In mathematics, any vector space, V, has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors. When applied to vector spaces of functions (which typically are infinite-dimensional), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in the study of functional analysis.
There are two types of dual spaces: the algebraic dual space, and the continuous dual space. The algebraic dual space is defined for all vector spaces. When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.
Other articles related to "dual space, space, dual, spaces":
... Let the vector space V have a basis, not necessarily orthogonal ... Then the dual space V* has a basis called the dual basis defined by the special property that Or, more succinctly, where δ is the Kronecker delta ... A linear functional belonging to the dual space can be expressed as a linear combination of basis functionals, with coefficients ("components") ui, Then ...
... The dual space H* is the space of all continuous linear functions from the space H into the base field ... defined by This norm satisfies the parallelogram law, and so the dual space is also an inner product space ... The dual space is also complete, and so it is a Hilbert space in its own right ...
... In analogy with the case of the algebraic double dual, there is always a naturally defined continuous linear operator Ψ V → V′′ from a normed space V into its continuous double ... Normed spaces for which the map Ψ is a bijection are called reflexive ... When V is a topological vector space, one can still define Ψ(x) by the same formula, for every x ∈ V, however several difficulties arise ...
... functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars ... In general, if V is a vector space over a field k, then a linear functional ƒ is a function from V to k, which is linear for all for all The set of ... This space is called the dual space of V, or sometimes the algebraic dual space, to distinguish it from the continuous dual space ...
... Let be a convex cone in a real vector space V equipped with an inner product ... The dual cone to C is the set This is also a convex cone ... If C is equal to its dual cone, C is called self-dual ...
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