Publication: Proximinality in L-p(mu,X).
Loading...
Full text at PDC
Publication Date
1998-05-02
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Academic Press-Elsevier Science
Citations
Exportar
Abstract
Let X be a Banach space and let Y be a closed subspace of X. Let 1 less than or equal to p less than or equal to infinity and let us denote by L-p(mu, X) the Banach space of all X-valued Bochner p-integrable (essentially bounded for p = infinity) functions on a certain positive complete sigma-finite measure space (Omega, Sigma, mu), endowed with the usual p-norm. In this paper we give a negative answer to the following question: "If Y is proximinal in X, is L-p(mu, Y) proximinal in L-p(mu, X)?" We also show that the answer is affirmative for separable spaces Y. Some consequences of this are obtained.
Description
UCM subjects
Unesco subjects
Keywords
Citation
W. Deeb and R. Khalil, Best approximation in L(X, Y), Math. Proc. Cambridge Philos. Soc. 104 (1988), 527-531.
J. Diestel and J. J. Uhl Jr., ``Vector Measures,'' Math. Surveys, Vol. 15, Amer. Math. Soc., Providence, 1977.
R. Holmes and B. Kripke, Smoothness of approximation, Michigan Math. J. 15 (1968), 225-248.
Zhibao Hu and Bor-Luh Lin, Extremal structure of the unit ball of Lp(+, X)*, J. Math. Anal. Appl. 200 (1996), 567-590.
R. Khalil, Best approximation in Lp(I, X), Math. Proc. Cambridge Philos. Soc. 94 (1983), 277-279.
R. Khalil and W. Deeb, Best approximation in Lp(I, X), II, J. Approx. Theory 59 (1989), 296-299.
R. Khalil and F. Saidi, Best approximation in L1(I, X), Proc. Amer. Math. Soc. 123 (1995), 183-190.
W. A. Light, Proximinality in Lp(S, Y), Rocky Mountain J. Math. 19 (1989), 251-259.
W. A. Light and E. W. Cheney, Some best approximation theorems in tensor product spaces, Math. Proc. Cambridge Philos. Soc. 89 (1981), 385-390.
W. A. Light and E. W. Cheney, ``Approximation Theory in Tensor Product Spaces,'' Lecture Notes in Math., Vol. 1169, Springer-Verlag, New York, 1985.
F. B. Saidi, On the smoothness of the metric projection and its applications to proximinality in Lp(S, Y), J. Approx. Theory 83 (1995), 205-219.
You Zhao-Yong and Guo Tie-Xin,Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in Lp(+, X),J.Approx. Theory 78 (1994), 314-320.