Tensor product and infinite dimensional vector space

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The discussion centers on the application of tensor products in linear maps between vector spaces, specifically questioning whether this holds for infinite dimensional vector spaces. An example from General Relativity illustrates how metrics can be expressed using tensor products. The original poster seeks clarification on the validity of this concept in the context of infinite dimensions. A reference to Wikipedia is provided as a potential source for further understanding. The conversation emphasizes the importance of tensor products in both finite and infinite dimensional settings.
ivl
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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!
 
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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)".
 

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