A note on pointwise best approximation (Q1266132)

Let \((\Omega ,A,\mu)\)be a \(\sigma -\)finite measure space with \(\mu (\Omega)>0\), \(0<p<\infty\) and \(Y\) is a separable subspace of a Banach space \(X\). In \textit{You Zhao-Yong} and \textit{Guo Tie-Xin} [J. Approximation Theory 78, No. 3, 314-320 (1994; Zbl 0808.41024)] was proved that \(Y\) is proximinal in \(X\) iff \( L^p(\mu ,Y)\) is proximinal in \(L^p(\mu ,Y)\). But there is a gap in the process of the proof of this theorem. In this paper a new proof of this theorem is given.