Mendoza Casas, José (1998) Proximinality in L-p(mu,X). Journal of Approximation Theory, 93 (2). pp. 331-343. ISSN 0021-9045
Restricted to Repository staff only until 31 December 2020.
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.
|Uncontrolled Keywords:||proximinal subspaces; best approximation in Lp (μ,X).|
|Subjects:||Sciences > Mathematics > Functional analysis and Operator theory|
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.
|Deposited On:||25 Oct 2012 08:05|
|Last Modified:||07 Feb 2014 09:37|
Repository Staff Only: item control page