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Best L∞-approximation of measurable, vector-valued functions

Published online by Cambridge University Press:  24 October 2008

Abdallah M. Al-Rashed  and
Richard B. Darst
Abdallah M. Al-Rashed
Affiliation:
Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
Richard B. Darst
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado, U.S.A.
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Abstract

Let (Ω, ,μ) be a probability space, and let be a sub-sigma-algebra of . Let X be a uniformly convex Banach space. Let A =L∞(Ω, , μ X) denote the Banach space of (equivalence classes of) essentially bounded μ-Bochner integrable functions g: Ω.→ X, normed by the function ∥.∥∞ defined for g ∈ A by

(cf. [6] for a discussion of this space). Let B = L∞(Ω, , μ X), and let f ε A. A sufficient condition for g ε B to be a best L∞-approximation to f by elements of B is established herein.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 3 , November 1984 , pp. 477 - 481
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

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