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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Parametrised Hilbert spaces; can we put a norm on the following space of Hilbert spaces?

I think that the concept you are looking for is that of a measurable field of Hilbert spaces. These are studied in detail in Dixmier's classic "von Neumann algebras" which is available on Google Book …
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