Definition (Dual space)

Let $X$ be a normed vector space. We define the dual space of $X$ as the set of linear bounded functions on $X$ to $\mathbb{R}$ or $\mathbb{C}$, denoted as $X^{*}$: $$X^{*}:=\{ f\in\Gamma(X;\mathbb{F}):\lVert f \rVert <\infty \}$$