The weak topology in a normed 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 of vectors of a normed space converges weakly if and only if it converges in the weak 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 for any normed space the weak topology is Hausdorff.

Let $X$ be a normed space. Prove that every functional $f \in X^{*}$ is continuous with respect to the weak topology. Prove that, conversely, every linear functional on $X$ continuous in the weak topology lies in $X^{*}$.

Let $X$ be a Banach space, and $U$ a convex set in it. Prove that $U$ is closed with respect to the norm if and only if $U$ is closed in the weak topology.

Give an example of a Banach space $X$ and a set $A$ in it that is closed with respect to the norm but not closed in the weak topology.

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

Give an example of a set $M$ in the space $l_{2}$ that is not open in the weak topology, but for which, for every sequence $x_{n}$ weakly converging to $x \in M$, all $x_{n}$, starting from some index, lie in $M$.

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

In the space $l_{p}, p \in (1, \infty )$, give an example of a sequence $\left\{ x_{n}\right\}_{1}^{\infty }$ such that $\left\| x_{n}\right\| \rightarrow \infty$, but the point 0 lies in the closure of the set $\left\{ x_{n}\right\}_{n = 1}^{\infty }$ in the weak topology.

Prove that for every function $x_{0} \in L_{1}[0,1]$ the set $\left\{ x \in L_{1}[0,1]:\left|x(t)\right| \leqslant \left|x_{0}(t)\right| \text{ for almost every } t \in [0,1]\right\}$ is weakly sequentially compact in $L_{1}[0,1]$.