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

V. P. Fonf; W. B. Johnson; A. M. Plichko; V. V. Shevchyk

doi:10.1090/S0002-9947-05-04083-3

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

V. P. Fonf, W. B. Johnson, A. M. Plichko, V. V. Shevchyk

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

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 ℒ_p spaces and quotients of ℒ_p spaces in terms of covering compact sets in X by operator ranges from ℒ_p spaces. A Banach space X is a ℒ₁ 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.

Original language	English
Pages (from-to)	1421-1434
Number of pages	14
Journal	Transactions of the American Mathematical Society
Volume	358
Issue number	4
DOIs	https://doi.org/10.1090/S0002-9947-05-04083-3
State	Published - 1 Jan 2006

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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

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