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\(G_{\delta}\)-embeddings in Hilbert space. (English) Zbl 0565.46011
It is shown that a separable Banach space X has the point of weak to norm continuity property (resp. the Radon-Nikodym property) if and only if there exists a compact \(G_{\delta}\)-embedding (resp. an \(H_{\delta}\)- embedding) from X into \(\ell_ 2\). This solves several questions of J. Bourgain and H. P. Rosenthal [J. Funct. Anal. 52, 149-188 (1983; Zbl 0541.46020)]. It is also shown that every non-relatively compact sequence in a Banach space with property (PC) has a difference subsequence which is a boundedly complete basic sequence. This solves a question of Pelczynski and extends some results of W. B. Johnson and H. P. Rosenthal [Stud. Math. 43, 77-92 (1972; Zbl 0213.393)]. Various related questions asked in the above Bourgain-Rosenthal reference and by G. A. Edgar and R. F. Wheeler [Pac. J. Math. 115, 317- 350 (1984; Zbl 0506.46007)] and N. Ghoussoub and H. P. Rosenthal [Math. Ann. 264, 321-332 (1983; Zbl 0511.46017)] are also settled.

46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46C99 Inner product spaces and their generalizations, Hilbert spaces
Full Text: DOI
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