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Note on a theorem of S. Mercourakis about weakly K-analytic Banach spaces. (English) Zbl 0687.54027
Let E be a Banach space with the weak topology, and let L be a subset of E. L is said to be weakly countably determined if there exist a subspace M of the irrationals and a USC map \(\Phi: M\to {\mathcal K}(L)\) (the hyperspace of compacta in L) such that \(\cup \Phi (M)=L\). This paper gives an alternative proof of the following theorem by S. Mercourakis [Trans. Am. Math. Soc. 300, 307-327 (1987; Zbl 0621.46018)]: Let L be a weakly countably determined subset of a Banach space E, of density \(\tau\). Then there exist a separable metric space M, a set \(\Lambda\) of cardinality \(\tau\), and a bounded linear continuous injection \(T: C_ p(L)\to \{x\in R^{M\times \Lambda}:\) for each compact \(C\subseteq M\) and every \(\epsilon >0\), \(| x(t,\lambda)| \geq \epsilon\) for at most finitely many \((t,\lambda)\in C\times \Lambda\}\). If L is K-analytic, one can take for M the space of irrationals.
Reviewer: F.van Engelen

MSC:
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
46B20 Geometry and structure of normed linear spaces
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