dual space

English

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

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.

Synonyms

• (vector space of linear functionals): algebraic dual space, dual vector space
• (vector space of continuous linear functionals): continuous dual space, continuous dual

Further reading

• Linear form on Wikipedia.Wikipedia