A study on coreflexive Banach Spaces

In this paper, we study non-reflexive Banach spaces $X$ for which the quotient space $X^{**}/X$ is reflexive. Such spaces were first introduced by James R.~Clark, where they were called coreflexive spaces. We show that a space $X$ is coreflexive if and only if every separable subspace $Y\subseteq X$ is coreflexive, provided that $X$ is w$^*$-sequently dense in its bidual $X^{**}$. We show that coreflexive spaces are stable under $\ell^{p}$-sum for $1<p<\infty$. We show that if $X$ is a coreflexive space such that $X^{**}/X$ is separable, then the space of Bochner $p$-integrable functions, $L^{p}(\mu,X)$ is coreflexive for $1<p<\infty$. We conclude by providing an alternative proof of the fact, in a quasi-reflexive space $X$, w-PC's of the unit ball $X_{1}$ continue to have the same property in all the higher even-order dual unit balls of $X$.