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plural dual spaces
(mathematics) The vector space which comprises the set of linear functionals of a given vector space. examples
(mathematics) The vector space which comprises the set of continuous linear functionals of a given topological vector space. quotations examples
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