Tensor product and infinite dimensional vector space

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SUMMARY

The discussion centers on the application of the tensor product in the context of linear maps between vector spaces, specifically addressing its validity in infinite dimensional vector spaces. It is established that every linear map can indeed be expanded using the tensor product, exemplified by the metric in General Relativity represented as g=\sum_{i,j}g^{ij}e_{i}\otimes e_{j}. The inquiry focuses on whether this principle holds true for infinite dimensional spaces, with a reference to Wikipedia for further clarification on the outer product.

PREREQUISITES
  • Understanding of linear maps between vector spaces
  • Familiarity with tensor products in linear algebra
  • Basic knowledge of General Relativity and its mathematical framework
  • Concept of infinite dimensional vector spaces
NEXT STEPS
  • Research the properties of tensor products in infinite dimensional spaces
  • Study the implications of linear maps in functional analysis
  • Explore the role of metrics in General Relativity
  • Review the section on outer products in advanced linear algebra texts
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Mathematicians, physicists, and students studying linear algebra, functional analysis, or General Relativity who seek to deepen their understanding of tensor products and their applications in infinite dimensional vector spaces.

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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