Weak-polynomial convergence on a Banach space
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- by J. A. Jaramillo and A. Prieto PDF
- Proc. Amer. Math. Soc. 118 (1993), 463-468 Request permission
Abstract:
We show that any super-reflexive Banach space is a $\Lambda$-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of $\kappa$-space (i.e., a Banach space where the weak-polynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space $Z$ is a $\kappa$-space with the approximation property, then the uniform algebra $A(B)$ on the unit ball of $Z$ generated by the weak-star continuous polynomials is not tight.References
- Raymundo Alencar, Richard M. Aron, and Seán Dineen, A reflexive space of holomorphic functions in infinitely many variables, Proc. Amer. Math. Soc. 90 (1984), no. 3, 407–411. MR 728358, DOI 10.1090/S0002-9939-1984-0728358-5
- Richard M. Aron and Carlos Hervés, Weakly sequentially continuous analytic functions on a Banach space, Functional analysis, holomorphy and approximation theory, II (Rio de Janeiro, 1981) North-Holland Math. Stud., vol. 86, North-Holland, Amsterdam, 1984, pp. 23–38. MR 771320, DOI 10.1016/S0304-0208(08)70820-X
- T. K. Carne, B. Cole, and T. W. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (1989), no. 2, 639–659. MR 986022, DOI 10.1090/S0002-9947-1989-0986022-0 P. G. Casazza and T. J. Shura, Tsirelson space, Lecture Notes in Math., vol. 1363, Springer-Verlag, Berlin and New York, 1980. J. F. Castillo and C. Sánchez, Weakly-p-compact, $p$-Banach-Saks and super-reflexive Banach spaces, preprint.
- Soo Bong Chae, Holomorphy and calculus in normed spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92, Marcel Dekker, Inc., New York, 1985. With an appendix by Angus E. Taylor. MR 788158
- Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
- Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
- Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9 T. W. Gamelin, Uniform algebras, Chelsea, New York, 1984.
- J. Globevnik, On interpolation by analytic maps in infinite dimensions, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 2, 243–252. MR 475859, DOI 10.1017/S0305004100054505
- Robert C. James, Super-reflexive spaces with bases, Pacific J. Math. 41 (1972), 409–419. MR 308752
- William B. Johnson, On finite dimensional subspaces of Banach spaces with local unconditional structure, Studia Math. 51 (1974), 225–240. MR 358306, DOI 10.4064/sm-51-3-225-240
- Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90
- Raymond A. Ryan, Dunford-Pettis properties, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 5, 373–379 (English, with Russian summary). MR 557405 D. Van Dulst, Reflexive and super-reflexive spaces, Math. Centre Tracts, vol. 102, North-Holland, Amsterdam, 1982.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 463-468
- MSC: Primary 46B99; Secondary 46G20, 46J15
- DOI: https://doi.org/10.1090/S0002-9939-1993-1126196-0
- MathSciNet review: 1126196