Metrizability of weak topologies on bounded sets

Let $X$ be a normed space, with $X^{*}$ separable. Prove that the weak topology in the unit ball of the space $X$ is metrizable.

Let $X$ be a separable normed space. Prove that the *-weak topology in the unit ball of the space $X^{*}$ is metrizable.

Let $X$ be a normed space, and $X^{*}$ separable (hence, by the assertion of problem 5.42, the space $X$ itself is separable). Prove that weak convergence on the unit ball of the space $X$ is metrizable.

Let $X$ be a normed space, and $X^{*}$ separable. Specify a metric $w(x, y)$, defined on all of $X$, such that a sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ converges weakly to $x \in X$ if and only if two conditions hold:

1. the sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ is bounded;

2. $w\left(x_{n}, x\right) \rightarrow 0$.

Let $X$ be a separable normed space. Prove that weak-* convergence on the unit ball of the space $X^{*}$ is metrizable.

Let $X$ be a separable normed space. Specify a metric $w(x, y)$, defined on all of $X^{*}$, such that a sequence $\left\{ f_{n}\right\}_{1}^{\infty } *$-weak converges to $f \in X^{*}$ if and only if two conditions hold:

1. the sequence $\left\{ f_{n}\right\}_{1}^{\infty }$ is bounded;

2. $w\left(f_{n}, f\right) \rightarrow 0$.

Let $X$ and $Y$ be Banach spaces, let the spaces $X$ and $Y^{*}$ (and hence also the space $Y$) be separable, and let $Y$ be reflexive. Prove that weak operator convergence on any bounded set of the space $\mathcal{B}(X, Y)$ is metrizable (compare with problems 7.31 and 7.46).

Let $B = \operatorname {ball} M[0,1]$ and for $\mu , \nu$ in $M[0,1]$ define

Show that $d$ is a metric on $M[0,1]$ that defines the weak-star topology on $B$ but not on $M[0,1]$.

If $\mathcal{X}$ is a Banach space and $\mathcal{X}^*$ is separable, show that

If $B = \operatorname {ball} l^\infty$, show that

defines a metric on $B$ and that this metric defines the weak-star topology on $B$.

Is the weak topology of the unit ball in a separable Hilbert space metrizable?

If the weak topology of the unit ball in a Hilbert space $\mathbf{H}$ is metrizable, must $\mathbf{H}$ be separable?

Find an example of a bounded subset $A$ of a normed linear space $X, \mathcal{F}$ a set of functionals in $X^{*}$ containing $\mathcal{F}_{0}$ as a dense subset of $\mathcal{F}$ (dense in the sense of the norm topology on $X^{*}$) such that $\mathcal{F}$ and $\mathcal{F}_{0}$ generate different weak topologies for $X$, but the same weak topology for $A$.

Complete the last step of the proof of Theorem 10 by showing that the inequalities (11) imply that the metric $\rho$ induces the $W$-weak topology.

Theorem 10: Let $X$ be a normed linear space and $W$ a separable subspace of $X^{*}$ that separates points in $X$. Then the $W$-weak topology on the closed unit ball $B$ of $X$ is metrizable. The proof constructs a countable dense subset $\left\{ \psi_{k}\right\}_{k=1}^{\infty }$ of $B^{*} \cap W$ (where $B^{*}$ is the closed unit ball of $X^{*}$) and defines the metric

for which, for each natural number $n$,

Suppose $X$ is a Fréchet space (or, more generally, a metrizable locally convex space). Prove the following statements: