What is meant by the dual $X^{*}$ of a normed space $X$ ? Show that $X^{*}$ is a Banach space.

Let $X=C^{1}(0,1)$, the space of functions $f:(0,1) \rightarrow \mathbb{R}$ possessing a bounded, continuous first derivative. Endow $X$ with the sup norm $\|f\|_{\infty}=\sup _{x \in(0,1)}|f(x)|$. Which of the following maps $T: X \rightarrow \mathbb{R}$ are elements of $X^{*}$ ? Give brief justifications or counterexamples as appropriate.

1. $T f=f\left(\frac{1}{2}\right)$;
2. $T f=\|f\|_{\infty}$
3. $T f=\int_{0}^{1} f(x) d x$
4. $T f=f^{\prime}\left(\frac{1}{2}\right)$.

Now suppose that $X$ is a (real) Hilbert space. State and prove the Riesz representation theorem. Describe the natural $\operatorname{map} X \rightarrow X^{* *}$ and show that it is surjective.

[All normed spaces are over $\mathbb{R}$. You may assume that if $Y$ is a closed subspace of a Hilbert space $X$ then $\left.X=Y \oplus Y^{\perp} .\right]$