## Dual Space

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.

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### Some articles on dual space:

**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

**dual**V′′, defined by ... 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 ...

**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, applying the functional to a basis vector ej yields due ...

... 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 all linear functionals from V to k, Homk(V,k), is itself a ... This

**space**is called the

**dual space**of V, or sometimes the algebraic

**dual space**, to distinguish it from the continuous

**dual space**...

... The

**dual space**H* is the

**space**of all continuous linear functions from the

**space**H into the base field ... norm, 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 ...

... 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**...

### Famous quotes containing the words space and/or dual:

“Our passionate preoccupation with the sky, the stars, and a God somewhere in outer *space* is a homing impulse. We are drawn back to where we came from.”

—Eric Hoffer (1902–1983)

“Thee for my recitative,

Thee in the driving storm even as now, the snow, the winter-day

declining,

Thee in thy panoply, thy measur’d *dual* throbbing and thy beat

convulsive,

Thy black cylindric body, golden brass and silvery steel,”

—Walt Whitman (1819–1892)