Added l^p sequence spaces.

src/fa/lp/duality.tex

@@ -164,7 +164,7 @@ After the duality of $L^p$ and $L^q$ is established for Hölder conjugate expone
 
     If $q < \infty$, then the \hyperref[Monotone Convergence Theorem]{theorem:mct} implies that $\norm{g}_{L^q(X; F)} \le \norm{\phi_g}_{L^p(X; E)^*}$. Otherwise,
     \[
-    \norm{g}_{L^\infty(X; H)} \le \sup_{n \in \natp}\norm{g_n}_{L^\infty(X; H)} \le \norm{\phi}_{L^1(X; H)^*}
+    \norm{g}_{L^\infty(X; H)} \le \sup_{n \in \natp}\norm{g_n}_{L^\infty(X; H)} \le \norm{\phi_g}_{L^1(X; H)^*}
     \]
     
     The above argument shows that the truncation argument was technically not required. By applying the truncated case again, $\norm{g}_{L^q(X; F)} = \norm{\phi_g}_{L^p(X; E)^*}$.

src/fa/lp/index.tex

@@ -3,3 +3,4 @@
 
 \input{./definition.tex}
 \input{./duality.tex}
+\input{./seq.tex}

src/fa/lp/seq.tex

@@ -0,0 +1,32 @@
+\section{$l^p$ Direct Sums}
+\label{section:lp-direct-sum}
+
+\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
+    \[
+    [l^p(I); X_i] = \bracs{x \in \prod_{i \in I}X_i \bigg | \sum_{i \in I}\norm{x_i}_{X_i}^p < \infty}
+    \]
+
+    equipped with the norm
+    \[
+    \norm{x}_{[l^p(I); X_i]} = \braks{\sum_{i \in I}\norm{x_i}_{X_i}^{p}}^{1/p}
+    \]
+\end{definition}
+
+\begin{definition}[$l^\infty$-Direct Product]
+\label{definition:l-infty-direct-product}
+    Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$, then the \textbf{$l^\infty$-direct product} of $\seqi{X}$ is the space
+    \[
+    [l^\infty(I); X_i] = \bracs{x \in \prod_{i \in I}X_i \bigg | \sup_{i \in I}\norm{x_i}_{X_i} < \infty}
+    \]
+
+    equipped with the norm
+    \[
+    \norm{x}_{[l^\infty(I); X_i]} = \sup_{i \in I}\norm{x_i}_{X_i}
+    \]
+\end{definition}
+
+
+
+