Weak Convergence and Weak Topology

Question

In discussing weak topology of a normed space $X$, a lemma is given as follows.

If $(x_n)$ is a sequence in $X$ converging weakly to $x$, then $x_n$ is bounded.

I understand the proof of this lemma. My question here is why this lemma is given in the context of weak topology. That is, how are weak topology and weak convergence related? Thank you!

Answer 1

Weak convergence is the convergence with relation to the weak topology.

If $ x_n \in X $ converges weakly to $x $ then $ x $ lies in weak-closure of $ X $.

The weak topology on $ X$ is induced by all sets of the form $ l^{-1}(U) $ where $ l \in X^*$ and $ U \subseteq \mathbb{R}$ open.