Tensor product and infinite dimensional vector space

  • Thread starter ivl
  • Start date
  • #1
ivl
27
0
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

[itex]g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}[/itex].

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

Any help is very much appreciated!
 

Answers and Replies

  • #2
ivl
27
0
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

[itex]g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}[/itex].

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)".
 

Related Threads on Tensor product and infinite dimensional vector space

  • Last Post
Replies
1
Views
2K
Replies
21
Views
9K
  • Last Post
Replies
12
Views
7K
  • Last Post
Replies
1
Views
1K
  • Last Post
2
Replies
33
Views
12K
Replies
8
Views
4K
Replies
10
Views
3K
Replies
5
Views
6K
Replies
4
Views
6K
Replies
3
Views
709
Top