Dual Space and Duality Pairing

The continuous dual X* and the pairing ⟨x*,x⟩=x*(x).

Dual Space and Duality Pairing

Let $X$ be a normed space over $\mathbb{R}$ .

The (continuous) dual space $X^\ast$ is the set of all continuous linear functionals $x^\ast :X\to\mathbb{R}$ (equivalently, all bounded linear functionals on $X$ ).

For $x^\ast \in X^\ast$ and $x\in X$ , the duality pairing is

\langle x^\ast ,x\rangle:=x^\ast (x).

This pairing is the standard language used in separation theorems such as separation by a closed hyperplane and strict separation .