On subsymmetric bases in Fréchet spaces

It is proved that any non-normable Fréchet space with a semi-symmetric absolute basis is isomorphic to the space of all scalar sequences. A similar result is shown for quasi-homogeneous absolute bases. It is also proved that any nuclear Fréchet space with a semi-subsymmetric basis is isomorphic to .

     

Reflexivity and sets of Fréchet subdifferentiability

We show that the sets of Fréchet subdifferentiability of Lipschitz functions on a Banach space are Borel if and only if is reflexive. This answers a question of L. Zajíček.

     

Extreme points, exposed points, differentiability points in CL-spaces

This paper presents a property of geometric and topological nature of Gateaux differentiability points and Fréchet differentiability points of almost CL-spaces. More precisely, if we denote by a maximal convex set of the unit sphere of a CL-space , and by the cone generated by , then all Gateaux differentiability points of are just n-s, and all Fréchet differentiability points of are (where n-s denotes the non-support points set of ).

     

Local power series quotients of commutative Banach and Fréchet algebras

We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fréchet algebra. In §2 we derive necessary conditions for a commutative Fréchet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has finite closed descent, the algebra cannot be simply a radical algebra with identity adjoined--it must have nontrivial representation theory; if the generating element does not have finite closed descent, then the algebra cannot be a Banach algebra, and the generating element must be locally nilpotent (but non-nilpotent) in an associated quotient algebra. In §3 we consider a fundamental situation which leads to local power series quotients. Let be a derivation on a commutative radical Fréchet algebra with identity adjoined. We show in Theorem 3.10 that if the discontinuity of is not concentrated in the (Jacobson) radical, then has a local power series quotient. The question of whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the affirmative by C. J. Read.

     

Contractible Fréchet algebras

A unital Fréchet algebra is called contractible if there exists an element such that and for all where is the canonical Fréchet -bimodule morphism. We give a sufficient condition for an infinite-dimensional contractible Fréchet algebra to be a direct sum of a finite-dimensional semisimple algebra and a contractible Fréchet algebra without any nonzero finite-dimensional two-sided ideal (see Theorem 1). As a consequence, a commutative lmc Fréchet -algebra is contractible if, and only if, it is algebraically and topologically isomorphic to for some . On the other hand, we show that a Fréchet algebra, that is, a locally -algebra, is contractible if, and only if, it is topologically isomorphic to the topological Cartesian product of a certain countable family of full matrix algebras.

     

Topological mixing and Hypercyclicity Criterion for sequences of operators

For a sequence of continuous linear operators on a separable Fréchet space , we discuss necessary conditions and sufficient conditions for to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect to the full sequence implies topological mixing; 3) topological mixing implies the Hypercyclicity Criterion with respect to some sequence that cannot be syndetic in general, and also implies condition (b) of the Hypercyclicity Criterion with respect to the full sequence. Applications to two examples of operators on the Fréchet space of entire functions are also discussed.

     

Generalized subdifferential of the distance function

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space with norms on and locally uniformly convex.

     

Fixed point theory for weakly inward Kakutani maps: The projective limit approach

New fixed point results are presented for weakly inward Kakutani condensing maps defined on a Fréchet space . The proofs rely on the notion of an essential map and viewing as the projective limit of a sequence of Banach spaces.

     

Derivations with large separating subspace

In his famous paper The image of a derivation is contained in the radical, Marc Thomas establishes the (commutative) Singer-Wermer conjecture, showing that derivations from a commutative Banach algebra to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on must lie in the radical of . In this paper, we exhibit discontinuous derivations on a commutative unital Fréchet algebra such that the separating subspace is the whole of . Thus, the situation on Fréchet algebras is markedly different from that on Banach algebras.

     

Metrizability vs. Fréchet-Uryshon property

In metrizable spaces, points in the closure of a subset are limits of sequences in ; i.e., metrizable spaces are Fréchet-Uryshon spaces. The aim of this paper is to prove that metrizability and the Fréchet-Uryshon property are actually equivalent for a large class of locally convex spaces that includes - and -spaces. We introduce and study countable bounded tightness of a topological space, a property which implies countable tightness and is strictly weaker than the Fréchet-Urysohn property. We provide applications of our results to, for instance, the space of distributions . The space is not Fréchet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.

     

Extremal characterizations of asplund spaces

We prove new characterizations of Asplund spaces through certain extremal principles in nonsmooth analysis and optimization. The latter principles provide necessary conditions for extremal points of set systems in terms of Fréchet normals and -normals.

     

Partial subdifferentials, derivates and Rademacher's Theorem

In this paper, we present new partial subdifferentiation formulas in nonsmooth analysis, based upon the study of two directional derivatives. Simple applications of these formulas include a new elementary proof of Rademacher's Theorem in , as well as some results on Gâteaux and Fréchet differentiability for locally Lipschitz functions in a separable Hilbert space.

