A Question of Valdivia on Quasinormable Fréchet Spaces

1991 ◽  
Vol 34 (3) ◽  
pp. 301-304 ◽  
Author(s):  
José Bonet
Keyword(s):  
Positive Answer ◽  
Null Sequence ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Strong Dual

AbstractIt is proved that a Fréchet space is quasinormable if and only if every null sequence in the strong dual converges equicontinuously to the origin. This answers positively a question raised by Valdivia. As a consequence a positive answer to a problem of Jarchow on Fréchet Schwartz spaces is obtained.

scholarly journals On the projective tensor product of Fréchet spaces

1991 ◽  
Vol 34 (2) ◽  
pp. 169-178 ◽  
Author(s):  
Juan C. Díaz ◽  
Juan A. López Molina
Keyword(s):  
Tensor Product ◽  
Sequence Space ◽  
Positive Answer ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Montel Space

We are concerned with the following problem. Let F be a Fréchet Montel space and let E be a Fréchet space with a certain property (P). When does it follow that the complete projective tensor product has the property (P)? (We consider the following properties: being Montel, reflexive, satisfying the density condition.) In this paper we provide a positive answer if F is a Montel generalized Dubinsky sequence space with decreasing steps.

Representations of the spaces Cm(Ω) ∩ Hk, p (Ω)

1996 ◽  
Vol 120 (3) ◽  
pp. 489-498 ◽  
Author(s):  
A. A. Albanese ◽  
G. Metafune ◽  
V. B. Moscatelli
Keyword(s):  
Banach Spaces ◽  
Function Space ◽  
Inductive Limit ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Complemented Subspaces ◽  
Complemented Subspace ◽  
Strong Dual

The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

Interpolation in Separable Frechet Spaces with Applications to Spaces of Analytic Functions

1975 ◽  
Vol 27 (5) ◽  
pp. 1110-1113 ◽  
Author(s):  
Paul M. Gauthier ◽  
Lee A. Rubel
Keyword(s):  
Sufficient Conditions ◽  
Fréchet Space ◽  
Complex Numbers ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear Functional ◽  
Interpolating Sequence ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Spaces Of Analytic Functions

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

scholarly journals Some results on asymptotically normable Frechet spaces

BIBECHANA ◽  
1970 ◽  
Vol 7 ◽  
pp. 39-43
Author(s):  
GK Palei ◽  
NP Sah
Keyword(s):  
Banach Spaces ◽  
Tensor Products ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Norm ◽  
Köthe Spaces ◽  
Köthe Space

In this paper, it is shown that the asymptotically normable spaces are the smallest class of Frechet spaces which contains the nuclear Kothe spaces with continuous norm, the Banach spaces and is closed under e-tensor products and sub-spaces. Again our main aim will be to construct an example of a Kothe space which is Montel, admits a continuous norm, but still is not asymptotically normable. Keywords: Asymptotically normable; Frechet space; Kothe space DOI: 10.3126/bibechana.v7i0.4043BIBECHANA 7 (2011) 39-43

scholarly journals Selfadjoint Operators and Generalized Central Algorithms in Frechet Spaces

2006 ◽  
Vol 13 (2) ◽  
pp. 363-382
Author(s):  
Soso Tsotniashvili ◽  
David Zarnadze
Keyword(s):  
Differential Operators ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Von Neumann ◽  
Selfadjoint Operators ◽  
Elliptic Differential Operators ◽  
Dense Set

Abstract The paper gives an extension of the fundamental principles of selfadjoint operators in Fréchet–Hilbert spaces, countable-Hilbert and nuclear Fréchet spaces. Generalizations of the well known theorems of von Neumann, Hellinger–Toeplitz, Friedrichs and Ritz are obtained. Definitions of generalized central and generalized spline algorithms are given. The restriction 𝐴∞ of a selfadjoint operator 𝐴 defined on a dense set 𝐷(𝐴) of the Hilbert space 𝐻 to the Frechet space 𝐷(𝐴∞) is substantiated. The extended Ritz method is used for obtaining an approximate solution of the equation 𝐴∞𝑢 = 𝑓 in the Frechet space 𝐷(𝐴∞). It is proved that approximate solutions of this equation constructed by the extended Ritz method do not depend on the number of norms that generate the topology of the space 𝐷(𝐴∞). Hence this approximate method is both a generalized central and generalized spline algorithm. Examples of selfadjoint and positive definite elliptic differential operators satisfying the above conditions are given. The validity of theoretical results in the case of a harmonic oscillator operator is confirmed by numerical calculations.

