*-weak topology in the dual space

Let $X$ be a normed space. Prove that the *-weak topology $* w$ on $X^{*}$ is generated by some family of seminorms (i.e., $(X^{*}, * w)$ is a polynormed space).

Prove that a sequence $\left\{ f_{n}\right\}_{1}^{\infty } \subset X^{*}$ converges *-weakly if and only if it converges in the *-weak topology.

Let $X$ be a finite-dimensional normed space. Prove that the *-weak topology in $X^{*}$ coincides with the norm topology.

Let $X$ be an infinite-dimensional normed space. Prove that the *-weak topology in $X^{*}$ is strictly weaker than the norm topology.

Let $X$ be a normed space. In the space $X^{*}$ consider the weak topology (it is introduced in the standard way, by means of functionals from $X^{* *}$). Prove that if $X$ is not reflexive, then in $X^{*}$ the *-weak topology is weaker than the weak topology. Prove that in the case of a reflexive space these topologies coincide.

Let $X$ be an infinite-dimensional normed space. Prove that every neighborhood of zero in the *-weak topology of the space $X^{*}$ contains an infinite-dimensional subspace.

Prove that for any normed space the *-weak topology on the dual space is Hausdorff.

Let $X$ be a normed space. Prove that for every linear functional $F$ on $X^{*}$, continuous in the *-weak topology, there exists a vector $x$ from $X$ such that $F(f) = f(x)$ for every $f \in X^{*}$.

Give an example of a normed space $X$ and a closed subspace $X_{0} \subset X^{*}$ such that $X_{0}$ is closed with respect to the norm, but not closed in the *-weak topology.

Let $X$ be a normed space. Prove that separability of $X^{*}$ with respect to the norm implies $*$-weak separability. Give a counterexample to the converse statement.

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 Banach space. Prove that the space $X^{*}$ is metrizable in the $*$-weak topology if and only if $\operatorname {dim} X < \infty$.

Prove the Banach-Alaoglu theorem 14.5 for a separable space $X$.

Theorem 14.5 (S. Banach, L. Alaoglu): let $X$ be a linear normed space. Then the closed unit ball of the space $X^{*}$ is compact in the $*$-weak topology.

Let $X$ be a normed space. Denote $Y = \pi (X)$, where $\pi : X \rightarrow X^{* *}$ is the canonical embedding. Prove that the map $\pi$ is an isomorphism (i.e., is continuous, bijective, and has an inverse map that is also continuous) of the topological spaces $(X, w)$ and $(Y, * w)$, where $X$ is the space with the weak topology, $Y$ is the space with the topology induced from $\left(X^{* *}, * w\right)$, and $* w$ is the *-weak topology. Prove that the closure of the subspace $Y$ in this topology is the whole space $X^{* *}$.

Prove Theorem 14.4 in the case when $X^{*}$ is separable.

Theorem 14.4 (S. Banach, 1932; L. Alaoglu, 1940): Let $X$ be a linear normed space. The closed unit ball of $X$ is compact in the weak topology if and only if $X$ is reflexive.

Prove that in an infinite-dimensional reflexive Banach space weak convergence does not coincide with convergence in norm.

Prove that a closed subspace of a reflexive normed space is reflexive.

Prove that any closed subspace of a reflexive normed space is a proximinal subspace (see Definition 2.17).

Definition 2.17: Let $X$ be a normed space and $Y$ a linear subspace of it. The subspace $Y$ is called a subspace of existence (proximinal subspace) if for every element $x \in X$ there exists in $Y$ an element of best approximation, i.e., a vector $y_{0} \in Y$ with $\operatorname {dist}(x, Y) = \left\| x-y_{0}\right\|$.

Let $X$ and $Y$ be Banach spaces, $A: X \rightarrow Y$ a linear operator, continuous with respect to the weak topologies. Prove that $A \in \mathcal{B}(X, Y)$ (compare with problem 7.77).

Prove that none of the spaces $c, c_{0}, C[0,1], L_{1}[0,1]$ is isometrically isomorphic to any space dual to a normed space.