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 The existence of a factorized unbounded operator between Fréchet spaces

2018 ◽  
Vol 13 (01) ◽  
pp. 2050017
Author(s):  
Ersin Kızgut ◽  
Murat Yurdakul
Keyword(s):  
Unbounded Operator ◽  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Linear Map ◽  
Infinite Dimensional ◽  
Locally Convex ◽  
Convex Spaces

For locally convex spaces [Formula: see text] and [Formula: see text], the continuous linear map [Formula: see text] is called bounded if there is a zero neighborhood [Formula: see text] of [Formula: see text] such that [Formula: see text] is bounded in [Formula: see text]. Our main result is that the existence of an unbounded operator [Formula: see text] between Fréchet spaces [Formula: see text] and [Formula: see text] which factors through a third Fréchet space [Formula: see text] ends up with the fact that the triple [Formula: see text] has an infinite dimensional closed common nuclear Köthe subspace, provided that [Formula: see text] has the property [Formula: see text].

scholarly journals Quasi-bounded sets

1990 ◽  
Vol 13 (3) ◽  
pp. 607-610
Author(s):  
Jan Kucera
Keyword(s):  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Bounded Set ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

It is proved in [1] & [2] that a set bounded in an inductivelimit E=indlim Enof Fréchet spaces is also bounded in someEniffEis fast complete. In the case of arbitrary locally convex spacesEnevery bounded set in a fast completeindlim Enis quasi-bounded in someEn, though it may not be bounded or even contained in anyEn. Every bounded set is quasi-bounded. In a Fréchet space every quasi-bounded set is also bounded.

A Short Proof of Vladimirskii′s Theorem on Precompact Perturbations in Locally Convex Spaces

1975 ◽  
Vol 18 (5) ◽  
pp. 649-655
Author(s):  
Le Quang Chu
Keyword(s):  
Compact Operator ◽  
Short Proof ◽  
Locally Convex Spaces ◽  
Fréchet Spaces ◽  
Continuous Operators ◽  
Locally Convex ◽  
Convex Spaces

Let T, P denote two continuous operators from E into F, where E and F are locally convex spaces. It is proved by L. Schwartz [8] and G. KÖthe [6] that if E and F are Fréchet spaces, T is a ϕ_-operator and P a compact operator, then T+P is a ϕ_-operator.

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 Categories whose objects are determined by their rings of endomorphisms

1976 ◽  
Vol 15 (1) ◽  
pp. 65-72
Author(s):  
Grigore Călugăreanu
Keyword(s):  
Banach Spaces ◽  
General Theorem ◽  
Locally Convex Spaces ◽  
Ring Isomorphism ◽  
Additive Category ◽  
Locally Convex ◽  
Convex Spaces

In an additive category A0, objects are said to be determined by their rings of endomorphisms if for each ring-isomorphism F of the rings of endomorphisms of two objects A, B in A0 there is an isomorphism f: A → B in A0 such that F(α) = fαf-1, for every endomorphism α of A. Considering.this problem in the context of closed categories (in Eilenberg and Kelly's sense), the author proves a general theorem which generalises results of Eidelheit (for real Banach spaces) and of Kasahara (for real locally convex spaces).

Linear Chaos on Fréchet Spaces

2003 ◽  
Vol 13 (07) ◽  
pp. 1649-1655 ◽  
Author(s):  
J. Bonet ◽  
F. Martínez-Giménez ◽  
A. Peris
Keyword(s):  
Linear Operators ◽  
Locally Convex Spaces ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Locally Convex ◽  
Continuous Linear Operators ◽  
Convex Spaces

This is a survey on recent results about hypercyclicity and chaos of continuous linear operators between complete metrizable locally convex spaces. The emphasis is put on certain contributions from the authors, and related theorems.

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.

Sign in / Sign up

Export Citation Format

Share Document