Added mental gymnastics for lp direct sums.

src/fa/lp/seq.tex

@@ -27,6 +27,41 @@
     \]
 \end{definition}
 
+\begin{proposition}
+\label{proposition:lp-direct-sum-gymnastics}
+    Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty]$. 
+    \begin{enumerate}
+        \item (\textbf{Hölder's Inequality}) Let $q \in [1, \infty]$ be the Hölder conjugate of $p$, $\seqi{Y}, \seqi{Z}$ be normed spaces, and $\seqi{\lambda}$ such that for each $i \in I$, $\lambda \in L^2(X_i, Y_i; Z)$.
+        
+        For each $x \in [l^p(I); X_i]$ and $y \in [l^q(I); Y_i]$, let $\lambda(x, y)_i = \lambda_i(x_i, y_i)$, then
+        \[
+        \norm{\lambda(x, y)}_{[l^1(I); Z_i]} \le \norm{x}_{[l^p(I); X_i]} \cdot \norm{y}_{[l^q(I); X_i]} \cdot \sup_{i \in I}\norm{\lambda_i}_{L^2(X_i, Y_i; Z_i)}
+        \]
+        \item (\textbf{Minkowski's Inequality}) For each $x, y \in [l^p(I); X_i]$,
+        \[
+        \norm{x + y}_{[l^p(I); X_i]} \le \norm{x}_{[l^p(I); X_i]} + \norm{y}_{[l^p(I); X_i]}
+        \]
+
+        \item (\textbf{Markov's Inequality}) If $p < \infty$, then for each $\alpha > 0$ and $x \in [l^p(I); X_i]$, 
+        \[
+        |\bracsn{i \in I|\ \norm{x}_{X_i} \ge \alpha}| \le \frac{1}{\alpha^p}\norm{f}_{[l^p(I); X_i]}^p
+        \]
+
+        In particular, $\bracs{i \in I|x_i \ne 0}$ is countable. 
+        \item For any $q \in [p, \infty]$, $[l^p(I); X_i] \subset [l^q(I); X_i]$, where for any $x \in [l^p(I); X_i]$, $\norm{x}_{[l^q(I); X_i]} \le \norm{x}_{[l^p(I); X_i]}$. 
+        \item If $X_i$ is a Banach space for all $i \in I$, then so is $[l^p(I); X_i]$. 
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1), (2), (3): By the classical \hyperref[Hölder's inequality]{theorem:holder}, \hyperref[Minkowski's inequality]{theorem:minkowski}, and \hyperref[Markov's inequality]{theorem:markov-inequality}.
+    
+    (4): For each $i \in I$, $\norm{x_i}_{X_i} \le \norm{x}_{[l^p(I); X_i]}$, so the result holds when $q = \infty$. 
+
+    If $q < \infty$, then by \autoref{proposition:lp-intersection-interpolation}, there exists $\lambda \in [p, q]$ such that 
+    \[
+    \norm{x}_{[l^p(I); X_i]} \le \norm{x}_{[l^p(I); X_i]}^{\lambda}\norm{x}_{[l^\infty(I); X_i]}^{1 - \lambda} \le \norm{x}_{[l^p(I); X_i]}
+    \]
+\end{proof}