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.[1]
In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dualspace for short) consisting of all linear forms...
of mathematics, the strong dualspace of a topological vector space (TVS) X {\displaystyle X} is the continuous dualspace X ′ {\displaystyle X^{\prime...
continuous anti-dualspace or simply the anti-dualspace of X {\displaystyle X} if no confusion can arise. When H {\displaystyle H} is a normed space then the...
strong dualspaces. Every normed space can be isometrically embedded onto a dense vector subspace of some Banach space, where this Banach space is called...
The dual of a given linear program (LP) is another LP that is derived from the original (the primal) LP in the following schematic way: Each variable in...
V {\displaystyle V} ), the dual set of B {\displaystyle B} is a set B ∗ {\displaystyle B^{*}} of vectors in the dualspace V ∗ {\displaystyle V^{*}} with...
In mathematics, a dual system, dual pair or a duality over a field K {\displaystyle \mathbb {K} } is a triple ( X , Y , b ) {\displaystyle (X,Y,b)} consisting...
Every bilinear form B on V defines a pair of linear maps from V to its dualspace V∗. Define B1, B2: V → V∗ by B1(v)(w) = B(v, w) B2(v)(w) = B(w, v) This...
is the strong dual of the strong dual of X {\displaystyle X} ) is a homeomorphism (or equivalently, a TVS isomorphism). A normed space is reflexive if...
automorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dualspace S ∗ {\displaystyle {\mathcal...
parallelogram law, and so the dualspace is also an inner product space where this inner product can be defined in terms of this dual norm by using the polarization...
numbers, and every finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally...
the continuous dualspace is the zero space. In the case of the counting measure on the natural numbers (producing the sequence space L p ( μ ) = ℓ p...
the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Let X {\displaystyle X} be a normed vector space with...
vector space over k with addition and scalar multiplication defined pointwise. This space is called the dualspace of V, or sometimes the algebraic dual space...
map between two vector spaces, defined over the same field, is an induced map between the dualspaces of the two vector spaces. The transpose or algebraic...
space for every point on a smooth manifold. Typically, the cotangent space, T x ∗ M {\displaystyle T_{x}^{*}\!{\mathcal {M}}} is defined as the dual space...
the origin. the strong dualspace X b ′ {\displaystyle X_{b}^{\prime }} of X {\displaystyle X} is normable. the strong dualspace X b ′ {\displaystyle X_{b}^{\prime...
a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual. Suppose that H is a topological vector space (TVS). A function...
and multiplication by a scalar, the linear forms form a vector space, called the dualspace of V, and usually denoted V* or V′. If v1, ..., vn is a basis...
initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal...
the algebraic dualspace of an R-module X. Let X and Y be R-modules. If u : X → Y is a linear map, then its algebraic adjoint or dual, is the map u# :...
Fréchet, establishes an important connection between a Hilbert space and its continuous dualspace. If the underlying field is the real numbers, the two are...
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