From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Newsgroups: sci.math
Subject: Re: Definition : Tensor Product
Date: 29 Nov 1998 13:45:46 -0500
In article <366165D3.72BE0146@nada.kth.se>,
x98-cgu wrote:
>Please, zumbody can explain me what a tensor product is ? Perhaps an
>example ?
My favourite example comes from functions on sets, finite or infinite. To
relate it to calculus situations:
Let x |-> f(x) be a function on a set S,
let y |-> g(y) be a function on a set T,
then (x,y) |-> f(x) * g(y) is a function of two variables, so it is
defined on the set S x T (cartesian product). And this can be called the
tensor product of these two functions. In a notation that suppresses
variables,
f (circled cross) g
is the function described above.
Then we bring in the terminology of vector spaces:
Suppose we had a space X of functions on S and a space Y of functions on
T. If we form all finite linear combinations of tensor products of pairs
of functions, one from X and the other from Y, we build the tensor product
X (circled cross) Y
of the spaces X and Y.
To be even more concrete, let
X be the space of polynomial functions in variable x, of degree
and representing a vector c_1 * e_1 + ... + c_n * e_n by a function c on
the set {1, 2, ... n} which assigns c(j)=c_j, j=1,...,n .
Then you can repeat the story of tensor products in terms of functions of
two variables.
The basis of the tensor product of two spaces, one with bases
and
will be a space with a basis
< e_j (circled cross) f_k : j from 1 to n and k from 1 to m >
Then you can prove what is often presented as a definition, namely that
every bilinear function of two variable vectors can be written uniquely as
a composition of tensor multiplication, followed by a linear function on
the tensor product. But this belongs perhaps to a more advanced course.
Hope some of it helps, ZVK(Slavek).