Rocky Mountain Journal of Mathematics

Eberlein compactness

Surjit Singh Khurana

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For finite measure space $(X, \A, \mu)$, a Banach space $E$ with $E^{\p}$ its dual, and a relatively countably compact $Q \subset (L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$, entirely different proofs are given of the results that (i)~$\overline{Q}$ is Eberlein compact, (ii)~the closed convex hull of $\overline{Q}$ in $(L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$ is also compact and (iii)~the closed convex hull of $\overline{Q}$ in $(L_{1}(E), \sigma(L_{1}(E), L_{\infty}(E^{\p})))$ and in $(L_{1}(E), \| \cdot\|_{1})$ are the same.

Article information

Rocky Mountain J. Math., Volume 44, Number 1 (2014), 179-187.

First available in Project Euclid: 2 June 2014

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Zentralblatt MATH identifier

Primary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 46A50: Compactness in topological linear spaces; angelic spaces, etc. 46E40: Spaces of vector- and operator-valued functions
Secondary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10]

Relatively countably compact Eberlein compact Grothendieck completeness theorem barycenter


Khurana, Surjit Singh. Eberlein compactness. Rocky Mountain J. Math. 44 (2014), no. 1, 179--187. doi:10.1216/RMJ-2014-44-1-179.

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