# Tensor product and infinite dimensional vector space

• ivl

#### ivl

Hi all,

it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product.

For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as

$g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}$.

However, does this statement hold true when the linear operator maps between infinite dimensional vector spaces?

Any help is very much appreciated!

ivl said:
Hi all,

it is of course true that every linear map between two vector spaces can be expanded by means of the tensor product.

For instance, the metric in General Relativity (mapping covectors to vectors) can be expanded as

$g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}$.

However, does this statement hold true when the linear operator maps between infinite dimensional vector spaces?

Any help is very much appreciated!

I found an answer in wikipedia, see:

http://en.wikipedia.org/wiki/Outer_product

towards the end of the section "Definition (abstract)".