Almost periodicity in Fréchet spaces

In this paper we deal with almost periodic functions with values in a Fréchet space. We apply obtained results to prove the existence of solutions of the initial value problem as well as the Volterra integral equation in this class of functions. We also introduce and investigate asymptotically almost periodic functions with values in a Fréchet space.     

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.

     

Existence of hypercyclic polynomials on complex Fréchet spaces

We show that every complex separable infinite dimensional Fréchet space admits hypercyclic polynomials of any degree. This result complements the analogous one for the linear case, due to Ansari, Bernal, Bonet and Peris.     

Quasi-reflexive Fréchet spaces and mean ergodicity

We characterize quasi-reflexive Fréchet spaces with a basis in terms of the properties of this basis. As a consequence we prove that a Fréchet space with a basis is quasi-reflexive of order one if and only if for every power bounded operator T, either T or T^′ is mean ergodic.     

Workshop lecture on products of Fréchet spaces

The general question, “When is the product of Fréchet spaces Fréchet?” really depends on the questions of when a product of α₄ Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α₄, and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α₃ Fréchet spaces and its subclass of ℵ₀-bisequential spaces. The latter is closed under countable products, the former not even under finite products. A number of fundamental results and open problems are recalled, some further highlighting the difference between being α₃ and Fréchet and being ℵ₀-bisequential.     

Tightness and distinguished Fréchet spaces

Valdivia invented a nondistinguished Fréchet space whose weak bidual is quasi-Suslin but not K-analytic. We prove that Grothendieck/Köthe's original nondistinguished Fréchet space serves the same purpose. Indeed, a Fréchet space is distinguished if and only if its strong dual has countable tightness, a corollary to the fact that a (DF)-space is quasibarrelled if and only if its tightness is countable. This answers a Cascales/K?kol/Saxon question and leads to a rich supply of (DF)-spaces whose weak duals are quasi-Suslin but not K-analytic, including the spaces Cc(κ) for κ a cardinal of uncountable cofinality. Our level of generality rises above (DF)- or even dual metric spaces to Cascales/Orihuela's class G. The small cardinals b and d invite a novel analysis of the Grothendieck/Köthe example, and are useful throughout.     

Fréchet-Urysohn for finite sets, II

We continue our study [G. Gruenhage, P.J. Szeptycki, Fréchet Urysohn for finite sets, Topology Appl. 151 (2005) 238-259] of several variants of the property of the title. We answer a question from that paper by showing that a space defined in a natural way from a certain Hausdorff gap is a Fréchet α₂ space which is not Fréchet-Urysohn for 2-point sets (FU₂), and answer a question of Hrušák by showing that under MAω₁, no such “gap space” is FU₂. We also introduce versions of the properties which are defined in terms of “selection principles”, give examples when possible showing that the properties are distinct, and discuss relationships of these properties to convergence in product spaces, to the αi-spaces of A.V. Arhangel'skii, and to topological games.     

Topological structure of solution sets in Fréchet spaces: The projective limit approach

In this paper we show that the solution set of certain Volterra inclusions defined between Fréchet spaces is a continuum. The proof relies on results in Banach spaces and on viewing a Fréchet space as a projective limit of a sequence of Banach spaces.     

Continuity of mappings between Fréchet spaces

For all linear and many nonlinear operators between Fréchet spaces, several continuity results are established. Especially, a new closed graph theorem is given.     

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, AR and R_δ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.     

Shrinking and boundedly complete Schauder frames in Fréchet spaces

We study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder frames on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented.     

Spaces of N-homogeneous polynomials between Fréchet spaces

Properties of the Nachbin-ported topology on spaces of N-homogeneous polynomials between Fréchet spaces are investigated by applying results on derived functors.     

Fixed point properties for semigroup of nonexpansive mappings on Fréchet spaces

In this paper, we shall establish a fixed point property on Fréchet spaces for left reversible semitopological semigroups generalizing some classical results.     

An inverse function theorem in Fréchet spaces

I present an inverse function theorem for differentiable maps between Fréchet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C², or even C¹, or even Fréchet-differentiable.     

Hypercyclic subspaces for Fréchet space operators

A continuous linear operator is hypercyclic if there is an x∈X such that the orbit {Tnx} is dense, and such a vector x is said to be hypercyclic for T. Recent progress show that it is possible to characterize Banach space operators that have a hypercyclic subspace, i.e., an infinite dimensional closed subspace H⊆X of, except for zero, hypercyclic vectors. The following is known to hold: A Banach space operator T has a hypercyclic subspace if there is a sequence (ni) and an infinite dimensional closed subspace E⊆X such that T is hereditarily hypercyclic for (ni) and Tni→0 pointwise on E. In this note we extend this result to the setting of Fréchet spaces that admit a continuous norm, and study some applications for important function spaces. As an application we also prove that any infinite dimensional separable Fréchet space with a continuous norm admits an operator with a hypercyclic subspace.     

Exact calculus of Fréchet subdifferentials for Hadamard directionally differentiable functions

We continue the study of the calculus of the generalized subdifferentials started in [V.F. Demyanov, V. Roshchina, Exhausters and subdifferentials in nonsmooth analysis, Optimization (2006) (in press)] and [V. Roshchina, Relationships between upper exhausters and the basic subdifferential in Variational Analysis, Journal of Mathematical Analysis and Applications 334 (2007) 261–272] and provide some basic calculus rules for the Fréchet subdifferentials via collections of compact convex sets associated with Hadamard directional derivative. The main result of this paper is the sum rule for the Fréchet subdifferential in the form of an equality, which holds for Hadamard directionally differentiable functions, and is of significant interest from the points of view of both theory and applications.     

The Fréchet Approximate Jacobian and Local Uniqueness in Variational Inequalities

We introduce the concept of Fréchet approximate Jacobian matrices for continuous vector functions and use it to establish some sufficient criteria for the local uniqueness of solutions to a variational inequality problem involving continuous, not necessarily locally Lipschitz functions. Examples are also given to illustrate the usefulness of our approach.     

Fréchet differentiability of the solutions of a semilinear abstract Cauchy problem

Sufficient conditions are found under which the solutions z(t;q) of a semilinear abstract Cauchy problem of the form are Fréchet differentiable with respect to the parameter q. An explicit form is provided for the sensitivity equation satisfied by the Fréchet derivative Dqz(t;q).