We know that for a space $X$ if $X$ is reflexive then $X^*$ (the dual space) is reflexive. We also know that if $X$ is a banach space then if $X^*$ is reflexive then $X$ is reflexive.
Is the requirement for $X$ to be banach necessary?
Can we find a non-reflexive space $Y$ such that $Y^*$ is reflexive?