A note on sequentially compact quasi-pseudo-metric spaces

It is asked in [3] whether sequential compactness is equivalent to compactness in a quasi-pseudo-metric space. In this note we give a counterexample and a proposition relating to this question.     

Arzela-Ascoli's theorem for Riemann-integrable functions on compact spaces

We recall that the ⁿ-valued Riemann integrable functions resp. more general: the Banach space (algebra) -valued Darboux integrable functions — on the compact support of a non negative Radon measure are a Banach space (algebra) with respect to the ess. sup. norm. It is shown that the Darboux integrable functions with a precompact range also form a Banach space (algebra). For this space we deduce a direct analogue of Arzela-Ascoli's theorem.     

Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

Questions on generalised Baire spaces

We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015).     

A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. I

Generalizing de Vries’ duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.     

The Baire category theorem in products of linear spaces and topological groups

A space is a Baire space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely metrizable linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E×F is not. If X is a completely metrizable linear space of weight ℵ₁ then X is the direct sum E⊕F of two linear subspaces E and F such that both E and F are Baire but E×F is not.     

Integral representation of belief measures on compact spaces

An integral representation theorem for outer continuous and inner regular belief measures on compact topological spaces is elaborated under the condition that compact sets are countable intersections of open sets (e.g. metric compact spaces). Extreme points of this set of belief measures are identified with unanimity games with compact support. Then, the Choquet integral of a real valued continuous function can be expressed as a minimum of means over the sigma-core and also as a mean of minima over the compact subsets. Similarly, for bounded measurable functions, the Choquet integral is expressed as min of means over the core, we prove in addition that it is a mean of infima over the compact subsets. Then, we obtain Choquet–Revuz' measure representation theorem and introduce the Möbius transform of a belief measure. An extension to locally compact and sigma-compact topological spaces is provided.     

An algebraic version of Tamano's theorem for countably compact spaces

We show that if X is countably compact but not compact then one can find a compact space K such that X⊕K does not embed closedly into any normal topological group.     

Unbounded order continuous operators on Riesz spaces

In this paper, using the concept of unbounded order convergence in Riesz spaces, we define new classes of operators, named unbounded order continuous (uo-continuous, for short) and boundedly unbounded order continuous operators. We give some conditions under which uo-continuity will be equivalent to order continuity of some operators on Riesz spaces. We show that the collection of all uo-continuous linear functionals on a Riesz space E is a band of \(E^\sim \).     

Multiscale Young measures in homogenization of continuous stationary processes in compact spaces and applications

We introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces. The latter are functions f○T:Rn×Q→Q with f○T(x,ω)=f(T(x)ω) where Q is a compact (Hausdorff topological) space, f∈C(Q) and T(x):Q→Q, x∈Rn, is an n-dimensional continuous dynamical system endowed with an invariant Radon probability measure μ. It can be easily shown that for almost all ω∈Q the realization f(T(x)ω) belongs to an algebra with mean value, that is, an algebra of functions in BUC(Rn) containing all translates of its elements and such that each of its elements possesses a mean value. This notion was introduced by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators, Mat. Zametki 33 (1983) 571-582, English transl. in Math. Notes 33 (1983) 294-300]. We then establish the existence of multiscale Young measures in the setting of algebras with mean value, where the compactifications of Rn provided by such algebras plays an important role. These parametrized measures are useful in connection with the existence of correctors in homogenization problems. We apply this framework to the homogenization of a porous medium type equation in Rn with a stationary continuous process as a stiff oscillatory external source. This application seems to be new even in the classical context of periodic homogenization.     

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

On Banach spaces whose dual balls are not weak∗ sequentially compact

Theorem 1. LetX be a Banach space. (a) IfX ^∗ has a closed subspace in which no normalized sequence converges weak^∗ to zero, thenl ₁ is isomorphic to a subspace ofX. (b) IfX ^∗ contains a bounded sequence which has no weak^∗ convergent subsequence, thenX contains a separable subspace whose dual is not separable. The second-named author was supported in part by NSF-MPS 72-04634-A03.     

Complementation in spaces of continuous functions on compact lines

We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.