dual space

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Noun

dual space (plural dual spaces)

1. (mathematics) The vector space which comprises the set of linear functionals of a given vector space.
2. (mathematics) The vector space which comprises the set of continuous linear functionals of a given topological vector space.
• 2011, David E. Stewart, Dynamics with Inequalities, Society for Industrial and Applied Mathematics, page 17,
The dual space of a Banach space ${\displaystyle X}$ is the vector space of continuous linear functions ${\displaystyle X\rightarrow \mathbb {R} }$, which are called functionals. Similar notation is used for duality pairing between the Banach space ${\displaystyle X}$ and its dual space ${\displaystyle X'}$: ${\displaystyle \left\langle u,v\right\rangle }$ is the result of applying the functional ${\displaystyle u\in X'}$ to ${\displaystyle v\in X}$: ${\displaystyle \left\langle u,v\right\rangle =u(v)}$ explicitly uses the fact that ${\displaystyle u}$ is a function ${\displaystyle X\rightarrow \mathbb {R} }$.

Usage notes

Defined for all vector spaces, the dual space may, for clarity, be called the algebraic dual space. When defined for a topological vector space, the (algebraic) dual space has a subspace that corresponds to continuous linear functionals and is called the continuous dual space or continuous dual — or simply the dual space.