Fixed typo.

src/fa/lp/definition.tex

@@ -129,9 +129,9 @@
 
 \begin{theorem}[{{\cite[III.6.5]{SchaeferWolff}}}]
 \label{theorem:l1-tensor}
-    Let $(X, \cm, \mu)$ be a measure space and $E$ be a Banach space over $\real \in \RC$, then the map $L^1(X; \real) \td{\otimes}_\mu E \to L^1(X; E)$ defined by extending
+    Let $(X, \cm, \mu)$ be a measure space and $E$ be a Banach space over $K \in \RC$, then the map $L^1(X; K) \td{\otimes}_\mu E \to L^1(X; E)$ defined by extending
     \[
-    L^1(X; \real) \times E \to L^1(X; E) \quad f \otimes x \mapsto x \cdot f
+    L^1(X; K) \times E \to L^1(X; E) \quad f \otimes x \mapsto x \cdot f
     \]
     
     is an isometric isomorphism. 
@@ -139,12 +139,12 @@
 \begin{proof}
     By (U) of the \hyperref[tensor product]{definition:tensor-product}, the given map admits a unique extension
     \[
-    M: L^1(X; \real) \otimes E \to L^1(X; E) \quad \sum_{j = 1}^n f_j \otimes x_j \mapsto \sum_{j = 1}^n x_j \cdot f_j
+    M: L^1(X; K) \otimes E \to L^1(X; E) \quad \sum_{j = 1}^n f_j \otimes x_j \mapsto \sum_{j = 1}^n x_j \cdot f_j
     \]
 
     Restricting $M$ to the simple functions yields a linear isomorphism
     \[
-    M: [L^1(X; \real) \cap \Sigma(X; \real)] \otimes E \to L^1(X; E) \cap \Sigma(X; E)
+    M: [L^1(X; K) \cap \Sigma(X; K)] \otimes E \to L^1(X; E) \cap \Sigma(X; E)
     \]
 
     For any $\phi \in L^1(X; E) \cap \Sigma(X; E)$, write
@@ -154,15 +154,15 @@
 
     then
     \[
-    \normn{M^{-1}\phi}_{L^1(X; \real) \otimes E} \le \sum_{y \in \phi(X) \setminus \bracs{0}} \norm{y}_E \cdot \mu\bracs{\phi = y} = \int \norm{\phi}_E d\mu = \norm{\phi}_{L^1(X; E)}
+    \normn{M^{-1}\phi}_{L^1(X; K) \otimes E} \le \sum_{y \in \phi(X) \setminus \bracs{0}} \norm{y}_E \cdot \mu\bracs{\phi = y} = \int \norm{\phi}_E d\mu = \norm{\phi}_{L^1(X; E)}
     \]
 
     On the other hand, for any representation $M^{-1}\phi = \sum_{j = 1}^n a_j \one_{A_j}$, 
     \[
-    \normn{\phi}_{L^1(X; E)} \le \sum_{j = 1}^n \norm{a_j}_E \mu(A_j) = \sum_{j = 1}^n \norm{a_j}_E \normn{\one_{A_j}}_{L^1(X; \real)}
+    \normn{\phi}_{L^1(X; E)} \le \sum_{j = 1}^n \norm{a_j}_E \mu(A_j) = \sum_{j = 1}^n \norm{a_j}_E \normn{\one_{A_j}}_{L^1(X; K)}
     \]
 
-    As this holds for all such representations, $\normn{\phi}_{L^1(X; E)} = \normn{M^{-1}\phi}_{L^1(X; \real) \otimes E}$. Therefore $M$ restricted to $[L^1(X; \real) \cap \Sigma(X; \real)] \otimes E$ is an isometry. By \autoref{proposition:lp-simple-dense}, $[L^1(X; \real) \cap \Sigma(X; \real)] \otimes E$ is dense in $L^1(X; \real) \widehat{\otimes}_\pi E$, and $L^1(X; E) \cap \Sigma(X; E)$ is dense in $E$. By the \hyperref[Linear Extension Theorem]{theorem:linear-extension-theorem-normed}, $M$ extends uniquely into the given map on $L^1(X; \real) \otimes E$, which then extends into an isometry $L^1(X; \real) \otimes E \to L^1(X; E)$. 
+    As this holds for all such representations, $\normn{\phi}_{L^1(X; E)} = \normn{M^{-1}\phi}_{L^1(X; K) \otimes E}$. Therefore $M$ restricted to $[L^1(X; K) \cap \Sigma(X; K)] \otimes E$ is an isometry. By \autoref{proposition:lp-simple-dense}, $[L^1(X; K) \cap \Sigma(X; K)] \otimes E$ is dense in $L^1(X; K) \widehat{\otimes}_\pi E$, and $L^1(X; E) \cap \Sigma(X; E)$ is dense in $E$. By the \hyperref[Linear Extension Theorem]{theorem:linear-extension-theorem-normed}, $M$ extends uniquely into the given map on $L^1(X; K) \otimes E$, which then extends into an isometry $L^1(X; K) \otimes E \to L^1(X; E)$. 
 \end{proof}