**Algebraic Dual Space**

Given any vector space *V* over a field *F*, the **dual space** *V** is defined as the set of all linear maps *φ*: *V* → *F* (linear functionals). The dual space *V** itself becomes a vector space over *F* when equipped with the following addition and scalar multiplication:

for all *φ*, *ψ* ∈ *V**, *x* ∈ *V*, and *a* ∈ *F*. Elements of the algebraic dual space *V** are sometimes called **covectors** or **one-forms**.

The pairing of a functional *φ* in the dual space *V** and an element *x* of *V* is sometimes denoted by a bracket: *φ*(*x*) = or *φ*(*x*) = ⟨*φ*,*x*⟩. The pairing defines a nondegenerate bilinear mapping : *V* × V* → *F*.

Read more about this topic: Dual Space

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

... In coding theory, the

**dual**code of a linear code is the linear code defined by where is a scalar product ... In linear algebra terms, the

**dual**code is the annihilator of C with respect to the bilinear form ... The dimension of C and its

**dual**always add up to the length n A generator matrix for the

**dual**code is a parity-check matrix for the original code and vice versa ...

...

**Dual**use goods are products and technologies normally used for civilian purposes but which may have military applications ... -European Commision, Trade Topics,

**Dual**Use “ ”

**Dual**-use is a term often used in politics and diplomacy to refer to technology which can be used for both peaceful and military aims ... More generally speaking,

**dual**-use can also refer to any technology which can satisfy more than one goal at any given time ...

**Algebraic Dual Space**- Quotient Spaces and Annihilators

... The annihilator of a subset is itself a vector

**space**... a vector subspace, then after identifying W with its image in the second

**dual space**under the double duality isomorphism V ≈ V** ... annihilator is a Galois connection on the lattice of subsets of a finite-dimensional vector

**space**...

...

**Dual**(album), an album of traditional Scottish and Irish music recorded by Éamonn Doorley, Muireann Nic Amhlaoibh, Julie Fowlis and Ross Martin, released 2008

**Dual**(film) "

**Dual**" (Heroes), an episode ...

... The road is a mixture of

**dual**carriageway and single carriageway, with the section east of Low Newton, Cumbria to the M6 being mainly

**dual**... Further

**dual**sections are south of Newby Bridge, south of Greenodd and south of Ulverston ...

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

“In the tale proper—where there is no *space* for development of character or for great profusion and variety of incident—mere construction is, of course, far more imperatively demanded than in the novel.”

—Edgar Allan Poe (1809–1849)

“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an *algebraic* formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”

—Henry David Thoreau (1817–1862)

“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)