dual space
Dual of a vector space; dual bases
Let be a vector space over a field . The dual of , denoted by , is the vector space of linear forms on , i.e. linear mappings . The operations in are defined pointwise:
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
by
It is easy to see that the are nonzero elements of and are independent. Thus is a basis of , called the dual basis of .
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
If is a topological vector space, the continuous dual of is the subspace of consisting of the continuous linear forms.
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.
Remarks
Linear forms are also known as linear functionals.
Another way in which a linear mapping can arise is via a bilinear form
The notions of duality extend, in part, from vector spaces to modules, especially free modules over commutative rings. A related notion is the duality in projective spaces.
Title | dual space |
Canonical name | DualSpace |
Date of creation | 2013-03-22 12:16:52 |
Last modified on | 2013-03-22 12:16:52 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 15 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 15A99 |
Synonym | algebraic dual |
Synonym | continuous dual |
Synonym | dual basis |
Synonym | reflexive |
Synonym | natural embedding |
Synonym | topological dual |
Related topic | DualHomomorphism |
Related topic | DoubleDualEmbedding |
Related topic | BanachSpace |
Related topic | Unimodular |
Related topic | LinearFunctional |
Related topic | BoundedLinearFunctionalsOnLpmu |