Geometry of Banach spaces and biorthogonal systems

S. J. Dilworth; Maria Girardi; W. Johnson

ArticleOriginal scientific text

Title

Geometry of Banach spaces and biorthogonal systems

Authors ¹, ¹, ²

Affiliations

Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A.

Abstract

A separable Banach space X contains

ℓ_{1}

isomorphically if and only if X has a bounded fundamental total

w c_{0} \cdot

-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total

w c_{0} \cdot

-biorthogonal system.

[DJ] W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. 45 (1973), 173-179.
[D1] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984.
[D2] J. Diestel, A survey of results related to the Dunford-Pettis property, in: Proc. Conf. on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, NC, 1979), Amer. Math. Soc., Providence, RI, 1980, 15-60.
[DU] J. Diestel and J. J. Uhl, Jr., Vector Measures, with a foreword by B. J. Pettis, Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977.
[DRT] P. N. Dowling, N. Randrianantoanina and B. Turett, Remarks on James's distortion theorems, Bull. Austral. Math. Soc. 57 (1998), 49-54.
[Dv] A. Dvoretzky, Some results on convex bodies and Banach spaces, in: Proc. Internat. Sympos. on Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem, 1961, 123-160.
[E] P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309-317.
[FS] C. Foiaş and I. Singer, On bases in C([0,1]) and L([0,1]), Rev. Roumaine Math. Pures Appl. 10 (1965), 931-960.
[GK] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
[H1] J. Hagler, Embeddings of $l^{1}$ into conjugate Banach spaces, Ph.D. dissertation, Univ. of California, Berkeley, CA, 1972.
[H2] J. Hagler, Some more Banach spaces which contain $ℓ_{1}$ , Studia Math. 46 (1973), 35-42.
[HJ] J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact, Israel J. Math. 28 (1977), 325-330.
[JR] W. B. Johnson and H. P. Rosenthal, On ω*-basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), 77-92.
[LT1] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I. Sequence Spaces, Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977.
[LT2] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Ergeb. Math. Grenzgeb. 97, Springer, Berlin, 1979.
[LT3] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer, Berlin, 1973,
[M] A. I. Markushevich, On a basis in the wide sense for linear spaces, Dokl. Akad. Nauk SSSR 41 (1943), 241-244.
[Mil] V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspekhi Mat. Nauk 26 (1971), no. 6, 73-149.
[OP] R. I. Ovsepian and A. Pełczyński, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in $L^{2}$ , Studia Math. 54 (1975), 149-159.
[P] A. Pełczyński, All separable Banach spaces admit for every ε>0 fundamental total and bounded by 1+ε biorthogonal sequences, ibid. 55 (1976), 295-304.
[P1] A. Pełczyński, On Banach spaces containing $L_{1} (μ)$ , ibid. 30 (1968), 231-246.
[S1] I. Singer, Bases in Banach Spaces. I, Grundlehren Math. Wiss. 154, Springer, New York, 1970.
[S2] I. Singer, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), 183-188.
[W] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, Cambridge, 1991.

Title

Geometry of Banach spaces and biorthogonal systems

Affiliations

Abstract

Bibliography