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

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- by V. P. Fonf, W. B. Johnson, A. M. Plichko and V. V. Shevchyk PDF
- Trans. Amer. Math. Soc.
**358**(2006), 1421-1434

## 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.## References

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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 - © Copyright 2005 by the authors
- 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
- MathSciNet review: 2186980