Hello, I had a question regarding unitary transformations. The most common definition I see for unitary transformations is defined as a transformation between Hilbert spaces that preserves inner products. I was wondering if all unitary transformations between Hilbert spaces (according to this definition) are necessarily bounded linear transformations. (i.e. [tex]U( \alpha x + \beta y ) = \alpha U x + \beta U y [/tex] and [tex] U \in \mathcal{L} (H_1 , H_2) [/tex].) I have been trying to prove this to myself for the last hour but can't seem to show this for some reason.(adsbygoogle = window.adsbygoogle || []).push({});

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# Are Unitary Transformations Always Linear?

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