Pointwise Best Approximation in the Space of Strongly Measurable Functions with Applications to Best Approximation in Lp(μ,X)

✍ Scribed by Y. Zhaoyong; G. Tiexin

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 259 KB
• Volume: 78
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1994.1081

✦ Synopsis

The object of this paper is to prove the following theorem: Let (Y) be a closed subspace of the Banach space (X,(S, \Sigma, \mu)) a (\sigma)-finite measure space, (L(S, Y)) (respectively, (L(S, X)) ) the space of all strongly measurable functions from (S) to (Y) (respectively, (X) ), and (p) a positive number. Then (L(S, Y)) is pointwise proximinal in (L(S, X)) if and only if (L^{p}(\mu, Y)) is proximinal in (L^{p}(\mu, X)). As an application of the theorem stated above, we prove that if (Y) is a separable closed subspace of the Banach space (X, p) is a positive number, then (L^{p}(\mu, Y)) is proximinal in (L^{p}(\mu, X)) if and only if (Y) is proximinal in (X). Finally, several other interesting results on pointwise best approximation are also obtained. 1994 Academic Press, Inc.