One-Form

A linear, real-valued Function of Vectors such that $\omega^1({\bf v})\mapsto \Bbb{R}$. Vectors and one-forms are Dual to each other because Vectors are Contravariant (``Kets'': $\left\vert\psi\right\rangle$) and one-forms are Covariant Vectors (``Bras'': $\left\langle\phi\right\vert$), so

$$\omega^1({\bf v})\equiv {\bf v}(\omega^1) \equiv \left\langl... ...}\right\rangle{} = \left\langle{\phi\vert\psi}\right\rangle{}.$$

The operation of applying the one-form to a Vector $\omega^1({\bf v})$ is called Contraction.

See also Angle Bracket, Bra, Differential k-Form, Ket