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- W1488589439 abstract "We investigate possible extensions of the classical Krein-v{S}mulian theorem to various weak topologies. In particular, we show that if $X$ is a WCG Banach space and $tau$ is any locally convex topology weaker than the norm-topology, then for every $tau$-compact norm-bounded set $H$, $overline{operatorname{conv}}^{,tau}H$ is $tau$-compact. In arbitrary Banach spaces, the norm-fragmentability assumption on $H$ is shown to be sufficient for the last property to hold. A new proof to the following result is given: If a Banach space does not contain a copy of $ell_1[0,1]$, then the Krein-v{S}mulian theorem holds for every topology $tau$ induced by a norming set of functionals. We conclude that in such spaces a norm-bounded set is weakly compact if it is merely compact in the topology induced by a boundary. On the other hand, the same statement is obtained for all $C(K)$ and $ell_1(Gamma)$ spaces." @default.
- W1488589439 created "2016-06-24" @default.
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- W1488589439 date "2003-10-01" @default.
- W1488589439 modified "2023-09-24" @default.
- W1488589439 title "On the Krein-Šmulian theorem for weaker topologies" @default.
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- W1488589439 doi "https://doi.org/10.1215/ijm/1258138086" @default.
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