I am trying to prove the following:
A Banach space $X$ is reflexive iff $X^*$ is reflexive.
Thus far, I have proven the forward direction:
Let $J_X:X\mapsto X^{**}$ be the mapping defined by $J_X(x)=x^{**}$ and let $J_{X^*}:X^*\mapsto X^{***}$ be defined by $J_{X^*}(x^*)=x^{***}$. The statement above is equivalent to showing that $J_X(X)=X^{**}$ if and only if $J_{X^*}(X^*)=X^{***}$. Suppose that $J_X(X)=X^{**}$. Let $y\in X^{***}=B(X^{**},\mathbb{R})$. We show that there is some $x^*\in X^*$ such that $J_{X^*}(x^*)=y$, thus showing that $J_{X^*}(X^*)=X^{***}$. Note that $yJ_X:X\mapsto \mathbb{R}$, and hence, $yJ_X\in B(X,\mathbb{R})=X^*$. Since $J_X(X)=X^{**}$, if $z\in X^{**}$, then there is some $x\in X$ such that $J_X(x)=z$. We have $$ y(z)=y(J_X(x))=(J_X(x))(yJ_X)=z(yJ_X)=(J_{X^*}(yJ_X))(z). $$ Thus, $y=J_{X^*}(yJ_X)$, implying that $X^*$ is reflexive.
I have no clue how to prove the other direction. I wanted to try a similar approach, but it's turning out to be kind of difficult (and ugly looking). I would greatly appreciate some help! Thanks!
