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In mathematics, vector space of linear forms

In mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on 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.

Given any vector space over a field , the (algebraic) dual space (alternatively denoted by or ) is defined as the set of all linear maps (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted or This pairing defines a nondegenerate bilinear mapping called the natural pairing.

There is a natural homomorphism from into the double dual , defined by for all In other words, if is the evaluation map defined by , then is defined as the map This map is always injective;
Indeed, the isomorphism of a finite dimensional vector space with its double dual is an archetypal example of a natural isomorphism. Infinite dimensional Hilbert spaces are not isomorphic to their algebraic double duals, but instead to their continuous double duals.

When dealing with topological vector spaces, the continuous linear functionals from the space into the base field (or ) are particularly important. This gives rise to the notion of the "continuous dual space" or "topological dual" which is a linear subspace of the algebraic dual space , denoted by For any finite dimensional normed vector space or topological vector space, such as Euclidean n space, the continuous dual and the algebraic dual coincide. This is however false for any infinite dimensional normed space, as shown by the example of discontinuous linear maps. Nevertheless, in the theory of topological vector spaces the terms "continuous dual space" and "topological dual space" are often replaced by "dual space". For a topological vector space its continuous dual space, or topological dual space, or just dual space (in the sense of the theory of topological vector spaces) is defined as the space of all continuous linear functionals
Important examples for continuous dual spaces are the space of compactly supported test functions and its dual the space of arbitrary distributions (generalized functions); the space of arbitrary test functions and its dual the space of compactly supported distributions; and the space of rapidly decreasing test functions the Schwartz space, and its dual the space of tempered distributions (slowly growing distributions) in the theory of generalized functions.

If X is a Hausdorff topological vector space (TVS), then the continuous dual space of X is identical to the continuous dual space of the completion of X.

If the dual of a normed space V is separable, then so is the space V itself. The converse is not true: for example, the space ℓ^( 1) is separable, but its dual ℓ^( ∞) is not.