Linear Functional - Dual Vectors and Bilinear Forms

Dual Vectors and Bilinear Forms

Every non-degenerate bilinear form on a finite-dimensional vector space V gives rise to an isomorphism from V to V*. Specifically, denoting the bilinear form on V by <, > (for instance in Euclidean space <v,w> = v•w is the dot product of v and w), then there is a natural isomorphism given by

The inverse isomorphism is given by where ƒ* is the unique element of V for which for all w ∈ V

The above defined vector v* ∈ V* is said to be the dual vector of v ∈ V.

In an infinite dimensional Hilbert space, analogous results hold by the Riesz representation theorem. There is a mapping V → V* into the continuous dual space V*. However, this mapping is antilinear rather than linear.