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On two classes of (F)-spaces. (English) Zbl 0621.46001
Let \(\| \cdot \|_ 1\leq \| \cdot \|_ 2\leq...\leq \| \cdot \|_ n\leq..\). be a fundamental system of seminorms on a Fréchet space E. \(\| y\|_ n^*=\sup \{| y(x)|:\| x\|_ n\leq 1\}\), \(x\in E\), \(y\in E'\), \(n=1,2,..\). The following conditions of D. Vogt are important in the theory of nuclear Fréchet spaces:
(DN) There exist \(k_ 0\in N\) such that for every \(k\in N\) there are \(n\in N\) and \(C>0\) with \(\| x\|^ 2_ k\leq C\| x\|_{k_ 0}\| x\|_ n\) for all \(x\in E.\)
(\(\Omega)\) For any \(p\in N\) there is \(q\in N\) such that for every \(k\in N\) there are \(m\in N\) and \(C>0\) with \(\| y\|_ q^{*1+m}\leq \| y\|^*_ k\| y\|_ p^{*m}\) for all \(y\in E'\) [Math. Z. 155, 109-117 (1977; Zbl 0337.46015), Stud. Math. 67, 225-240 (1980; Zbl 0464.46010)].
Let \(K:=R\) or C. If M is a set and a(.) a function in M such that a(t)\(\geq 1\), \(t\in M\), then \[ \Lambda^{\infty}=\Lambda^{\infty}(M,a)=\{x\in K^ M:\| x\|_ k=\sup_{t}| x(t)| a^ k(t)<\infty \text{ for all } k\} \]
\[ \Lambda^ 1=\Lambda^ 1(M,a)=\{x\in K^ M:\| x\|_ k=\sum_{t}| x(t)| a^ k(t)<\infty \text{ for all } k\}. \] (s) is the Fréchet space of rapidly decreasing sequences. The author proves the following results:
(a) E has property (DN) if and only if E is isomorphic to a subspace of a space \(\Lambda^{\infty}.\)
(b) The class (DN) is the smallest class of Fréchet spaces, which contains the Banach spaces and (s) and which is closed under subspaces and \({\hat \otimes}_{\epsilon}.\)
(c) The following are equivalent: (1) E has property (DN) and E’ is strictly separable in its weak*-topology; (2) E is isomorphic to a subspace of \(\ell^{\infty}{\hat \otimes}_{\epsilon}(s)\); (3) is isomorphic to a subspace of \({\mathcal B}(R)\); (4) E is isomorphic to a subspace of some \(\Lambda^{\infty}(N,a).\)
(c) E has property \((\Omega)\) if and only if E is isomorphic to a quotient of a space \(\Lambda^ 1.\)
(d) The class \((\Omega)\) is the smallest class of Fréchet spaces, which contains the Banach spaces and (s) and which is closed under quotient spaces and \({\hat \otimes}_{\pi}.\)
(e) The following are equivalent: (1) E has property (\(\Omega)\) and is separable; (2) E is isomorphic to a quotient of \(\ell^ 1{\hat \otimes}_{\pi}(s)\); (3) E is isomorphic to a quotient of \({\mathcal D}_{L^ 1}({\mathbb{R}})\); (4) E is isomorphic to a quotient of some \(\Lambda^ 1(N,a)\).
Reviewer: M.Valdivia

MSC:
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
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