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A ``nonlinear'' proof of Pitt's compactness theorem

Author(s): M. Fabian; V. Zizler
Journal: Proc. Amer. Math. Soc. 131 (2003), 3693-3694.
MSC (2000): Primary 46B25
Posted: July 9, 2003
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Abstract: Using Stegall's variational principle, we present a simple proof of Pitt's theorem that bounded linear operators from $\ell_q$ into $\ell_p$ are compact for $1\le p<q<+\infty$.

References:

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M. Fabian, P. Habala, P. Hájek, V. Montesinos Santaluciá, J. Pelant, and V. Zizler, Functional analysis and infinite-dimensional geometry, CMS Books in Mathematics, Springer-Verlag, New York, 2001. MR 2002f:46001

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J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin, 1977. MR 58:17766

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R. R. Phelps, Convex functions, monotone operators, and differentiability, Lecture Notes in Math. No. 1364, 2nd Edition, Springer-Verlag, Berlin, 1993. MR 94f:46055

4.
Ch. Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Annalen 236 (1978), 171-176. MR 80a:46022

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Ch. Stegall, Optimization and differentiation in Banach spaces, Linear Algebra and Appl. 84 (1986), 191-211. MR 88a:49005

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Additional Information:

M. Fabian
Affiliation: Mathematical Institute, Czech Academy of Sciences, Zitná 25, 11567 Praha 1, Czech Republic
Email: fabian@math.cas.cz

V. Zizler
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: vzizler@math.ualberta.ca

DOI: 10.1090/S0002-9939-03-07200-9
PII: S 0002-9939(03)07200-9
Keywords: $\ell_p$ space, compact operator, variational principle
Received by editor(s): April 6, 2001
Posted: July 9, 2003
Additional Notes: Supported by grants GA CR 201-98-1449, AV 1019003, and NSERC 7926
Communicated by: Jonathan M. Borwein
Copyright of article: Copyright 2003, American Mathematical Society

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