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Covering a compact set in a Banach space by an operator range of a Banach space with basis

Author(s): V. P. Fonf; W. B. Johnson; A. M. Plichko; V. V. Shevchyk
Journal: Trans. Amer. Math. Soc. 358 (2006), 1421-1434.
MSC (2000): Primary 46B28; Secondary 46B15, 46B25, 46B50
Posted: September 9, 2005
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Abstract | References | Similar articles | Additional information

Abstract: A Banach space $X$ has the approximation property if and only if every compact set in $X$ is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for $\mathcal{L}_p$spaces and quotients of $\mathcal{L}_p$ spaces in terms of covering compact sets in $X$ by operator ranges from $\mathcal{L}_p$ spaces. A Banach space $X$is a $\mathcal{L}_1$ space if and only if every compact set in $X$ is contained in the closed convex symmetric hull of a basic sequence which converges to zero.

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Additional Information:

V. P. Fonf
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel --- and --- Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: fonf@black.bgu.ac.il

W. B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: johnson@math.tamu.edu

A. M. Plichko
Affiliation: Instytut Matematyki, Politechnika Krakowska im. Tadeusza Kosciuszki, ul. Warszawska 24, Krakow 31-155, Poland
Email: aplichko@usk.pk.edu.pl

V. V. Shevchyk
Affiliation: Sebastian-Kneipp Gasse, 7, Augsburg 86152, Germany
Email: vshevchyk@hotmail.com

DOI: 10.1090/S0002-9947-05-04083-3
PII: S 0002-9947(05)04083-3
Received by editor(s): September 7, 2001
Received by editor(s) in revised form: July 9, 2002
Posted: September 9, 2005
Additional Notes: The second author was supported in part by NSF DMS-9900185, DMS-0200690, Texas Advanced Research Program 010366-0033-20013, and the U.S.-Israel Binational Science Foundation
The third author was supported in part by the DAAD Foundation
Copyright of article: Copyright 2005, by the authors

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