Chart of how the mathematical spaces are related? (soft question) When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how these are related to one another (I'm sure some, or even most, are not).  For example, how is Hilbert space related to a vector space, and how does that relate to a vector field?  I haven't had a formal course on functional analysis, so forgive me if this is a nonsense question, but is there a table of some sort of which space is a subspace of another, or relations among the common spaces?  The chart in the back of Casella and Burger's Statistical Inference text would be of the flavor I am thinking, which is a trimmed version of this chart on page 3.
Thanks in advance!
 A: A Hilbert space is a vector space (in general on complex numbers) endowed with an inner product and complete (as metric space) under the norm induced by the inner product. So a Hilbert space is a vector space, is normed and hence is a metric space (then is a normal space, with the language of topology) and finally is complete as metric space, so is a Banach space. (Note that is a metric space since a norm always induces a metric.)
For what concernes functional analysis, we can state that the most general family is that of topological vector spaces (TVS). (However, they are a small set in the greatest set of topological spaces). In the TVS, we can recognize the two big families of metrizable TVS and locally convex TVS (LCTVS). These classes intersect but do not coincide. All of the spaces of interest are subset of the intersection. A broad class in intersection of LCTVS and metrizable TVS is that of Frechét spaces. Banach spaces are the largest set of interest in Frechét spaces and, in Banach spaces, a tiny but very important set is that of Hilbert spaces.
Probably, in some book a similar table is drawn (even if I don't know one where such a table is, so to speak, a graphical object), but I think you can depict it by yourself with the preceeding indications.
A: Isn't all that you want, but see the diagram https://www.math.ucdavis.edu/~greg/topvs.png in https://mathoverflow.net/questions/8443/barrelled-bornological-ultrabornological-semi-reflexive-how-are-these-us.
