countably normed spaces and countably normed spaces [closed]

Question

Why locally convex spaces are not presented as countably normed spaces i.e an infinite sequence of norms (see Generalized functions Tome 2 by Gelfand and Chilov) in the western mathematical references ? I think in my humble opinion that Gelfand's presentation is more intuitive and is a direct generalization of normed spaces. Is it correct to say that 2-normed spaces (introduced by Gahler and White) are spaces with a continuous family of norms ?

Answer 1

First, as Pietro Majer notes, certainly not every topological vector space is countably normed...

But, yes, books in English called "functional analysis" seem to mostly ignore "rigged Hilbert spaces" (a.k.a., "scales of Hilbert spaces") as ways to present (projective) limits and inductive limits (colimits) of Hilbert spaces. But many modern PDE sources do use (Levi-) Sobolev spaces, perhaps without letting on that one could take limits and colimits usefully.

I myself have found rigged Hilbert spaces very useful as a finer descriptive device than merely test-function/distribution and Schwartz-function/tempered-distribution. Even the discussion of convergence of Fourier series is saner and more robust in the context of $L^2$ (Levi-) Sobolev spaces and the Sobolev imbedding theorem, rather than feeling an obligation to fret over continuous functions whose Fourier series don't converge to them in sup norm. In contrast, the Sobolev space that is strictly smaller but includes into $C^o$ consists of continuous functions whose Fourier series do converge to them in $C^o$ also, by Sobolev imbedding.

I do also think that the discussion of nuclear Frechet spaces is more intelligible when slightly restricted to address (projective) limits of Hilbert spaces with Hilbert-Schmidt transition maps. (My course notes at http://www.math.umn.edu/~garrett/m/fun/ systematically take this viewpoint.)