EXISTENCE OF A KIND OF BEST SIMULTANEOUS APPROXIMATIONS IN $L_p(\Omega,\Sigma, X)$

Abstract

Let $X$ be a Banach space, $Y$ a nonempty locally weakly compact closed convex subset of $X$, $(\Omega,\Sigma,\mu)$ a complete positive $\sigma$-finite measure space and $\Sigma_0$ a sub-$\sigma$-algebra of $\Sigma$. This paper gives existence results of best simultaneous approximations to two functions in $L_p(\Omega,\Sigma,X)$ from $L_p(\Omega,\Sigma,Y)/L_p(\Omega,\Sigma_0,Y)$ if $\overline{{\rm span}\, Y}$/and $\overline{{\rm span}\, Y}^*$ has/have the Radon-Nikodym property.