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The Lindelöf property in Banach spaces. (English) Zbl 1038.54008

If \((M,\rho)\) is a metric space with \(\rho\) bounded and \(I\) is a set, then one can consider several topologies on the product \((M,\rho)^I\): (i) the product topology of pointwise convergence; (ii) the uniform metric; (iii) the \(\gamma\)-topology \(\gamma(I)\), i.e., the topology of uniform convergence on countable subsets of \(I\). The main result of the paper is the following: If \(K\) is a pointwise compact subset of \((M,\rho)^I\), then \(K\) is fragmented by the uniform metric if and only if \((K,\gamma(I))\) is Lindelöf. Moreover, in this case the countable power of \((K,\gamma(I))\) is Lindelöf, too.
The main theorem has several corollaries: (1) If \(X\) is a Banach space and \(K\subset X^*\) is weak* compact and weakly Lindelöf, then the weak* closure of the convex hull of \(K\) is weakly Lindelöf. (This answers a question asked by M. Talagrand [Pac. J. Math. 81, 239–251 (1979; Zbl 0367.54004)].) (2) A new characterization of Radon-Nikodým compacta. (3) Unified proofs of some known results by J. Orihuela [Progress in functional analysis, Peníscola/Spain, North-Holland Math. Stud. 170, 279–291 (1992; Zbl 0789.46015)] and by P. R. Meyer [Duke Math. J. 33, 33–39 (1966; Zbl 0138.17602)]. (4) Unified proof of the existence of projectional resolution of the identity in WCG spaces and in duals to Asplund spaces.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
46B99 Normed linear spaces and Banach spaces; Banach lattices
46A50 Compactness in topological linear spaces; angelic spaces, etc.
54C35 Function spaces in general topology
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