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.

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 Fixed point theory for Mönch-type maps defined on closed subsets of Fréchet spaces: the projective limit approach

2005 ◽  
Vol 2005 (17) ◽  
pp. 2775-2782 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Jewgeni H. Dshalalow ◽  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces

New Leray-Schauder alternatives are presented for Mönch-type maps defined between Fréchet spaces. The proof relies on viewing a Fréchet space as the projective limit of a sequence of Banach spaces.

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

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 Dual characterization of the Dieudonne-Schwartz theorem on bounded sets

1983 ◽  
Vol 6 (1) ◽  
pp. 189-192 ◽  
Author(s):  
C. Bosch ◽  
J. Kucera ◽  
K. McKennon
Keyword(s):  
Inductive Limit ◽  
Projective Limit ◽  
Fréchet Spaces ◽  
Inductive Limits ◽  
Counter Example ◽  
Bounded Sets ◽  
Strong Dual

The Dieudonné-Schwartz Theorem on bounded sets in a strict inductive limit is investigated for non-strict inductive limits. Its validity is shown to be closely connected with the problem of whether the projective limit of the strong duals is a strong dual itself. A counter-example is given to show that the Dieudonné-Schwartz Theorem is not in general valid for an inductive limit of a sequence of reflexive, 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 Leray-Schauder results for multivalued nonlinear contractions defined on closed subsets of a Fréchet space

2006 ◽  
Vol 2006 ◽  
pp. 1-8
Author(s):  
Ravi P. Agarwal ◽  
Donal O'Regan
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Multivalued Contractions ◽  
Nonlinear Contractions

New Leray-Schauder results are presented for multivalued contractions defined on subsets of a Fréchet spaceE. The proof relies on fixed point results in Banach spaces and on viewingEas the projective limit of a sequence of Banach spaces.

scholarly journals On Universal Schauder Bases in Non-Archimedean Fréchet Spaces

2004 ◽  
Vol 47 (1) ◽  
pp. 108-118 ◽  
Author(s):  
Wiesław Śliwa
Keyword(s):  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Complemented Subspace ◽  
Countable Type ◽  
Schauder Bases

AbstractIt is known that any non-archimedean Fréchet space of countable type is isomorphic to a subspace of . In this paper we prove that there exists a non-archimedean Fréchet space U with a basis (un) such that any basis (xn) in a non-archimedean Fréchet space X is equivalent to a subbasis (ukn) of (un). Then any non-archimedean Fréchet space with a basis is isomorphic to a complemented subspace of U. In contrast to this, we show that a non-archimedean Fréchet space X with a basis (xn) is isomorphic to a complemented subspace of if and only if X is isomorphic to one of the following spaces: c0, c0 × . Finally, we prove that there is no nuclear non-archimedean Fréchet space H with a basis (hn) such that any basis (yn) in a nuclear non-archimedean Fréchet space Y is equivalent to a subbasis (hkn) of (hn).

Darboux chart on projective limit of weak symplectic Banach manifold

2015 ◽  
Vol 12 (07) ◽  
pp. 1550072 ◽  
Author(s):  
Pradip Mishra
Keyword(s):  
Banach Space ◽  
Symplectic Structure ◽  
Reflexive Banach Space ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Banach Manifold ◽  
Boundedness Condition ◽  
Projective System ◽  
Banach Manifolds

Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

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.

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