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.
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