Dual space

In mathematics, every vector space

V

{\displaystyle V}

has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on

V

,

{\displaystyle V,}

together with the vector space structure of pointwise addition and scalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space.

When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces.