Theorem 14.2.4.label Let $(X, \cm, \mu)$ be a measure space, $K \in \RC$, $H$ be a Hilbert space over $K$, $p, q \in [1, \infty]$ be Hölder conjugates such that one of the following holds:
- (a)
$p \in (1, \infty)$ and $q \in (1, \infty)$.
- (b)
$p = 1$, $q = \infty$, and $\mu$ is $\sigma$-finite.
For each $g \in L^{q}(X, \cm, \mu; H)$, let
then the mapping
is a conjugate linear isometric isomorphism.
Proof, [Theorem 6.15, Fol99]. By Theorem 14.2.3, the given map is isometric. Thus it is sufficient to show that it is surjective. Let $\phi \in L^{p}(X; H)^{*}$.
(Finite): First suppose that $\mu$ is finite, then $\Sigma(X, \cm; H) \subset L^{p}(X; H)$, and $\phi$ induces an $H$-valued measure on $(X, \cm)$, absolutely continuous with respect to $\mu$. By the Radon-Nikodym Theorem, there exists $g \in L^{1}(X; H)$ such that for each $f \in \Sigma(X, \cm; H)$,
By Theorem 14.2.3, $g \in L^{q}(X; H)$.
(Arbitrary): In the case of (a), by Lemma 14.2.2, there exists a $\sigma$-finite set $A \in \cm$ such that for each $f \in L^{p}(X; H)$, $\dpn{f, \phi}{L^p(X; H)}= \dpn{\one_A \cdot f, \phi}{L^p(X; H)}$. In the case of (b), $A = X$ is a $\sigma$-finite set satisfying the same restriction condition.
Let $\seq{A_n}\subset \cm$ such that $\mu(A_{n}) < \infty$ for all $n \in \natp$, and $A = \bigsqcup_{n \in \natp}A_{n}$. By the finite case, there exists $\seq{g_n}\subset L^{q}(X; H)$ such that for each $n \in \natp$ and $f \in L^{p}(X; H)$,
Let $g = \sum_{n = 1}^{\infty} g_{n}$. If $q < \infty$, then $g \in L^{q}(X; H)$ by the Monotone Convergence Theorem. Otherwise,
For every $f \in L^{p}(X; H)$,
by the Dominated Convergence Theorem.
Therefore the mapping is surjective, and hence an isomorphism.$\square$
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