scholarly journals Distinguishedness of weighted Fréchet spaces of continuous functions

1992 ◽  
Vol 35 (2) ◽  
pp. 271-283 ◽  
Author(s):  
Françoise Bastin
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Locally Compact ◽  
Spaces Of Continuous Functions ◽  
Locally Compact Hausdorff Space

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

An integro-differential equation from population genetics and perturbations of differentiable semigroups in Fréchet spaces

1991 ◽  
Vol 118 (1-2) ◽  
pp. 63-73 ◽  
Author(s):  
Reinhard Bürger
Keyword(s):  
Differential Equation ◽  
Population Genetics ◽  
Quantitative Trait ◽  
Existence And Uniqueness ◽  
Directional Selection ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Uniqueness Of Solutions ◽  
Integro Differential Equation

SynopsisExistence and uniqueness of solutions of an integro-differential equation that arises in population genetics are proved. This equation describes the evolution of type densities in a population that is subject to mutation and directional selection on a quantitative trait. It turns out that a certain Fréchet space is the natural framework to show existence and uniqueness. One of the main steps in the proof is the investigation of perturbations of generators of differentiable semigroups in Fréchet spaces.

On the Weak Basis Theorem in F-spaces

1974 ◽  
Vol 26 (6) ◽  
pp. 1294-1300 ◽  
Author(s):  
Joel H. Shapiro
Keyword(s):  
Banach Spaces ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex ◽  
Convex Spaces

It is well-known that every weak basis in a Fréchet space is actually a basis. This result, called the weak basis theorem was first given for Banach spaces in 1932 by Banach [1, p. 238], and extended to Fréchet spaces by Bessaga and Petczynski [3]. McArthur [12] proved an analogue for bases of subspaces in Fréchet spaces, and recently W. J. Stiles [18, Corollary 4.5, p. 413] showed that the theorem fails in the non-locally convex spaces lp (0 < p < 1). The purpose of this paper is to prove the following generalization of Stiles' result.

scholarly journals Weakly compactly generated Frechet spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 721-724 ◽  
Author(s):  
Surjit Singh Khurana
Keyword(s):  
Measurable Function ◽  
Weak Topology ◽  
Measure Space ◽  
Finite Measure ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Weakly Compact ◽  
Finite Measure Space ◽  
Compactly Generated

It is proved that a weakly compact generated Frechet space is Lindelöf in the weak topology. As a corollary it is proved that for a finite measure space every weakly measurable function into a weakly compactly generated Frechet space is weakly equivalent to a strongly measurable function.

scholarly journals The Exponential Representation of Holomorphic Functions of Uniformly Bounded Type

2004 ◽  
Vol 76 (2) ◽  
pp. 235-246
Author(s):  
Thai Thuan Quang
Keyword(s):  
Entire Function ◽  
Holomorphic Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Exponential Form ◽  
Absolute Basis ◽  
Uniformly Bounded

AbstractIt is shown that if E, F are Fréchet spaces, E ∈ (Hub), F ∈ (DN) then H(E, F) = Hub(E, F) holds. Using this result we prove that a Fréchet space E is nuclear and has the property (Hub) if and only if every entire function on E with values in a Fréchet space F ∈ (DN) can be represented in the exponential form. Moreover, it is also shown that if H(F*) has a LAERS and E ∈ (Hub) then H(E × F*) has a LAERS, where E, F are nuclear Fréchet spaces, F* has an absolute basis, and conversely, if H(E × F*) has a LAERS and F ∈ (DN) then E ∈ (Hub).

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