From 813cff3c8163503cd5630a0ece9c39101f7b6a3f Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 18 Mar 2026 17:31:07 -0400 Subject: [PATCH] Fixed typo. --- src/fa/lp/definition.tex | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/src/fa/lp/definition.tex b/src/fa/lp/definition.tex index 828f67e..9a90bd3 100644 --- a/src/fa/lp/definition.tex +++ b/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}