Let $X$ be separable Banach space and $\{x\_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y\_n=\frac{1}{n}\sum\_{i=1}^{n}{x\_i}$, then (together with the Krein and Eberlein-Smulian theorems), we can assume that there exists a **subsequence** of $\{y\_n\}$ converges weakly to some element $y \in X$.


We suppose that every **weak limit** point of $\{y\_n\}$ must equal $y$. 


Can we say that $\{y\_n\}$ must converge weakly to $y$?



Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$
If $y\_n$ does not converge weakly to $y$ then there is a weakly open neighborhood $U$ of $y$ and a subsequence $y\_{n\_k} \notin U$. By weak compactness this subsequence has a weak limit point $z \notin U$. But $z$ is then also a weak limit point of the original sequence $y\_n$, contradicting uniqueness.


