Let $X$ be a normed vector space over the real numbers.

(a) Define the dual space $X^{*}$ of $X$ and prove that $X^{*}$ is a Banach space. [You may use without proof that $X^{*}$ is a vector space.]

(b) The Hahn-Banach theorem states the following. Let $X$ be a real vector space, and let $p: X \rightarrow \mathbb{R}$ be sublinear, i.e., $p(x+y) \leqslant p(x)+p(y)$ and $p(\lambda x)=\lambda p(x)$ for all $x, y \in X$ and all $\lambda>0$. Let $Y \subset X$ be a linear subspace, and let $g: Y \rightarrow \mathbb{R}$ be linear and satisfy $g(y) \leqslant p(y)$ for all $y \in Y$. Then there exists a linear functional $f: X \rightarrow \mathbb{R}$ such that $f(x) \leqslant p(x)$ for all $x \in X$ and $\left.f\right|_{Y}=g$.

Using the Hahn-Banach theorem, prove that for any non-zero $x_{0} \in X$ there exists $f \in X^{*}$ such that $f\left(x_{0}\right)=\left\|x_{0}\right\|$ and $\|f\|=1$.

(c) Show that $X$ can be embedded isometrically into a Banach space, i.e. find a Banach space $Y$ and a linear map $\Phi: X \rightarrow Y$ with $\|\Phi(x)\|=\|x\|$ for all $x \in X$.