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

In the setting of The Real Numbers: Standing Notation and Background, let $E$ with norm $\lVert\cdot\rVert$ be a real normed space and let $E^{*}$ be its dual space.

(Weak-star convergence)¶ A sequence $(\ell_{m})_{m\in\mathbb{N}}$ in $E^{*}$ converges weak-star to $\ell\in E^{*}$ if, for every $v\in E$, the sequence of real numbers $(\ell_{m}(v))_{m\in\mathbb{N}}$ converges to $\ell(v)$.