Covering a compact set in a Banach space by an operator range of a Banach space with basis
Authors:
V. P. Fonf, W. B. Johnson, A. M. Plichko and V. V. Shevchyk
Journal:
Trans. Amer. Math. Soc. 358 (2006), 1421-1434
MSC (2000):
Primary 46B28; Secondary 46B15, 46B25, 46B50
DOI:
https://doi.org/10.1090/S0002-9947-05-04083-3
Published electronically:
September 9, 2005
MathSciNet review:
2186980
Full-text PDF Free Access
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
MR Author ID:
190586
Email:
fonf@black.bgu.ac.il
W. B. Johnson
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
MR Author ID:
95220
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
Received by editor(s):
September 7, 2001
Received by editor(s) in revised form:
July 9, 2002
Published electronically:
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
Article copyright:
© Copyright 2005
by the authors
