Weak-Star Convergence in the Dual of a Real Normed Space

In the setting of Locally Compact Metric Spaces and Compactly Supported Functions: Standing Notation, let $X$ be a real normed space with dual space $X^{*}$, let $(F_{m})_{m\in\mathbb{N}}$ be a sequence in $X^{*}$, and let $F\in X^{*}$.

(i) Weak-star convergence. $(F_{m})$ converges weak-star to $F$, written $F_{m}\overset{*}{\rightharpoonup}F$, if the real sequences $(F_{m}(x))_{m\in\mathbb{N}}$ converge as follows:

$$F_{m}(x)\to F(x)\qquad\text{for every }x\in X.$$