Hilbert Spaces deep concepts, Study notes of Functional Analysis

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Hilbert Spaces 15 SEP 2021 - TAGS: ANALYSIS- QUAL-PREP Hilbert Spaces are banach spaces whose norms come from an inner product. This is fantastic, because inner product spaces are a very minimal amount of structure for the amount of geometry they buy us. Beacuse of the new geoemtric structure, many of the pathologies of banach spaces are absent from the theory of hilbert spaces, and the rigidity of the category of hilbert spaces (there is a complete cardinal invariant describing hilbert spaces up to isometric isomorphism) makes it extremely easy to make an abstract hilbert space concrete. Moreover, this “concretization” is exactly the genesis of the fourier transform! With that introduction out of the way, let’s get to it! Scanned with CamScanner