Dual of a vector space; dual bases
for , and .
is isomorphic to if and only if the dimension of is finite. If not, then has a larger (infinite) dimension than ; in other words, the cardinal of any basis of is strictly greater than the cardinal of any basis of .
Even when is finite-dimensional, there is no canonical or natural isomorphism . But on the other hand, a basis of does define a basis of , and moreover a bijection . For suppose . For each from to , define a mapping
The dual of is called the second dual or bidual of . There is a very simple canonical injection , and it is an isomorphism if the dimension of is finite. To see it, let be any element of and define a mapping simply by
is linear by definition, and it is readily verified that the mapping from to is linear and injective.
Dual of a topological vector space
A normed vector space is said to be reflexive if the natural embedding is an isomorphism. For example, any finite dimensional space is reflexive, and any Hilbert space is reflexive by the Riesz representation theorem.
Linear forms are also known as linear functionals.
Another way in which a linear mapping can arise is via a bilinear form
|Date of creation||2013-03-22 12:16:52|
|Last modified on||2013-03-22 12:16:52|
|Last modified by||Daume (40)|