RÉSUMÉ. Dans cet article, nous présentons de nouvelles formules de sousdifférentiation partielle en analyse nonlisse, basées sur l'étude de deux dérivées directionnelles. Une simple application de ces formules nous permet d'obtenir une nouvelle preuve élémentaire du théorème de Rademacher dans , ainsi que certains résultats sur la différentiabilité Gâteaux ou Fréchet des fonctions localement Lipschitz sur un espace de Hilbert séparable.

     

Projectively bounded Fréchet measures

A scalar valued set function on a Cartesian product of -algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.

     

Bump functions and differentiability in Banach spaces

We show that if a Banach space admits a continuous symmetrically Fréchet subdifferentiable bump function, then is an Asplund space.

     

On concepts of directional differentiability

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Fréchet, and Hadamard are singled out from the general framework of -directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed.     

Généralisation du critère de Beurling-Nyman pour l'hypothèse de Riemann

Nous généralisons dans cet article le critère de Beurling-Nyman, qui concerne la fonction de Riemann, à une large classe de séries de Dirichlet. Nous établissons donc une correspondance entre la densité d'un certain sous-espace de fonctions dans et la localisation des zéros d'une série de Dirichlet. Nous utilisons pour obtenir ce résultat la structure de l'espace de Hardy du demi-plan.

ABSTRACT. We generalise Beurling-Nyman's criterion, already known for the Riemann function, to a larger class of Dirichlet series. We reveal a link between the density of some subspace of functions in and the localization of the zeros of a Dirichlet series. To do so, we use the structure of the Hardy space of the half-plan.

     

Construction de certaines opérades et bigèbres associées aux polytopes de Stasheff et hypercubes

Stasheff polytopes, introduced by Stasheff in his study of -spaces, are linked to associativity. The direct sum of their cellular complexes is the underlying complex of the operad which describes homotopy associative algebras. In particular, there exists a quasi-isomorphism .

Here, we define on the direct sum of their dual cellular complexes the structure of a differential graded operad. This construction extends the dendriform operad of Loday, which corresponds to the vertices of the polytopes. We also define the structure of a differential graded operad on the direct sum of the dual cellular complexes of the hypercubes. We define a quasi-isomorphism from to each of these operads.

We also define non-differential variants of the two preceding operads and a morphism from to each of these operads. We show that the free algebras have a coproduct which turns them into bialgebras.

RÉSUMÉ. Les polytopes de Stasheff, introduits pour l'étude des -espaces, sont liés à l'associativité. La somme directe de leurs complexes cellulaires forme le complexe sous-jacent à l'opérade qui décrit les algèbres associatives à homotopie près. En particulier, il existe un quasi-isomorphisme .

Ici, on munit la somme directe des duaux de leurs complexes cellulaires d'une structure d'opérade différentielle graduée. Cette construction généralise l'opérade des algèbres dendriformes de Loday, qui correspond aux sommets des polytopes. On munit aussi la somme directe des duaux des complexes cellulaires des hypercubes d'une structure d'opérade différentielle graduée. On définit un quasi-isomorphisme de dans chacune de ces deux opérades.

On construit également des variantes non différentielles des deux opérades précédentes. On définit un morphisme de dans chacune de ces opérades et on montre que les algèbres libres sont munies d'un coproduit coassociatif qui en fait des bigèbres.

     

Fréchet-Urysohn spaces in free topological groups

Let and be respectively the free topological group and the free Abelian topological group on a Tychonoff space . For every natural number we denote by () the subset of () consisting of all words of reduced length . It is well known that if a space is not discrete, then neither nor is Fréchet-Urysohn, and hence first countable. On the other hand, it is seen that both and are Fréchet-Urysohn for a paracompact Fréchet-Urysohn space . In this paper, we prove first that for a metrizable space , () is Fréchet-Urysohn if and only if the set of all non-isolated points of is compact and is Fréchet-Urysohn if and only if is compact or discrete. As applications, we characterize the metrizable space such that is Fréchet-Urysohn for each and is Fréchet-Urysohn for each except for . In addition, however, there is a first countable, and hence Fréchet-Urysohn subspace of () which is not contained in any (). We shall show that if such a space is first countable, then it has a special form in (). On the other hand, we give an example showing that if the space is Fréchet-Urysohn, then it need not have the form.

     

Slices in the unit ball of the symmetric tensor product of $\mathcal{C}(K)$ and L1(μ)

We prove that for the cases (K infinite) and X=L ₁(μ) (μ σ-finite and atomless) it holds that every slice of the unit ball of the N-fold symmetric tensor product of X has diameter two. In fact, we prove more general results for weak neighborhoods relative to the unit ball. As a consequence, we deduce that the spaces of N-homogeneous polynomials on those classical Banach spaces have no points of Fréchet differentiability. Dedicated to Angel Rodríguez Palacios on the occasion of his 60th birthday.     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.