scholarly journals Quasianalytic functionals and projective descriptions

2004 ◽  
Vol 94 (2) ◽  
pp. 249 ◽  
Author(s):  
José Bonet ◽  
Reinhold Meise
Keyword(s):  
Laplace Transform ◽  
Convex Subset ◽  
Inductive Limit ◽  
Entire Functions ◽  
Fréchet Spaces ◽  
Open Convex ◽  
Open Convex Subset

The topology of the weighted inductive limit of Fréchet spaces of entire functions in $N$ variables which is obtained as the Fourier Laplace transform of the space of quasianalytic functionals on an open convex subset of $\mathrm{R}^N$ cannot be described by means of weighted sup-seminorms.

Amenability and invariant subspaces

1974 ◽  
Vol 18 (2) ◽  
pp. 200-204 ◽  
Author(s):  
Anthony To-Ming Lau
Keyword(s):  
Convex Subset ◽  
Invariant Subspaces ◽  
Complex Field ◽  
Open Convex ◽  
Amenable Semigroup ◽  
Linear Action ◽  
Banach Theorem ◽  
Amenable Semigroups ◽  
Open Convex Subset ◽  
Invariant Element

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

Théorème d'existence pour des problèmes variationnels non convexes

1987 ◽  
Vol 107 (1-2) ◽  
pp. 43-64 ◽  
Author(s):  
J. P. Raymond
Keyword(s):  
Convex Subset ◽  
Existence Results ◽  
Open Convex ◽  
Image Position ◽  
Open Convex Subset

SynopsisOn donne dans cet article un théorème d'existence de solutions lipschitziennes pour des problèmes du type:où Ω est un ouvert convexe borné de ℝn, n ≧ 2, p ≧ 2, aucune hypothèse de convexité n'est faite sur g ou f. On étend de la sorte des ŕesultats d'existence obtenus en dimension 1.where Ω is a bounded open convex subset of ℝn, n ≧ 2, p ≧ 2; we suppose no assumption of convexity on g or f. In this way we extend existence results proved in dimension 1.)

scholarly journals The differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Inductive Limit ◽  
Asplund Space ◽  
Fréchet Spaces ◽  
Linear Spaces ◽  
Asplund Spaces ◽  
Continuous Convex Function ◽  
Mazur’S Theorem ◽  
Topological Linear ◽  
Differentiability Properties

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

scholarly journals A dual differentiation space without an equivalent locally uniformly rotund norm

2004 ◽  
Vol 77 (3) ◽  
pp. 357-364 ◽  
Author(s):  
Petar S. Kenderov ◽  
Warren B. Moors
Keyword(s):  
Dense Subset ◽  
Continuum Hypothesis ◽  
Open Convex ◽  
Residual Subset ◽  
Continuous Convex Function ◽  
Dual Differentiation ◽  
Fréchet Differentiable ◽  
Rotund Norm ◽  
Open Convex Subset ◽  
The Continuum

AbstractA Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.

Sufficient sets in weighted Fréchet spaces of entire functions

2013 ◽  
Vol 54 (4) ◽  
pp. 575-587 ◽  
Author(s):  
A. V. Abanin ◽  
V. A. Varziev
Keyword(s):  
Entire Functions ◽  
Fréchet Spaces

scholarly journals Separable determination of Fréchet differentiability of convex functions

1995 ◽  
Vol 52 (1) ◽  
pp. 161-167 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Fréchet Differentiability ◽  
Continuous Convex Function ◽  
Separability Condition ◽  
Frechet Differentiability ◽  
Open Convex Subset ◽  
Insight Into

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Fréchet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

scholarly journals On the characterisation of Asplund spaces

1982 ◽  
Vol 32 (1) ◽  
pp. 134-144 ◽  
Author(s):  
J. R. Giles
Keyword(s):  
Banach Space ◽  
Direct Proof ◽  
Asplund Space ◽  
Open Convex ◽  
Asplund Spaces ◽  
Separable Subspace ◽  
Nikodym Property ◽  
Continuous Convex Function ◽  
Fréchet Differentiable ◽  
Open Convex Subset

AbstractA Banach space is an Asplund space if every continuous convex function on an open convex subset is Fréchet differentiable on a dense G8 subset of its domain. The recent research on the Radon-Nikodým property in Banach spaces has revealed that a Banach space is an Asplund space if and only if every separable subspace has separable dual. It would appear that there is a case for providing a more direct proof of this characterisation.

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.

scholarly journals A Newton-type method and its application

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
V. Antony Vijesh ◽  
P. V. Subrahmanyam
Keyword(s):  
Existence And Uniqueness ◽  
Functional Integral ◽  
Existence And Uniqueness Theorem ◽  
Open Convex ◽  
Type Method ◽  
Nonlinear Functional ◽  
Recent Theorem ◽  
Urysohn Operator ◽  
Functional Integral Equations ◽  
Open Convex Subset

We prove an existence and uniqueness theorem for solving the operator equationF(x)+G(x)=0, whereFis a continuous and Gâteaux differentiable operator and the operatorGsatisfies Lipschitz condition on an open convex subset of a Banach space. As corollaries, a recent theorem of Argyros (2003) and the classical convergence theorem for modified Newton iterates are deduced. We further obtain an existence theorem for a class of nonlinear functional integral equations involving the Urysohn operator.

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