Let $X$ be a normed linear space ( or Banach space ). The linear operator $J : X \to (X^{*})^{*}$ defined by$$ J(x)[\psi] = \psi(x) \ \operatorname{for all} x\in X , \psi \in X^{*}$$
is called the natural embedding of $X$ into $(X^*)^*$. The space $X $ is said to be reflexive provided $J(X)=(X^{*})^{*}$.
Now let $B$ be the closed unit ball of $X$, and $B^{**}$ the closed unit ball of $X^{**}$. My question is, if $J(B)=B^{**}$, then $X$ is reflexive ? It seems that this reductibility occurs in the proof of the Eberlein-Smulian Theorem, Royden's Real Analysis, Fourth edition, Section 15.3, Theorem 8.