On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces
Authors:
S. Argyros and V. Farmaki
Journal:
Trans. Amer. Math. Soc. 289 (1985), 409427
MSC:
Primary 46B20; Secondary 46C10
MathSciNet review:
779073
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Abstract 
References 
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Abstract: A characterization of weakly compact subsets of a Hilbert space, when they are considered as subsets of spaces with an unconditional basis, is given. We apply this result to renorm a class of reflexive 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 .
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 [1]
 A. Abramovich, Weakly compact sets in topological spaces, Theor. Funktsii Funktsional. Anal. I Prilozhen 15 (1972), 2735.
 [2]
 D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 3546. MR 0228983 (37:4562)
 [3]
 S. Argyros, On nonseparable Banach spaces, Trans. Amer. Math. Soc. 210 (1982), 193216. MR 642338 (84f:46018)
 [4]
 Y. Benyamini and T. Starbird, Embedding weakly compact sets into Hilbert space, Israel J. Math. 23 (1976), 137141. MR 0397372 (53:1231)
 [5]
 Y. Benyamini, M. E. Rudin and M. Wage, Continuous image of weakly compact subsets of Banach spaces, Pacific J. Math. 70 (1977), 309324. MR 0625889 (58:30065)
 [6]
 W. J. Davies, T. Figiel, W. B. Johnson and A. Pełczynski, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311327. MR 0355536 (50:8010)
 [7]
 M. Day, R. C. James and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23 (1971), 10511059. MR 0287285 (44:4492)
 [8]
 J. Diestel, Geometry of Banach spaces, Selected Topics, Lecture Notes in Math., vol. 485, SpringerVerlag, Berlin and New York. MR 0461094 (57:1079)
 [9]
 W. F. Eberlein, Weak compactness in Banach spaces. I, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 5153. MR 0021239 (9:42a)
 [10]
 T. Figiel, W. Johnson and L. Tzafriri, On lattices and spaces having local unconditional structure with application to Lorentz functions spaces, J. Approx. Theory 13 (1975), 395412. MR 0367624 (51:3866)
 [11]
 A. L. Garkavi, On the Chebyshev center of a set in a normed space, Investigations on Contemporary Problems in the Constructive Theory of Function, Moscow, 1981, pp. 328331. MR 0188752 (32:6188)
 [12]
 R. C. James, Super reflexive spaces with bases, Pacific J. Math. 41 (1972), 409419. MR 0308752 (46:7866)
 [13]
 D. N. Kutzarova and S. L. Troyanski, Reflexive Banach spaces without equivalent norms which are uniformly convex or uniformly differential in every direction, Studia Math. 72 (1982), 9295. MR 665893 (83k:46024)
 [14]
 J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Lecture Notes in Math., vol. 338, SpringerVerlag, Berlin and New York. MR 0415253 (54:3344)
 [15]
 R. Phelps, Lectures in Choquet's theorem, Van Nostrand Math. Studies, Vol. 7, Van Nostrand, Princeton, N. J.
 [16]
 H. P. Rosenthal, The hereditary problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83111. MR 0417762 (54:5810)
 [17]
 N. Shanin, On the product of topological space, Trudy Mat. Inst. Akad. Nauk. SSSR 24 (1948), 112 pages (Russian).
 [18]
 S. L. Troyanski, On uniform convexity and smoothness in every direction in nonseparable Banach spaces with an unconditional basis, C. R. Acad. Bulgare Sci. 30 (1977), 12431246. MR 0500080 (58:17787)
 [19]
 , On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173180. MR 0306873 (46:5995)
 [20]
 V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. 87 (1971), 533. MR 0300060 (45:9108)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507790739
PII:
S 00029947(1985)07790739
Article copyright:
© Copyright 1985
American Mathematical Society
