ON THE DUNFORD-PETTIS CRITERION FOR UNIFORMLY INTEGRABLE SETS IN GENERALIZED LEBESGUE-BOCHNER SPACE $L^1(E,(X_\vartheta,\|.\|))$
Abstract
Let $(X,\|.\|,\vartheta)$ be a bitopological vector space such
that $(X,\vartheta)$ is a topological vector space, $(X,\|.\|)$ is
a reflexive normed space, the unit ball ${\cal B}_1(X)$ is closed
in $(X,\vartheta)$ and sequentially complete under the topology
$\vartheta$. Let $I=[\alpha,\beta]$ be an interval of $R$. Denote
by $l(X_{\vartheta},R)$ the space of all sequentially continuous
linear mapping from $X_\vartheta$ to $R$.
We prove that if a subset ${\cal K}$ of
$L^1(I,(X_\vartheta,\|.\|))$ is sequentially relatively
compact in
$(L^1(I,(X_\vartheta,\|.\|)),\sigma(L^1(I,(X_\vartheta,\|.\|)),L^{\infty}(I,(l(X_{\vartheta},R),\|.\|_{X'}))))$,\linebreak then it is weakly uniformly integrable.
that $(X,\vartheta)$ is a topological vector space, $(X,\|.\|)$ is
a reflexive normed space, the unit ball ${\cal B}_1(X)$ is closed
in $(X,\vartheta)$ and sequentially complete under the topology
$\vartheta$. Let $I=[\alpha,\beta]$ be an interval of $R$. Denote
by $l(X_{\vartheta},R)$ the space of all sequentially continuous
linear mapping from $X_\vartheta$ to $R$.
We prove that if a subset ${\cal K}$ of
$L^1(I,(X_\vartheta,\|.\|))$ is sequentially relatively
compact in
$(L^1(I,(X_\vartheta,\|.\|)),\sigma(L^1(I,(X_\vartheta,\|.\|)),L^{\infty}(I,(l(X_{\vartheta},R),\|.\|_{X'}))))$,\linebreak then it is weakly uniformly integrable.
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