

Dual spaces
Name: jan
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
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
"covariant  /contravariant tensor fields").
Replies:
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 bilinear 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 bilinear 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.
jlu
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Update: June 2012

