Added section regarding the duality of L^p spaces.

src/fa/lp/definition.tex

@@ -23,11 +23,13 @@
 
 \begin{definition}[Hölder conjugates]
 \label{definition:holder-conjugates}
-    Let $p, q \in (1, \infty)$, then $p$ and $q$ are \textbf{Hölder conjugates} if
+    Let $p, q \in [1, \infty]$, then $p$ and $q$ are \textbf{Hölder conjugates} if
     \[
     \frac{1}{p} + \frac{1}{q} = 1
     \]
 
+    under the identification that $1/\infty = 0$. 
+
 \end{definition}
 
 \begin{lemma}
@@ -40,15 +42,15 @@
 \end{lemma}
 
 
-\begin{theorem}[Hölder's Inequality, {{\cite[6.2]{Folland}}}]
+\begin{theorem}[Hölder's Inequality]
 \label{theorem:holder}
-    Let $(X, \cm, \mu)$ be a measure space, $E, F$ be a normed vector spaces, $p, q \in [1, \infty]$. If $p, q$ are Hölder conjugates or if $p = 1$ and $q = \infty$, then for any $f \in \mathcal{L}^p(X; E)$ and $g \in \mathcal{L}^q(X; F)$,
+    Let $(X, \cm, \mu)$ be a measure space, $E, F$ be a normed vector spaces, $p, q \in [1, \infty]$ be Hölder conjugates, then for any $f \in \mathcal{L}^p(X; E)$ and $g \in \mathcal{L}^q(X; F)$,
     \[
     \int \norm{f}_E \norm{g}_F d\mu \le \norm{f}_{L^p(X; E)}\norm{g}_{L^q(X; F)}
     \]
 
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof, {{\cite[Theorem 6.2]{Folland}}}. ]
     First suppose that $p = 1$ and $q = \infty$. In this case,
     \[
     \int \norm{f}_E \norm{g}_F d\mu \le \norm{g}_{L^\infty(X; F)}\int \norm{f}_Ed\mu = \norm{f}_{L^1(X; E)}\norm{g}_{L^\infty(X; F)}