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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!