Existence of weak limit for bouded sequence $\{y_n\}$ such that for every weak limit point $\{y_n\}$ must equal $y$

Question

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 $y$?

Answer 1

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.