An F-space is a completely metrizable topological vector space, i.e. the vector topology is induced by a complete metric. A Fréchet space is, by definition, a locally convex F-space.


It is known that all (infinite dimensional) separable Fréchet spaces are homeomorphic to $l\_2$, the space of square summable sequences of real numbers. (See *e.g.* Anderson, R. D.; Bing, R. H., *A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines.* Bull. Amer. Math. Soc. 74, 1968, 771–792.)


More precisely any such space, including $l\_2$ is homeomorphic to a countable infinite product of copies of real lines.



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I am interested in any reference or recent literature on the topology (homeomorphism class) of F-spaces. 


More precisely, I am interested in the following examples: $L\_p([0,1])$ or $l\_p$ with $0<p<1$, which are in fact quasi-Banach spaces. Meaning, the p-sum/integral defines a quasi-norm only, that is $||f+g||\_p\leq K(||f||\_p+||g||\_p)$, holds only for a uniform constant $K>1$. Nevertheless $d(f,g)=||f-g||\_p^p$ is a complete metric inducing the vector topology.


This question arose when wondering if $l\_p$ has a structure of a Hilbert manifold.



Are separable F-spaces (completely metrizable topological vector space) homeomorphic to $l_2$?
There is a famous linear metric space constructed by R. Cauty [Un espace métrique linéaire qui n'est pas un rétracte absolu, Fund. Math. 146 (1994)] whose completion is a  **separable $F$-space which is not an AR**. I do not have Cauty's paper handy but the latter fact is stated on the first page of  [Cauty's space enhanced](http://www.sciencedirect.com/science/article/pii/S0166864111003257)  in
[Topology Appl. 159 (2012), no. 1, 28–33]. 


Since $\ell\_2$ is an AR, the above $F$-space is not homeomorphic to $\ell\_2$.


Incidentally, a separable $F$-space is homeomorphic to $\ell\_2$ if and only if it is a non-locally-compact AR, see Corollary 5.2.2 in [Absorbing sets in Infinite-Dimensional Manifolds, T. Banakh, T. Radul, M. Zarichnyi]. 


