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Now, you're confusing me. The [itex]\varphi_n[/itex] make a countable basis of the vector space [itex]V'[/itex], which is impossible for Banach spaces. Iirc it's a consequence of Baire category theorem.. now that I think about it, maybe P1 more generally can also be shown with BCT.I didn't read yet in detail but what do you mean with "basis"? A topological basis? In that case ##l^2(\Bbb{N})## is separable and thus has a countable basis, yet has infinite dimension. So I don't quite see how you would get a contradiction.

I edited the post to emphasise it's a basis in the sense of vector spaces.

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