Isomorphism is defined as a bijective linear operator, while isometry is inherently injective but not necessarily surjective. But if we have a surjective isometry, can we say it must be an isomorphism?
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Yes you can.... – Ethan Bolker Sep 24 '18 at 21:38
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1Provided you mean linear isometry when you say isometry. See https://math.stackexchange.com/questions/59374/example-of-a-non-linear-isometry for examples of isometries that aren't linear – Niall Taggart Sep 24 '18 at 21:42
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How is this actually proven? Specifically the multiplication preserving aspect of the homomorphism in the case of algebras. Or does the result not extend to algebras? – Tavin Dec 20 '22 at 02:02