i]^*}$. 
+    
+\end{proof}