Weak$^*$-weak points of continuity and very smooth points in Bochner function spaces $L^{p}(μ, X)$

Abstract

Suppose that $([0,1],\mathcal{A},μ)$ is the Lebesgue measure space and $1< p<\infty$ with $\frac{1}{p}+\frac{1}{q}=1$. We prove that an unit vector $h\in L^{p}(μ, X)^*$ is a weak$^*$-weak point of continuity of $S(L^{p}(μ, X)^*)$ if and only if $\frac{h(t)}{\|h(t)\|}$ is a weak$^*$-weak point of continuity of $S(X^*)$ for almost every $t\in [0, 1]$ with $h(t)\neq 0$, where $X$ is a Banach space such that $X^*$ is separable. As an application of this result, under the same hypothesis on $X$, we prove that a point $f\in L^p(μ, X)_{1}$ is a very smooth point if and only if $\frac{f(t)}{\|f(t)\|}\in S(X)$ is a very smooth point for almost every $t\in [0, 1]$ with $f(t)\neq 0$. We give several examples to show that these results does not hold for $p=1$.