I need your help with this problem that I founded it in a lecture notes. Then, the problem says:
Let $ X $ be a normed space. Show that if a sequence $ (x_n) _ {n \in \mathbb {N}} $ in $ X $ converges weakly to $ x $ then $ x\in Y $, where $ Y $ is the closure of the vector space generated by $\{x_n : n \in \mathbb{N} \} $.