On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces

Argyros, S.; Farmaki, V.

doi:10.1090/S0002-9947-1985-0779073-9

On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces
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by S. Argyros and V. Farmaki PDF

Trans. Amer. Math. Soc. 289 (1985), 409-427 Request permission

Abstract:

A characterization of weakly compact subsets of a Hilbert space, when they are considered as subsets of $B$-spaces with an unconditional basis, is given. We apply this result to renorm a class of reflexive $B$-spaces by defining a norm uniformly convex in every direction. We also prove certain results related to the factorization of operators. Finally, we investigate the structure of weakly compact subsets of ${L^1}(\mu )$.

References

Weakly compact sets in topological

spaces

15

D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35–46. MR 228983, DOI 10.2307/1970554
Spiros A. Argyros, On nonseparable Banach spaces, Trans. Amer. Math. Soc. 270 (1982), no. 1, 193–216. MR 642338, DOI 10.1090/S0002-9947-1982-0642338-2
Y. Benyamini and T. Starbird, Embedding weakly compact sets into Hilbert space, Israel J. Math. 23 (1976), no. 2, 137–141. MR 397372, DOI 10.1007/BF02756793
Y. Benyamini, M. E. Rudin, and M. Wage, Continuous images of weakly compact subsets of Banach spaces, Pacific J. Math. 70 (1977), no. 2, 309–324. MR 625889
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 0355536, DOI 10.1016/0022-1236(74)90044-5
M. M. Day, R. C. James, and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canadian J. Math. 23 (1971), 1051–1059. MR 287285, DOI 10.4153/CJM-1971-109-5
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
W. F. Eberlein, Weak compactness in Banach spaces. I, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 51–53. MR 21239, DOI 10.1073/pnas.33.3.51
T. Figiel, W. B. Johnson, and L. Tzafriri, On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces, J. Approximation Theory 13 (1975), 395–412. MR 367624, DOI 10.1016/0021-9045(75)90023-4
A. L. Garkavi, On the Čebyšev center of a set in a normed space, Studies of Modern Problems of Constructive Theory of Functions (Russian), Fizmatgiz, Moscow, 1961, pp. 328–331 (Russian). MR 0188752
Robert C. James, Super-reflexive spaces with bases, Pacific J. Math. 41 (1972), 409–419. MR 308752
D. N. Kutzarova and S. L. Troyanski, Reflexive Banach spaces without equivalent norms which are uniformly convex or uniformly differentiable in every direction, Studia Math. 72 (1982), no. 1, 91–95. MR 665893, DOI 10.4064/sm-72-1-91-95
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253

Lectures in Choquet’s theorem

Haskell P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83–111. MR 417762

On the product of topological space

24

S. L. Trojanski, Uniform convexity and smoothness in every direction in nonseparable Banach spaces with unconditional bases, C. R. Acad. Bulgare Sci. 30 (1977), no. 9, 1243–1246 (Russian). MR 500080
S. L. Troyanski, On locally uniformly convex and differentiable norms in certain non-separable Banach spaces, Studia Math. 37 (1970/71), 173–180. MR 306873, DOI 10.4064/sm-37-2-173-180
V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 33 pp. (errata insert). MR 300060