Definition of "dual space"

The dual space of a Banach space X {\displaystyle X} is the vector space of continuous linear functions X → R {\displaystyle X\rightarrow \mathbb {R} } , which are called functionals. Similar notation is used for duality pairing between the Banach space X {\displaystyle X} and its dual space X ′ {\displaystyle X'} : ⟨ u , v ⟩ {\displaystyle \left\langle u,v\right\rangle } is the result of applying the functional u ∈ X ′ {\displaystyle u\in X'} to v ∈ X {\displaystyle v\in X} : ⟨ u , v ⟩ = u ( v ) {\displaystyle \left\langle u,v\right\rangle =u(v)} explicitly uses the fact that u {\displaystyle u} is a function X → R {\displaystyle X\rightarrow \mathbb {R} } .

2011, David E. Stewart, Dynamics with Inequalities, Society for Industrial and Applied Mathematics, page 17