Weak topologies versus the norm topology

Prove that in a finite-dimensional normed space the weak topology coincides with the norm topology.

Prove that in an infinite-dimensional normed space every neighborhood of zero in the weak topology contains an infinite-dimensional subspace.

Prove that in an arbitrary infinite-dimensional normed space the weak topology is always strictly weaker than the norm topology (that is, every set open in the weak topology is open in the norm topology, but not conversely) ¹.

Prove that a normed space is separable with respect to the norm if and only if it is separable 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.

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

Is every weakly separable Hilbert space separable?

The weak topology of an infinite-dimensional Hilbert space is not metrizable.

Let $L$ be an infinite-dimensional normed space. Prove that the weak topology in $L$ does not coincide with the strong topology.

Let $X$ be an infinite dimensional normed linear space. Prove that every weakly open subset of $X$ is unbounded.

Let $X$ be a finite dimensional linear space. Show that the weak and strong topologies on $X$ are the same.

Show that every non-empty weakly open subset of an infinite dimensional normed linear space is unbounded with respect to the norm.

Show that the weak topology on an infinite dimensional normed linear space is not first countable.

Show that any non-empty weakly open set in an infinite dimensional normed linear space is unbounded with respect to the norm.

Use the Baire Category Theorem and the preceding problem to show that the weak topology on an infinite dimensional Banach space is not metrizable by a complete metric.

Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be two compact, Hausdorff topologies on a set $\mathcal{S}$ for which $\mathcal{T}_{1} \subseteq \mathcal{T}_{2}$. Show that $\mathcal{T}_{1} = \mathcal{T}_{2}$.

Let $X$ be an infinite-dimensional Fréchet space. Prove that $X^{*}$, with its weak*topology, is of the first category in itself.

Let $X$ be a topological vector space on which $X^{*}$ separates points. Prove that the weak*-topology of $X^{*}$ is metrizable if and only if $X$ has a finite or countable Hamel basis. (See Exercise 1, Chapter 2 for the definition).