Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Smoothness and weak$ \sp{\ast}$ sequential compactness


Authors: James Hagler and Francis Sullivan
Journal: Proc. Amer. Math. Soc. 78 (1980), 497-503
MSC: Primary 46B05
DOI: https://doi.org/10.1090/S0002-9939-1980-0556620-4
MathSciNet review: 556620
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If a Banach space E has an equivalent smooth norm, then every bounded sequence in $ {E^\ast}$ has a $ {\text{weak}^\ast}$ converging subsequence. Generalizations of this result are obtained.


References [Enhancements On Off] (What's this?)

  • [1] E. Asplund, Fréchet differentiablity of convex functions, Acta Math. 121 (1968), 31-47. MR 0231199 (37:6754)
  • [2] E. Bishop and R. R. Phelps, Support functionals of convex sets, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 27-35. MR 0154092 (27:4051)
  • [3] E. Cech and B. Pospisil, Sur les espaces compacts, Publ. Fac. Sci. Univ. Masaryk 258 (1938), 1-14.
  • [4] J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not $ wea{k^\ast}$ sequentially compact, Israel J. Math. 28 (1977), 325-330. MR 0482086 (58:2173)
  • [5] J. Hagler and E. Odell, A Banach space not containing $ {l^1}$ whose dual ball is not $ wea{k^\ast}$ sequentially compact, Illinois J. Math. 22 (1978), 290-294. MR 0482087 (58:2174)
  • [6] R. Haydon, On Banach spaces which contain $ {l^1}(\tau )$ and types of measures on compact spaces, Israel J. Math. 28 (1977), 313-324. MR 0511799 (58:23514)
  • [7] K. John and V. Zizler, On rough norms on Banach spaces (preprint). MR 500126 (80d:46028)
  • [8] V. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51-63. MR 0115076 (22:5879)
  • [9] D. G. Larman and R. R. Phelps, Gateaux differentiability of convex functions on Banach spaces, J. London Math. Soc. (to appear). MR 545208 (80m:46017)
  • [10] E. E. Leach and J. H. M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120-126. MR 0293394 (45:2471)
  • [11] R. R. Phelps, Support cones in Banach spaces and their applications, Advances in Math. 13 (1974), 1-19. MR 0338741 (49:3505)
  • [12] -, Convex functions on real Banach spaces, unpublished lecture notes.
  • [13] H. P. Rosenthal, A characterization of Banach spaces containing $ {l^1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413. MR 0358307 (50:10773)
  • [14] -, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 803-831. MR 499730 (80d:46023)
  • [15] C. Stegall, The Radon-Nikodym property in conjugate Banach spaces. II (preprint). MR 603779 (82k:46030)
  • [16] S. Troyanski, Example of a smooth space whose conjugate has no strictly convex norm, Studia Math. 35 (1970), 305-309. MR 0271708 (42:6589)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B05

Retrieve articles in all journals with MSC: 46B05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0556620-4
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society