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On strongly WCG Banach spaces. (English) Zbl 0637.46011

For a Banach space X, we consider a stronger version of the WCG property. We investigate the property that X is strongly weakly compactly generated (SWCG): i.e. there is a weakly compact subset K of X such that for each weakly compact subset L of X and \(\epsilon >0\), there is a positive integer n with L a subset of \(nK+\epsilon B\) (B the unit ball of X).
Every SWCG space is weakly sequentially complete; however, an example due to Batt and Hiermeyer shows that a separable weakly sequentially complete space need not be SWCG. An example due to Pisier reveals that the injective and projective tensor product of two SWCG spaces need not be SWCG. A translation of the SWCG property to locally convex topologies on a Banach space reveals that the space of vector valued Bochner integrable functions \(L_ 1(\mu,X)\) is strongly \(\sigma (L_ 1(\mu,X)\), \(L_{\infty}(\mu,X\) *))-compactly generated if and only if the underlying \(\sigma\)-finite measure space is purely atomic and X is SWCG or X is reflexive.
Reviewer: G.Schlüchtermann

MSC:

46B10 Duality and reflexivity in normed linear and Banach spaces
46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

[1] [AH] Azimi, P., Hagler, J.: Examples of hereditarilyl 1 Banach spaces failing the Schur property. Pac. J. Math.122, 287-297 (1986) · Zbl 0609.46012
[2] [Ak] Akemann, C.: The dual space of an operator algebra. Trans. Am. Math. Soc.126, 286-302 (1967) · Zbl 0157.44603 · doi:10.1090/S0002-9947-1967-0206732-8
[3] [Ba] Batt, J.: On weak compactness in spaces of vector-valued measures and Bochner integrable functions in connection with the Radon-Nikodym property of Banach spaces. Rev. Roum. Math. Pures Appl.19, 285-304 (1974) · Zbl 0276.28013
[4] [BD] Bourgain, J., Delbaen, F.: A class of specialL ?-spaces. Acta Math.145, 155-176 (1980) · Zbl 0466.46024 · doi:10.1007/BF02414188
[5] [BH1] Batt, J., Hiermeyer, W.: Weak compactness in the space of Bochner integrable functions. Unpublished manuscript, 1980
[6] [BH2] Batt, J., Hiermeyer, W.: On compactness inL p(?,X) in the weak topology and in the topology ?(L p(?,X),L q(?,X’)). Math. Z.182, 409-423 (1983) · Zbl 0501.46010 · doi:10.1007/BF01179760
[7] [BS] Batt, J., Schlüchtermann, G.: Eberlein compacts inL 1(X). Studia Math.83, 239-250 (1986) · Zbl 0555.46014
[8] [D1] Diestel, J.:L X 1 is weakly compactly generated ifX is. Proc. Am. Math. Soc.48, 508-510 (1975) · Zbl 0299.46035
[9] [D2] Diestel, J.: Sequences and series in Banach spaces. New York: Springer 1984
[10] [Din] Dinculeanu, N.: Weak compactness and uniform convergence of operators in spaces of Bochner integrable functions. J. Math. Anal. Appl.109, 372-387 (1985) · Zbl 0603.46040 · doi:10.1016/0022-247X(85)90157-X
[11] [DU] Diestel, J., Uhl, J.: Vector measures. Am. Math. Soc. Surveys15 (1977) · Zbl 0369.46039
[12] [Gro] Grothendieck, A.: Sur les applications lineaires faiblement compactes d’espaces du typeC(K). Can. J. Math.5, 129-173 (1953) · Zbl 0050.10902 · doi:10.4153/CJM-1953-017-4
[13] [Li] Lindenstrauss, J.: Weakly compact sets-their topological properties and the Banach spaces they generate. Ann. Math. Studies69, 235-293 (1972)
[14] [Pe] Pelczynski, A.: Banach spaces of analytic functions and absolutely summing operators. CBMS Regional Conf. Series, V.30, Am. Math. Soc., 1977
[15] [Pi] Pisier, G.: Counterexamples to a conjecture of Grothendieck. Acta Math.151, 181-208 (1983) · Zbl 0542.46038 · doi:10.1007/BF02393206
[16] [R1] Rosenthal, H.: A characterization of Banach spaces containingl 1. Proc. Nat. Acad. Sci. USA71, 2411-2413 (1974) · Zbl 0297.46013 · doi:10.1073/pnas.71.6.2411
[17] [R2] Rosenthal, H.: The heredity problem for weakly compactly generated Banach spaces. Compos. Math.28, 83-111 (1974) · Zbl 0298.46013
[18] [Sa] Saab, E., Saab, P.: On Pelczynski’s properties (V) and (V :). Pac. J. Math.125, 205-210 (1986) · Zbl 0561.46011
[19] [Scha] Schaefer, H.: Topological vector spaces. New York: Springer 1971 · Zbl 0212.14001
[20] [Schl] Schlüchtermann, G.: Der Raum der Bochner-integrierbaren FunktionenL 1(?,X) und die ?(L 1(?,X),L ?(?,X’))-Topologie. Thesis, Universität München, 1986
[21] [Tak] Takesaki, M.: Theory of operator algebras I. New York: Springer 1979 · Zbl 0436.46043
[22] [Tal] Talagrand, M.: Weak Cauchy sequences inL 1(E). Am. J. Math.106, 703-724 (1984) · Zbl 0579.46025 · doi:10.2307/2374292
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