Consider the following problem, Exercise 5.24 in Folland's Real Analysis 2nd Edition:
Let $X$ be a Banach space. (a) Let $\widehat X, \widehat{X^*}$ be the natural images of $X, X^*$ in $X^{**}, X^{***}$, and let $$ \widehat X^0 = \{F \in X^{***} \ : \ F|_{\widehat X} = 0\}. $$ Show that $\widehat{X^*} \cap \widehat X^0 = \{0\}$ and that $\widehat{X^*} + \widehat X^0 = X^{***}$. (b) Show that $X$ is relfexive iff $X^*$ is reflexive.
My doubt is in the second part of item (a) ($\widehat{X^*} + \widehat X^0 = X^{***}$) and in how to prove (b) using part (a) (which is what I suppose the author wants us to do).
Any hints would be the most appreciated.
For completenes, here is what I did for the first part of item (a):
Let $F \in \widehat{X^*} \cap \widehat{X}^0$. Then $F(\widehat x) = 0$ for all $x \in X$. But $F = \widehat f$ for some $f \in X^*$. So we have that $$ \widehat{f}(\widehat x) = \widehat{x}(f) = f(x) = 0 \quad \forall x \in X, $$ hence $f = 0$ and therefore $F = 0$. Then $\widehat{X^*} \cap \widehat{X}^0= \{0\}$.
Thanks in advance and kind regards.
Here is my solution:
(a) Let $\hat{f}\in\hat{X}^*$ with $\hat{f}\vert_{\hat{X}}=0$. We will have that $\hat{f}(\hat{x})=0$ for all $x\in X$. Since $\hat{f}(G)=G(f)$ for any $G\in X^{***}$, we will have that $\hat{f}(\hat{x})=\hat{x}(f)=f(x)$, so $f=0$. Now if $F\in X^{***}$, we consider a functional $\phi:X\to\mathbb{C}$ such that $\phi(x)=F(\hat{x})$ for any $x\in X$ (we define it like this). We can easily see that $\phi$ is indeed linear and since $\|\hat{x}\|=\|x\|$ for any $x$ (this is By Hahn-Banach), we have that $\phi\in X^*$. Thus $\hat{\phi}\in\hat{X}^*$ and therefore $F=\hat{\phi}+(F-\hat{\phi})$. We will show that $F-\hat{\phi}\in \hat{X^0}$. Indeed, if $\hat{x}\in\hat{X}$ we will have that $F(\hat{x})-\hat{\phi}(\hat{x})=F(\hat{x})-\hat{x}(\phi)=F(\hat{x})-\phi(x)=0$.
(b) If $X$ is reflexive then $\hat{X}=X^{**}$ so $\hat{X}^0=0$ so $\hat{X}^*=X^{***}$. Conversely, if $\hat{X}^*=X^{***}$, we have that $\hat{X}^0=0$. Since $X$ is Banach, $\hat{X}$ is a closed subspace o f $X^{**}$ (as isometrically isomorphic to $X$). By Hahn-Banach, if $\hat{X}\neq X^{**}$ and $F\in X^{**}\setminus\hat{X}$, we may find $G\in X^{***}$ with $G\vert_{\hat{X}}=0$ and $G(F)=\text{dist}(F,\hat{X})$. But then $G\in\hat{X}^0$, so $G=0$, so $F\in\hat{X}$, a contradiction.