Let $X$ be a normed vector space over $\mathbb{R}$. Define the dual space $X^{*}$ and show directly that $X^{*}$ is a Banach space. Show that the map $\phi: X \rightarrow X^{* *}$ defined by $\phi(x) v=v(x)$, for all $x \in X, v \in X^{*}$, is a linear map. Using the Hahn-Banach theorem, show that $\phi$ is injective and $|\phi(x)|=|x|$.

Give an example of a Banach space $X$ for which $\phi$ is not surjective. Justify your answer.