Definition (3.2.1)

Let $X$ be a normed vector space and $X^{*}$ its dual. A sequence $\{ x_{n} \}_{n\in \mathbb{N}}\subset X$ is said to converge weakly to $x\in X$ if $$f(x_{n})\to f(x),\quad\forall f\in X^{*}$$

Definition (3.2.2)

A sequence $\{ f_{n} \}_{n\in \mathbb{N}}\subset X^{*}$ is said to converge in the weak* sense to $f\in X^{*}$ if $$f_{n}(x)\to f(x),\quad\forall x\in X$$