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A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0 such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq B\|f\|^{2} $$ for all $f\in H$.

Some books define a frame for just "Hilbert space" and not mentioning the "separability". Is there any difference between these two cases?

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    That's strange! No one have any idea about this problem!2012-10-15
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    All that I can add is: some books take it as the definition that a Hilbert Space is separable, this being the most interesting case. Perhaps this helps?!2012-10-15
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    @Sebastian: Thank you. I was wondering why a "separable" Hilbert space is most interesting than just Hilbert space?2012-10-16
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    Sorry this is a bit late. You can prove that any two separable Hilbert Spaces are isomorphic, by the appropriate sense of isomorphism. This gives nice results such as $L^2([0,1])$ being equivalent to $l^2$ the space of square bounded sequences2012-10-16

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