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I say two Hilbert spaces (over the complex numbers) A and B are isomorphic when there exists a vector space isomorphism from A to B, and they are isometric if there exists an isometric vector space isomorphism from A to B.

Of course A and B being isometric implies A and B being isomorphic, but does the converse always hold?

If not, does it at least hold for separable Hilbert spaces?

Francisco
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    The question you have asked is the same as the one at https://math.stackexchange.com/questions/547838/are-two-hilbert-spaces-with-the-same-algebraic-dimension-their-hamel-bases-have?rq=1 but you seem to not be familiar with the more basic fact that a Hilbert space is determined up to isometric isomorphism by the cardinality of an orthonormal basis. (In particular, any two separable infinite-dimensional Hilbert spaces are isometrically isomorphic, so the separable case is trivial.) – Eric Wofsey Jun 13 '22 at 21:11
  • Oh, I see! That is helpful. I was playing with equivalence classes of projections on Hilbert spaces and noticed that for it I required their images be isometric, rather than isomorphic, for them to be equivalent (on the equivalent class I am playing with), so I wondered if I could use isomorphisms even still. And now knowing their isometry classes are defined up to an orthonormal basis cardinality, I think I can progress with what I am doing! Thanks! – Francisco Jun 13 '22 at 22:16

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