Abstract
Let \(X\) be a Banach space and \(G\) be a closed subspace of \(X\). Let us denote by \(L^{\infty}\left( \mu,X\right) \) the Banach space of all \(X\)-valued essentially bounded functions on a \(\sigma\)-finite complete measure space \(\left( \Omega,\Sigma,\mu\right) .\) In this paper we show that if \(G\) is separable, then \(L^{\infty}\left( \mu,G\right) \) is simultaneously proximinal in \(L^{\infty}\left( \mu,X\right) \) if and only if \(G\) is simultaneously proximinal in \(X.\)
Publisher
Academia Romana Filiala Cluj
Reference21 articles.
1. A.P. Bosznoy, A remark on simultaneous approximation, J. Approx. Theory, 28 (1978), pp. 296-298.
2. A.S. Holland, B.N. Sahney and J.Tzimbalario, On best simultaneous approximation, J. Indian Math. Soc., 40 (1976), pp. 69-73.
3. C. B. Dunham, Simultaneous Chebyshev approximation of functions on an interval, Proc. Amer. Math. Soc., 18 (1967), pp. 472-477, https://doi.org/10.1090/s0002-9939-1967-0212463-6
4. Chong Li, On best simultaneous approximation, J. Approx. Theory, 91 (1998), pp. 332-348.
5. E. Abu-Sirhan, Best simultaneous approximation in Lp(I,X), Inter. J. Math. Analysis, 3 (2009) no. 24, pp. 1157-1168.