zbMATH — the first resource for mathematics

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)
Citations:
Zbl 0337.46015; Zbl 0464.46010
Full Text:
References:
  K. D.Bierstedt, R.Meise and W. H.Summers, Köthe sets and Köthe sequence spaces. In: Functional Analysis, Holomorphy and Approximation Theory; J. A. Barroso (ed.), North-Holland Math. Studies71, 27-90 (1982). · Zbl 0504.46007  A. Grothendieck, Sur les espaces (F) et (DF). Summa Brasil. Math.3, 57-112 (1954). · Zbl 0058.09803  A.Grothendieck, Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc.16 (1955). · Zbl 0123.30301  H. Komatsu, Ultradistributions I, Structure theorems and a characterization. J. Fac. Sc. Univ. Tokyo, Sec. IA20, 25-105 (1973). · Zbl 0258.46039  S. G. Krein, On an interpolation theorem in operator theory (Russian). Dokl. Akad. Nauk SSSR130, 491-494 (1960), English transl.: Soviet Math. Dokl.1, 61-64 (1960).  S. G. Krein, On the concept of a normal scale of spaces (Russian). Dokl Akad. Nauk SSSR132, 510-513 (1960), English transl.: Soviet Math. Dokl.1, 586-589 (1960).  S. G. Krein andY. I. Petunin, Scales of Banach spaces (Russian). Usp. Math. Nauk (2)21, 89-168 (1966), English transl: Russian Math. Surveys, (2)21, 85-159 (1966).  R.Meise and D.Vogt, A characterization of quasi-normable Fréchet spaces. To appear in Math. Nachr. · Zbl 0583.46002  B. S. Mityagin, Non Schwartzian power series spaces. Math. Z.182, 303-310 (1983). · Zbl 0541.46007  V. P. Palamodov, Homological methods in the theory of locally convex spaces (Russian). Usp. Math. Nauk (1)26, 3-66 (1971), English transl.: Russian Math. Surveys (1)26, 1-64 (1971).  H. J. Petzsche, Die Nuklearität der Ultradistributionsräume und der Satz vom Kern I, II. Manuscripta math.24, 133-171 (1978),27, 221-251 (1979). · Zbl 0373.46052  A.Pietsch, Nuclear locally convex spaces. Ergeb. Math. Grenzgeb.66, Berlin 1972. · Zbl 0308.47024  M. Valdivia, On quasi-normable echolon spaces. Proc. Edinburgh Math. Soc.24, 73-80 (1981). · Zbl 0451.46006  M.Valdivia, Topics in locally convex spaces. Amsterdam 1982. · Zbl 0489.46001  D. Vogt, Charakterisierung der Unterräume vons. Math. Z.155, 109-117 (1977). · Zbl 0337.46015  D.Vogt, Subspaces and quotients of (s). In: Functional Analysis, Surveys and Recent Results; K. D. Bierstedt, B. Fuchssteiner (eds.), North-Holland Math. Studies27, 167-187 (1977).  D. Vogt, Ein Isomorphiesatz für Potenzreihenräume. Arch. Math.38, 540-548 (1982). · Zbl 0477.46014  D.Vogt, Sequence space representations of spaces of test functions and distributions. In: Functional analysis, holomorphy, and approximation theory; G. Zapata (ed.), Lecture Notes in Pure and Appl. Math.83, 405-443, New York 1983.  D. Vogt, Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist. J. Reine Angew. Math.345, 182-200 (1983). · Zbl 0514.46003  D.Vogt, On the functors Ext1 (E, F) for Fréchet spaces. Preprint.  D. Vogt undM. J. Wagner, Charakterisierung der Quotientenräume vons und eine Vermutung von Martineau. Studia Math.67, 225-240 (1980). · Zbl 0464.46010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.