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.
Can this result be extended to (any) non-locally convex F-spaces?
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.