Definition talk:Compact Linear Transformation

I'm looking at all this work on Hilbert spaces, and seeing a lot of language in common with both linear algebra and topology. There's obvious contextual overlap. How easy would it be to exploit the parallelism, maybe be (a) establishing that a Hilbert space is a particular type of vector space(?) and therefore obeys various stuff which all vector spaces obeys, and (b) whichever aspects of it are also consistent with it being treated as a topological space 9? I haven't a clue what I'm talking about), and therefore etc. --prime mover 17:40, 23 February 2012 (EST)

The linear algebra parallelism is obvious as a Hilbert space of dimension $n$ is isomorphic to $\R^n$ or $\C^n$ (depending on the base field chosen). By definition, a Hilbert space is a vector space with an inner product on it (wrt which' (is that correct English? 'whose' seems false here) metric it has to be complete). The virtue of considering Hilbert (and Banach) spaces is that the continuous linear transformations are characterised by more direct properties of the inner product cq. norm.

In conclusion, a Hilbert space is simply the nicest abstraction of $\R^n$ to infinite dimensional systems. The next step is a Banach space (dropping the orthogonality of the inner product); after that, the realm gets messy and dark (AFAIK). --Lord_Farin 17:57, 23 February 2012 (EST)

The restriction to Hilbert spaces is unnecessary, my lecture notes and Rynne&Youngson work just with normed spaces. Will probably have separate definitions for Hilbert Space and Normed Spaces. (since Hilbert spaces don't have a norm immediately) This might mean temporary discrepancies but I will try to resolve them. More of a self-note but if anyone has better ideas do say. Caliburn (talk) 17:09, 11 August 2021 (UTC)