Gδ-embeddings in Hilbert space

AbstractIt 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 Gδ-embedding (resp. an Hδ-embedding) from X into l2. This solves several questions of J. Bourgain and H. P. Rosenthal (J. Funct. Anal. 52 (1983)). 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 (Studia Math. 43 (1972), 77–92). Various related questions asked in the above Bourgain-Rosenthal reference and by G. A. Edgar and R. F. Wheeler (Pac. J. Math. 115 (1984)) and N. Ghoussoub and H. P. Rosenthal (Math. Ann. 264 (1983), 321–332) are also settled