scholarly journals On the closed subspaces of universal series in Banach spaces and Fréchet spaces

2010 ◽  
Vol 198 (2) ◽  
pp. 121-145 ◽  
Author(s):  
Stéphane Charpentier
Keyword(s):  
Banach Spaces ◽  
Fréchet Spaces ◽  
Universal Series

Automatic Continuity for Linear Functions Intertwining Continuous Linear Operators on Frechet Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 518-530 ◽  
Author(s):  
Marc P. Thomas
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Linear Operators ◽  
Linear Functions ◽  
General Situation ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Continuity Theory ◽  
Continuous Linear Operators

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

scholarly journals THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓp+ AND Lp−

2016 ◽  
Vol 59 (2) ◽  
pp. 273-287 ◽  
Author(s):  
ANGELA A. ALBANESE ◽  
JOSÉ BONET ◽  
WERNER J. RICKER
Keyword(s):  
Banach Spaces ◽  
Point Spectrum ◽  
Fréchet Spaces ◽  
Cesàro Operator ◽  
Continuous Norm ◽  
Spectrum Point ◽  
Mean Ergodicity ◽  
Cesaro Operator

AbstractThe classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

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

Fractional powers of operators and Riesz fractional integrals

1989 ◽  
Vol 112 (3-4) ◽  
pp. 237-247 ◽  
Author(s):  
S. E. Schiavone
Keyword(s):  
Banach Spaces ◽  
Fractional Integral ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fréchet Spaces ◽  
Test Functions ◽  
Fractional Powers ◽  
Fractional Integral Operators ◽  
Fractional Powers Of Operators

SynopsisIn this paper, a theory of fractional powers of operators due to Balakrishnan, which is valid for certain operators on Banach spaces, is extended to Fréchet spaces. The resultingtheory is shown to be more general than that developed in an earlier approach by Lamb, and is applied to obtain mapping properties of certain Riesz fractional integral operators on spaces of test functions.

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 On properly separable quotients of strict (LF) Spaces

1989 ◽  
Vol 47 (2) ◽  
pp. 307-312 ◽  
Author(s):  
W. J. Robertson
Keyword(s):  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Inductive Limit ◽  
General Question ◽  
Fréchet Spaces ◽  
Infinite Dimensional

AbstractAll known Banach spaces have an infinite-dimensional separable quotient and so do all nonnormable Fréchet spaces, although the general question for Banach spaces is still open. A properly separable topological vector space is defined, in such a way that separable and properly separable are equivalent for an infinite-dimensional complete metrisable space. The main result of this paper is that the strict inductive limit of a sequence of non-normable Fréchet spaces has a properly separable quotient.

scholarly journals S-Barrelled Topological Vector Spaces*

1978 ◽  
Vol 21 (2) ◽  
pp. 221-227 ◽  
Author(s):  
Ray F. Snipes
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Fréchet Spaces ◽  
Topological Vector Spaces ◽  
Barrelled Spaces ◽  
Steinhaus Theorem ◽  
Locally Convex

N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called “espaces tonnelés” or “barrelled spaces.” These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8].

On the Dimension of a Complete Metrizable Topological Vector Space

1977 ◽  
Vol 20 (2) ◽  
pp. 271-272 ◽  
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Spaces ◽  
Topological Vector Space ◽  
Fréchet Spaces ◽  
Convex Case ◽  
Space Case ◽  
Locally Convex ◽  
Elegant Proof ◽  
Metrizable Topological Vector Space

The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.

scholarly journals Fixed points of cone compression and expansion multimaps defined on Fréchet spaces: The projective limit approach

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Multivalued Maps ◽  
Compression And Expansion

We present a generalization of the cone compression and expansion results due to Krasnoselskii and Petryshyn for multivalued maps defined on a Fréchet space E. The proof relies on fixed point results in Banach spaces and viewing E as the projective limit of a sequence of Banach spaces.

scholarly journals Fréchet AL-spaces have the Dunford-Pettis property

1998 ◽  
Vol 58 (3) ◽  
pp. 383-386 ◽  
Author(s):  
J.C. Díaz ◽  
A. Fernández ◽  
F. Naranjo
Keyword(s):  
Banach Spaces ◽  
Positive Cone ◽  
Weak Order ◽  
Fréchet Spaces ◽  
Order Unit ◽  
Continuous Norm

A Fréchet lattice E is an AL-space if its topology can be defined by a family of lattice seminorms that are additive in the positive cone of E. Grothendieck proved that AL-Banach spaces have the Dunford-Pettis property. This result was recently extended by Fernández and Naranjo to AL-Fréchet spaces with a continuous norm and weak order unit. In this note we show how to remove both hypotheses.

Sign in / Sign up

Export Citation Format

Share Document