Mazur's lemma

Mazur's theorem — Let ${\displaystyle (X,\lVert \cdot \rVert )}$ be a normed vector space and let ${\displaystyle \left\{x_{j}\right\}_{j\in \mathbb {N} }\subset X}$ be a sequence converges weakly to some ${\displaystyle x\in X}$.

Then there exists a sequence ${\displaystyle \left\{y_{k}\right\}_{k\in \mathbb {N} }\subset X}$ made up of finite convex combination of the ${\displaystyle x_{j}}$'s of the form

$${\displaystyle y_{k}=\sum _{j\geq k}\lambda _{j}^{(k)}x_{j}}$$

such that ${\displaystyle y_{k}\to x}$ strongly that is ${\displaystyle \lVert y_{k}-x\rVert \to 0}$.