# APPLICATION OF BISHOP-PHELPS THEOREM IN THE APPROXIMATION THEORY

Volume 3, Issue 2, pp 144 - 147 Publication Date: May 14, 2010
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### Authors

R. ZARGHAMI - University of Tabriz-Faculty of Mathematical Sciences, Tabriz, Iran

### Abstract

In this paper we apply the Bishop-Phelps Theorem to show that if $X$ is a Banach space and $G\subseteq X$ is a maximal subspace so that $G^\perp = \{x^* \in X^*\mid x^*(y) = 0; \forall y \in G\}$ is an L-summand in $X^*$, then $L^1(\Omega,G)$ is contained in a maximal proximinal subspace of $L^1(\Omega,X)$.

### Keywords

• Bishop-Phelps Theorem
• support point
• proximinality
• L-projection.

### References

[1] E. Bishop, R. R. Phelps, The support functionals of a convex set, Proc. Symposia in Pure Math. AMS. 7 (1963), 27-35.
[2] P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Math. 1574, Springer, Berlin, Heidelberg, New York, (1993).
[3] F. Hiai and H. Umegaki, Integrals, conditional expection, and martingales multivalued functions, J. Multivariate Anal. 7 (1977), 149-182.
[4] R. Khalil and F. Said, Best approximation in $L^1(­\Omega,X)$, Proceeding of the Amer. Math. Soc. 1 (1999), 183-189.
[5] P. Mani. A characterization of convex set, Handbook of convex geometry (1993).
[6] R. R. Phelps, The Bishop-Phelps Theorem in complex spaces: an open problem, Pure App. Math. 131 (1991), 337-340.
[7] M. Sababheh, R. Khalil, Remarks on remotal sets in vector valued function spaces, The J. Nonlinear Sci. Appl., (2009), no. 1, 1-10.
[8] I. Sadeqi, Support functionals and their relation to the RNP, IJMMS, 16 (2004), 827-832.