Duality

Show that wk is the smallest topology on $\mathcal{X}$ such that each $x^*$ in $\mathcal{X}^*$ is continuous.

Show that wk* is the smallest topology on $\mathcal{X}^*$ such that for each $x$ in $\mathcal{X}$, $x^* \mapsto \left\langle x, x^* \right\rangle$ is continuous.

Prove Theorem 1.3.

Let $\mathcal{X}$ be a complex LCS and let $\mathcal{X}^*_{\mathbb {R}}$ denote the collection of all continuous real linear functionals on $\mathcal{X}$. Use the elements of $\mathcal{X}^*_{\mathbb {R}}$ to define seminorms on $\mathcal{X}$ and let $\sigma (\mathcal{X}, \mathcal{X}^*_{\mathbb {R}})$ be the corresponding topology. Show that $\sigma (\mathcal{X}, \mathcal{X}^*) = \sigma (\mathcal{X}, \mathcal{X}^*_{\mathbb {R}})$.

Prove the remainder of Proposition 1.7.

If $A \subseteq \mathcal{X}$, show that $A$ is weakly bounded if and only if $A^\circ$ is absorbing in $\mathcal{X}^*$.

Let $\mathcal{X}$ be a normed space and let $\left\{ x_n\right\}$ be a sequence in $\mathcal{X}$ such that $x_n \to x$ weakly. Show that there is a sequence $\left\{ y_n\right\}$ such that $y_n \in \operatorname {co}\left\{ x_1, x_2, \dots , x_n\right\}$ and $\left\| y_n - x\right\| \to 0$.

If $\mathcal{H}$ is a Hilbert space and $\left\{ h_n\right\}$ is a sequence in $\mathcal{H}$ such that $h_n \to h$ weakly and $\left\| h_n\right\| \to \left\| h\right\|$, then $\left\| h_n - h\right\| \to 0$. (The same type of result is true for $L^p$-spaces if $1 < p < \infty$. See W.P. Novinger [1972].)

If $\mathcal{X}$ is a normed space show that the norm on $\mathcal{X}$ is lower semicontinuous for the weak topology and the norm of $\mathcal{X}^*$ is lower semicontinuous for the weak-star topology.

Suppose $\mathcal{X}$ is an infinite-dimensional normed space. If $S = \left\{ x \in \mathcal{X} : \left\| x\right\| = 1\right\}$, then the weak closure of $S$ is $\left\{ x : \left\| x\right\| \leq 1\right\}$.