×

zbMATH — the first resource for mathematics

\(G_{\delta}\)-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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] {\scJ. Bourgain and G. Pisier}, preprint, 1983.
[2] Bourgain, J; Delbaen, F, A class of special \(L\)_{∞}-spaces, Acta math., 145, 155-176, (1980) · Zbl 0466.46024
[3] Bourgain, J; Rosenthal, H.P, Martingales valued in certain subspaces of L1, Israel J. math., 37, 54-75, (1980) · Zbl 0445.46015
[4] Bourgain, J; Rosenthal, H.P, Geometrical implications of certain finite dimensional decompositions, Bull. soc. math. belg., 32, 57-82, (1980) · Zbl 0463.46011
[5] Bourgain, J; Rosenthal, H.P, Applications of the theory of semi-embeddings to Banach space theory, J. funct. anal., 52, (1983) · Zbl 0541.46020
[6] {\scJ. Diestel}, personal communication.
[7] Diestel, J; Uhl, J, Vector measures, Amer. math. soc. surveys, 15, (1977) · Zbl 0369.46039
[8] Edgar, G.A; Wheeler, R.F, Topological properties of Banach spaces, Pac. J. math., 115, (1984) · Zbl 0506.46007
[9] Ghoussoub, N, Some remarks concerning G_{δ}-embeddings and semi-quotient maps, ()
[10] Ghoussoub, N; Rosenthal, H.P, Martingales, G_{δ}-embeddings and quotients of L1, Math. ann., 264, 321-332, (1983) · Zbl 0511.46017
[11] Ghoussoub, N; Maurey, B, Counterexamples to several problems concerning G_{δ}-embeddings, (), 409-412 · Zbl 0559.46007
[12] Ghoussoub, N; Maurey, B, On the Radon-Nikodym property in function spaces, (), to appear · Zbl 0587.46014
[13] Hurewicz, W, Relative perfekte tule von punktinengen und mengen (A), Fund. math., 12, 78-109, (1928) · JFM 54.0097.06
[14] Johnson, W.B; Lindenstrauss, Y, Examples of \(L\)_{1}-spaces, Ark. mat., 18, 101-106, (1980) · Zbl 0464.46024
[15] Johnson, W.B; Rosenthal, H.P, On weak^{∗}-sequences and their applications to the study of Banach spaces, Studia math., 43, 77-92, (1972) · Zbl 0213.39301
[16] Lotz, H.P; Peck, N.T; Porta, H, Semi-embedding of Banach spaces, (), 233-240 · Zbl 0405.46013
[17] Lindenstrauss, Y; Stegall, C, Examples of separable spaces which do not contain l1 and whose duals are not separable, Studia math., 54, 81-105, (1974) · Zbl 0324.46017
[18] Lindenstrauss, Y; Tzafriri, L, Classical Banach spaces I, (1977), Springer-Verlag New York · Zbl 0362.46013
[19] {\scH. P. Rosenthal}, personal communication.
[20] Rosenthal, H.P, Geometric properties related to the Radon-Nikodym property, () · Zbl 0537.46026
[21] Talagrand, M, La structure du espaces de Banach réticulés ayant la propriété de Radon-Nikodym, Israel J. math., 44, 3, (1983) · Zbl 0523.46016
[22] Talagrand, M, Un espace de Banach réticulé qui a presque la propriété de Radon-Nikodym, (1982), preprint · Zbl 0537.46028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.