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 [Proc. Amer. Math. Soc. 36 (1972), pp. 421–427] where they were called coreflexive spaces. In Theorem 5, 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 \(1<p<\infty \) . In Theorem 14, 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 \(X_{1}\) continues to have the same property in all the higher even-order dual unit balls of X.
