A linear real-valued function ω^1 of vectors v such that ω^1(v)↦R. Vectors (i.e., contravariant vectors or "kets" left bracketing bar ψ〉) and one-forms (i.e., covariant vectors or "bras" 〈ϕ right bracketing bar ) are dual to each other. Therefore ω^1(v) congruent v(ω^1) congruent 〈ω^1, v〉 = 〈ϕ|ψ〉. The operation of applying the one-form to a vector ω^1(v) is called tensor contraction.

angle bracket | bra | comma derivative | contravariant vector | covariant vector | differential k-form | ket | vector