Dual Space - Algebraic Dual Space

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: VF (linear functionals). The dual space V* itself becomes a vector space over F when equipped with the following addition and scalar multiplication:

begin{align} & (varphi + psi)(x) = varphi(x) + psi(x) \ & (a varphi)(x) = a left(varphi(x)right) end{align}

for all φ, ψV*, xV, and aF. 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* × VF.

Read more about this topic:  Dual Space

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