dual space (original) (raw)

V is isomorphic to V∗ if and only if the dimension ofV is finite. If not, then V∗ has a larger (infinite) dimension than V; in other words, the cardinal of any basis of V∗ is strictly greater than the cardinal of any basis of V.

Even when V is finite-dimensional, there is no canonical or natural isomorphism V→V∗. But on the other hand, a basisℬ of V does define a basis ℬ∗ of V∗, and moreover abijection ℬ→ℬ∗. For supposeℬ={b1,…,bn}. For each i from 1 to n, define a mapping

by

It is easy to see that the βi are nonzero elements of V∗and are independent. Thus {β1,…,βn} is a basis ofV∗, called the dual basis of ℬ.

If V is a topological vector space, the continuous dual V′ of V is the subspace of V∗ consisting of the continuous linear forms.

A normed vector space V is said to be reflexive if the natural embedding V→V′′ is an isomorphism. For example, any finite dimensional space is reflexive, and any Hilbert space is reflexive by the Riesz representation theorem.

Remarks