On asymptotically normable Fréchet spaces

Abstract

Let E be a Fréchet space, $\| \;\|_1≤ \|\;\| ≤ \ldots $ a fundamental system of seminorms on E and U, $ U_k= {x∈ E: \|x\|_k≤ 1}$ for every k. E is called asymptotically normable, if there is a $k₀$, such that for every $k ≥ k₀$, there is a p so that the seminorms $\|\;\|_{k₀}$ and $\|\;\|_k$ define equivalent topologies on $U_p$. It is easy to see that in this case $\|\;\|_{k₀}$, is in fact a norm. This class of spaces appears in investigations about the structure of Fréchet spaces and about the behaviour of their operators as a natural counterpart of the class of quasi-normable spaces introduced by Grothendieck [4].While the quasi-normable Fréchet spaces E are those which admit an $ω$-type condition (see[7]) and for which there exists a nontrivial Fréchet space F with $Ext¹ (F, E) = 0$ (see [8], [9], [13], [14]), the asymptotically normable Fréchet spaces E are those which admit a DN-type condition (see [13] and below) and for which there exists a nontrivial Fréchet space F with $Ext¹ (E, F) = 0$. Nontrivial here could mean: an infinite dimensional nuclear Köthe space. In [7] it is shown that the quasi-normable spaces are the quotient spaces of standard spaces of the form $$λ(A, E) =\Big{ x = x_j)_j∈ E^ℕ:\|x\|_k=∑_j\|x_j\|_ka_{j,k}<+∈fty\; \textrm{for all } \Big}$$ where E is a Banach space and $A = (a_{j,k})$ a matrix with $0k and $$∑_j\frac{a_{j,k}}{a_{j,k+1}}<+∈fty .$$We show that the asymptotically normable spaces are the subspaces of these standard spaces. They are the smallest class of Fréchet spaces which contains the nuclear Köthe spaces with continuous norm, the Banach spaces and is closed under $ε$-tensor products and subspaces.The main tool for that is Theorem 3.3. For Schwartz spaces asymptotic normability coincides with countable normability in the sense of Gelfand-Shilov. We show by an example that this even for Montel spaces is not the case.