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.
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Update: June 2012