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Right and Left Weak Approximation Properties in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Changsun Choi ,
Ju Myung Kim  and
Keun Young Lee
Changsun Choi
Affiliation:
Division of Applied Mathematics, KAIST, Daejeon 305-701, Korea e-mail: cschoi@kaist.ac.krnorthstar21@hanmail.net
Ju Myung Kim
Affiliation:
Division of Applied Mathematics, KAIST, Daejeon 305-701, Korea e-mail: cschoi@kaist.ac.krnorthstar21@hanmail.net
Keun Young Lee
Affiliation:
National Institute for Mathematical Sciences, Yuseong-gu, Daejeon 305-340, Korea e-mail: kjm21@nims.re.kr
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Abstract

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New necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

Keywords

46B2846B10approximation propertyquasi approximation propertyweak approximation property
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 1 , 01 March 2009 , pp. 28 - 38
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Casazza, P. G., Approximation Properties. In: Handbook of the geometry of Banach spaces, Vol. 1, North-Holland, Amsterdam, 2001, pp. 271–316.Google Scholar
[2] Choi, C. and Kim, J. M.,Weak and quasi approximation properties in Banach spaces. J. Math. Anal. Appl. 316(2006), no. 2, 722–735.Google Scholar
[3] Diestel, J., Sequences and series in Banach spaces. Graduate Texts in Mathematics 92, Springer-Verlag, New York, 1984.Google Scholar
[4] Feder, M. and Saphar, P., Spaces of compact operators and their dual spaces. Israel J. Math. 21(1975), no. 1, 38–49.Google Scholar
[5] Figiel, T., Factorization of compact operators and applications to the approximation problem. Studia Math. 45(1973), 191–210.Google Scholar
[6] Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16(1955), no. 16.Google Scholar
[7] Kim, J. M., Dual problems for weak and quasi approximation properties. J. Math. Anal. Appl. 321(2006), 569–575.Google Scholar
[8] Kim, J. M., On relations between weak approximation properties and their inheritances to subspaces. J. Math. Anal. Appl. 324(2006), no. 1, 721–727.Google Scholar
[9] Kim, J. M., Compact adjoint operators and approximation properties. J. Math. Anal. Appl. 327(2007), 257–268.Google Scholar
[10] Lindenstrauss, J., On nonseparable reflexive Banach spaces. Bull. Amer. Math. Soc. 72(1966), 967–970.Google Scholar
[11] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces. I. Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin, 1977.Google Scholar
[12] Megginson, R. E., An Introduction to Banach space theory. Graduate Texts in Mathematics 183, Springer-Verlag, New York, 1998.Google Scholar
[13] Oja, E. and Pelander, A., The approximation property in terms of the approximability of weak*-weak continuous operators. J. Math. Anal. Appl. 286(2003), no. 2, 713–723.Google Scholar