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Fixed typo.
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Bokuan Li
2026-03-18 17:31:07 -04:00
parent 1cb04f668b
commit 813cff3c81
1 changed files with 7 additions and 7 deletions
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src/fa/lp/definition.tex
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@@ -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}