Dual Space and Duality Pairing

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

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).

Applications

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