Theorem 14.3.4.label Let $\seqi{X}$ be normed vector spaces over $K \in \RC$ and $p \in [1, \infty)$ and $q \in (1, \infty]$ be Hölder conjugates. For each $y \in [l^{q}(I); X_{i}^{*}]$, let
then the mapping
is an isometric isomorphism.
Proof. Let $\phi \in [l^{p}(I); X_{i}]^{*}$, then there exists $y \in [l^{\infty}(I); X_{i}^{*}]$ such that for each $i \in I$ and $x \in X_{i}$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[l^p(I); X_i]}= \dpn{x_i, y_i}{X_i}$.
Since the $q = \infty$ case has been ruled out, assume that $q \in (1, \infty)$. For each $\alpha \in (0, 1)$, there exists $x \in [l^{\infty}(I); X_{i}]$ with $\norm{x_i}_{X_i}\le 1$ and $\dpn{x_i, y_i}{X_i}\ge \alpha \norm{y_i}_{X_i^*}$. For each $J \subset I$ finite and $i \in I$, let $F_{J}(i) = \one_{J}(i) \cdot \norm{y_i}_{X_i^*}^{q - 1}$, then by Lemma 14.1.4,
so
As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^{q}(I); X_{i}^{*}]$ with $\norm{y}_{[l^q(I); X_i^*]}= \norm{\phi}_{[l^p(I); X_i]^*}$.$\square$
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