Mazur's lemma (weak limits and convex hulls)

Let $X$ be an NLS. A set $M \subset X$ is called weakly closed if for any sequence $\left\{ x_{n}: n \geq 1\right\} \subset M$ such that $x_{n} \xrightarrow {w} x \in X$, it follows that $x \in M$. Prove that

Let $X$ be a Banach space, and let the system $\left\{ e_{n}\right\}_{1}^{\infty }$ be weakly complete in it, i.e., for every vector $x \in X$ there exists a sequence $\left\{ x_{k}\right\}_{1}^{\infty } \subset \operatorname {Lin}\left\{ e_{n}\right\}_{1}^{\infty }$, weakly convergent to $x$. Prove that then the system $\left\{ e_{n}\right\}_{1}^{\infty }$ is complete in $X$ in the usual sense.

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.

(S. Mazur, 1933). Let a sequence $x_{n}$ converge weakly to $x$ in a normed space $X$. Prove that $x \in \overline{\operatorname {conv}\left\{ x_{n}\right\}_{1}^{\infty }}$.

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 $\left\| y_n - x\right\| \to 0$.

Every weakly closed set is strongly closed, but the converse is not true. Nevertheless every subspace of a Hilbert space (i.e., every strongly closed linear manifold) is weakly closed.

Prove that a subspace of a normed linear space is closed if and only if it is weakly closed.

Put $f_{N}(t) = N^{-1} \sum_{n = 1}^{N 2} e^{i n t}$. Prove that $f_{N} \rightarrow 0$ weakly in $L^{2}(-\pi , \pi )$.

By Theorem 3.13, some sequence of convex combinations of the $f_{N}$ converges to 0 in the $L^{2}$-norm. Find such a sequence. Show that $g_{N} = N^{-1}\left(f_{1}+\cdots +f_{N}\right)$ will not do.