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  <x98-cgu@nada.kth.se> 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 <m

and let 

 Y be the space of polynomial functions in variable y, of degree <n

then X is generated by m functions 1, x, x^2, ... x^(m-1),
and  Y is generated by n functions 1, y, y^2, ... x^(n-1),

The space X (circled cross) Y is then generated by m*n functions

  x^j * y^k  where 0 <= j < m  and  0 <= k < n.

For a finite dimensional space represented by other means, you can make
it "look like" the space of functions on a finite set, picking a basis 

 <e_1, e_2, ... , e_n>

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

 <e_1, e_2, ... , e_n> and <f_1, f_2, ... , f_m>

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).