# Dual space

## In mathematics, vector space of linear forms / From Wikipedia, the free encyclopedia

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In mathematics, any vector space *$V$* has a corresponding **dual vector space** (or just **dual space** for short) consisting of all linear forms on *$V,$* 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. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis.

Early terms for *dual* include *polarer Raum* [Hahn 1927], *espace conjugué*, *adjoint space* [Alaoglu 1940], and *transponierter Raum* [Schauder 1930] and [Banach 1932]. The term *dual* is due to Bourbaki 1938.^{[1]}