Bi-dual norm in a Banach space

Question

Suppose $X$ is a Banach space, $X^*$ its dual and $X^{**}$ the dual of the dual.

Then, for $x\in X$, we can define $F_x \in X^{**}$ as

$$ F_x(\phi) = \phi(x) $$ for every $\phi \in X^*$.

Then, it is clear that $\|F_x\| \leq \| x \|$, but does the equality hold?

Answer 1

$\|F_x\|=\sup_{\|\phi\|=1}|F_x(\phi)|$.

There exists $f_x\in X^*$ such that $\|f_x\|=1$ and $f_x(x)=\|x\|$. To see this, consider the linear functional $g:Vect(x)\rightarrow \mathbb R$ defined by $g({x\over\|x\|})=1$, which satisfies $g(x)=\|x\|$ and $\|g\|=1$. The Hahn-Banach theorem implies that you extend $g$ to $f_x$ with $\|f_x\|=1$.

This implies that $\|F_x\|\geq |f_x(x)|=\|x\|$.