Added l^p sequence spaces.

src/fa/lp/seq.tex

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+\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}
+
+
+
+