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.

Read more about Dual Space: Algebraic Dual Space, Continuous Dual Space

### Other articles related to "dual space, space, dual, spaces":

**Dual Space**in Finite Dimensions

... 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 ...

**Dual Space**- Double Dual

... 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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