Suppose you've got a linear map U between two Hilbert spaces H1 and H2. If U preserves the inner product - that is, [itex](Ux,Uy)_2 = (x,y)_1[/itex] for all x and y in H1 - is it necessarily unitary? Or are there inner product-preserving linear mappings that aren't one-to-one or onto?(adsbygoogle = window.adsbygoogle || []).push({});

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# Inner product-preserving map that isn't unitary?

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B Product rule OR Partial differentiation |

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