Mazur's lemma

Remarks

Fix $n$ and let $C_n$ be the norm closure of the convex hull of $\set{x_k : k \ge n}$. By the Hahn–Banach separation theorem, a norm-closed convex set is weakly closed, so $C_n$ is weakly closed. Since $x_k \to x$ weakly and $x_k \in C_n$ for $k \ge n$, the limit $x$ lies in $C_n$. Hence some convex combination $y_n$ of finitely many $x_k$ with $k \ge n$ satisfies $\norm{y_n - x} < 1/n$.

In the sequence space $\ell^2$, the unit vectors $e_n$ converge weakly to $0$ but $\norm{e_n} = 1$ for all $n$. The averages $y_n = \frac{1}{n}(e_n + e_{n+1} + \cdots + e_{2n-1})$ have $\norm{y_n} = 1/\sqrt{n}$, so they tend to $0$ in norm, as the lemma predicts.

Going deeper

For a student

The lemma is a quantitative form of the fact that a convex set has the same closure in the norm topology and in the weak topology. It is used in the calculus of variations and in partial differential equations: one obtains a weak limit from compactness. Then one uses the lemma to pass to strong limits of averages, which interact well with convex functionals. For example, it shows that a convex lower semicontinuous functional on a Banach space is weakly lower semicontinuous.