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Rosenberger, Bernd

doi:10.1002/mana.19720520112

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…If (iii) holds and ë satisfies (H 4), then s in the proof of Lemma l we obtain that for each t/e ^ there exists a Fe ^ such that Fc U and ((5 n (F, £/))" e ë 1/Ã . Finally, if (H4) holds, in view of the equivalence (i) o (iii) of Theorem l, our definition is in accordance with Rosenberger's definition of ö-nuclear spaces [12] (see also II, § 2) and with Ligaud's definition of diametrally nuclear, topological vector spaces [9], since in the latter two cases ë c l 1 . But then <5"(F, ß7) = á ð (Ã êé/ ) for all «eM and (i) follows.…”

Section: ë-Nuclear Spacesmentioning

confidence: 61%

“…If ^(á 9 ) = Ë(á, ö ), then there must exist a fc e ^/ for which Replacing « by n 1/q in this inequality we obtain first that and hence, using (12) and (13), so that (ii) holds with m -k 2 . Obviously ö~é (-W ö' 1 (--) for all n e N. Choose 5>0 such that \ n / \ n J / _ / l \ \ »* ( i \ Then a Standard argument yields ( ö 1 ( -1 1 ^ S ö M -l for all n, and hence (iii) V \ n // \ w / (iii) => (ii).…”

Section: ö-Nuclearitymentioning

confidence: 92%

“…In [12] Rosenberger introduced the class of </>-nuclear spaces, defined s follows. A locally convex space E is ö-nudear (ö e Ö) if it has a basis ^U of neighbourhoods of 0 such that for each U E <% tbere-exists Ve #U with Fc U and (<5"(F, U))" e É ö .…”

Section: ö-Nuclearitymentioning

confidence: 99%

“…includes, for example, the ö-nuclear spaces of Rosenberger [12]. In § 3 we present the problem and show that under a mild, technical hypothesis (H 5) on A, it has a solution remarkably similar to that for the case of nuclear spaces (Theorem 3).…”

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confidence: 92%

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On the existence of universal -nuclear Fréchet spaces.

1978

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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IntroductionSince its introduction by Grothendieck [4], the theory of nuclear spaces has acquired increasing importance in functional analysis. More recently, several generalizations and variations of this theory have been studied by many authors, culminating in the unifying theory of A-nuclearity presented in [3] (where numerous references to previous work can be found). In the theory of nuclearity a crucial role is played by the sequence space l 1 and, in essence, A-nuclearity is obtained through replacing l 1 by a sequence space A satisfying some suitable conditions. It is then natural to ask which properties of nuclear spaces go over to the more general setting of A-nuclear spaces. Given the enormous ränge of problems covered by such a question, in this paper we choose one such problem and give a complete solution to it, äs well äs present the form taken by the solution in the case of concrete sequence spaces encountered in the applications.We statt with the following observation. By the celebrated theorem of Kömura and Kömura [6], the Frechet space s of rapidly decreasing sequences is a universal generator for the variety of nuclear spaces and hence S N is a universal, nuclear Frechet space, in the sense that every nuclear Frechet space is isomorphic to a closed subspace of s". On the other band, in [10], Theorem 10 we proved that there is no universal ^-nuclear Frechet space and, so far, nobody has exhibited a Frechet space which is a universal generator for the variety of s-nuclear spaces (the reason will, of course, be apparent later).These considerations led us to consider the problem of characterising those sequence spaces (within a broad admissible class) admitting a universal, A-nuclear Frochet space. In this paper we solve this problem under minimal assumptions on and give some interesting applications. The results have been announced in [15].The paper is divided into two parts. Part I develops the general theory; we prefer to introduce the various hypothesis on the sequence space precisely at the point where they are needed. Starting with a preliminary section on sequence spaces, we proceed in § 2 with the definition and elementary properties of A-nuclear spaces. Our definition of A-nuclear spaces is slightly stronger than that adopted in [3] in the sense that our A-nuclear spaces are contained within the class of A-nuclear spaces äs defined in [3]. Furthermore, the two definitions coincide for a large class of sequence spaces A, while our definition Brought to you by | University of California Authenticated Download Date | 6/8/15 9:39 AM with (/") and (e n ) orthonormal Systems in E v and Å õ respectively, so that the map T vu is pseudo A-nuclear. Finally, if (H 4') is satisfied, (i) follows from (iv) by Lemma 2. Journal f r Mathematik. Band 301 2 Brought to you by | University of California Authenticated Download Date | 6/8/15 9:39 AM Ã ao Ú and form the space # = < { : Ó eJ{J < oo >. Arguing by contradiction, suppose that / 2 c # ; t «=i J then there exists c>0 such that oo / oo

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Section: ë-Nuclear Spacesmentioning

confidence: 61%

Section: ö-Nuclearitymentioning

confidence: 92%

Section: ö-Nuclearitymentioning

confidence: 99%

mentioning

confidence: 92%

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On the existence of universal -nuclear Fréchet spaces.

1978

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The above result can be used to prove, using the diametral dimensional approach, that finite products of nuclear or <£-nuclear (see [8]) or Schwartz spaces are respectively nuclear, «^-nuclear or Schwartz spaces. The following result is of value in what follows in the next section of this paper.…”

Section: Diametral Dimensions and Cartesian Productsmentioning

confidence: 94%