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 \(\Lambda^ 1(N,a)\).