Let $X$ be a Banach space.

(a) Define the dual space $X^{\prime}$, giving an expression for $\|\Lambda\|_{X^{\prime}}$ for $\Lambda \in X^{\prime}$. If $Y=L^{p}\left(\mathbb{R}^{n}\right)$ for some $1 \leqslant p<\infty$, identify $Y^{\prime}$ giving an expression for a general element of $Y^{\prime}$. [You need not prove your assertion.]

(b) For a sequence $\left(\Lambda_{i}\right)_{i=1}^{\infty}$ with $\Lambda_{i} \in X^{\prime}$, what is meant by: (i) $\Lambda_{i} \rightarrow \Lambda$, (ii) $\Lambda_{i} \rightarrow \Lambda$ (iii) $\Lambda_{i} \stackrel{*}{\rightarrow} \Lambda$ ? Show that (i) $\Longrightarrow$ (ii) $\Longrightarrow$ (iii). Find a sequence $\left(f_{i}\right)_{i=1}^{\infty}$ with $f_{i} \in$ $L^{\infty}(\mathbb{R})=\left(L^{1}(\mathbb{R})\right)^{\prime}$ such that, for some $f, g \in L^{\infty}\left(\mathbb{R}^{n}\right)$ :

$$f_{i} \stackrel{*}{\rightarrow} f, \quad f_{i}^{2} \stackrel{*}{\rightarrow} g, \quad g \neq f^{2} .$$

(c) For $f \in C_{c}^{0}\left(\mathbb{R}^{n}\right)$, let $\Lambda: C_{c}^{0}\left(\mathbb{R}^{n}\right) \rightarrow \mathbb{C}$ be the map $\Lambda f=f(0)$. Show that $\Lambda$ may be extended to a continuous linear map $\tilde{\Lambda}: L^{\infty}\left(\mathbb{R}^{n}\right) \rightarrow \mathbb{C}$, and deduce that $\left(L^{\infty}\left(\mathbb{R}^{n}\right)\right)^{\prime} \neq L^{1}\left(\mathbb{R}^{n}\right)$. For which $1 \leqslant p \leqslant \infty$ is $L^{p}\left(\mathbb{R}^{n}\right)$ reflexive? [You may use without proof the Hahn-Banach theorem].