Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Dual spaces
Name: jan
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995

Given a vector space V, there exists a vector space called "the dual of V", denoted V*. Which properties does V* have compared to V, that make V* important in some cases? (I am thinking of differential geometry, and of "co-variant - /contra-variant tensor fields").

A nice text is the book by Halmos.

Given a vector space defined over a field, such as the real numbers, the dual space is usually defined as the set of all bi-linear mappings of the vectors onto the field. This is unnecessarily abstract, so let us do it this way (for finite spaces):

Let V be a vector space with basis elements e1, . . . e_n, defined on the field of complex numbers. Let W be another vector space of the same dimension, with basis elements f1, . . . , fn, also defined over the complex numbers. We can make W into a "dual space" of V by defining a scalar product with w in W and v in V which is bi-linear in w and v and which satisfies = 1 if i = j and 0 otherwise. In most practical cases, W is just the complex conjugate of V.


Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory