Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.     

Completions of spaces of functions on compact spaces with respect to the Hausdorff uniformity

For bicompact X, the topology of uniform convergence on C(X) coincides with the Vietoris topology on the function graphs. The Vietoris topology on C(X) in turn is generated by the Hausdorff uniformity which is complete only for finite X. The supplement C_H(X) of the space C(X) with respect to the Hausdorff uniformity consists of lower semi-continuous multivalued mappings Φ : X → ℝ with compact fibers. The paper studies the spaces C_H(X) in the spaces of multivalued mappings. It is proved that C_H(X) is a Q-manifold provided that X is a Peano continuum. Several results on the metrizability on bicompacta having the form C_H(X, I) are also obtained. Bibliography: 27 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 213–235, 1995.     

Hyperbolicity outside a compact set and homogeneous spaces

Summary We study complex manifolds, which satisfy the following property: there exists a compact K such that M-K is Kobayashi-hyperbolic. In section 2, we study in some detail the case when M is homogeneous under the action of a complex Lie group. In section 3, we give some results when M is homogeneous under the action of a real Lie group of holomorphic transformations. Entrata in Redazione il 15 luglio 1997.     

Smoothness of functions with respect to a set

In this note we consider the properties of functions which are smooth with respect to congruent sets. We find necessary and sufficient conditions which these congruent sets must satisfy in order that asymptotic differentiability, together with the smoothness relative to a set, will imply ordinary differentiability almost everywhere. We consider the case of measurable and continuous functions.Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 431–436, March, 1974.     

Self-accessibility of a set with respect to a multivalued field

LetF be a multivalued field on the manifoldM and letN be a submanifold ofM, possibly with boundary. We give a sufficient condition for the self-accessibility property of the pair (N, F), that is:N is contained in the interior of the attainable set fromN at a time smaller or equal toT, for everyT>0 (at the timeT, ifN is compact). To obtain such a condition, a modification of Petrov's implicit function theorem is proved (Ref. 1). Finally, a necessary condition for the self-accessibility property is given.This work was performed under the auspices of the Consiglio Nazionale delle Ricerche, Rome, Italy.     

Complex interpolation of Orlicz spaces with respect to a vector measure

We apply the Calderón interpolation methods to Orlicz and weakly Orlicz function spaces with respect to a Banach‐space‐valued measure defined on a σ‐algebra. The results we obtain generalize those in the case of Banach lattices of p‐integrable and weakly p‐integrable functions with respect to such a vector measure.     

Van der Corput sets with respect to compact groups

We study the notion of van der Corput sets with respect to general compact groups.     

Almost Chebyshev set with respect to bounded subsets

The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively). Project supported by the National Natural Science Foundation of China, Natural Science Foundation of Zhejiang Province, and the State Major Key Project for Basic Researchers of China.     

Holomorphic Sobolev spaces associated to compact symmetric spaces

Using Gutzmer's formula, due to Lassalle, we characterise the images of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.     

The mobility of Riemannian spaces with respect to conformal mappings onto Einstein spaces

We give an estimate of the first lacuna in the distribution of mobility degrees r of n-dimensional (pseudo-)Riemannian spaces with respect to conformal mappings onto Einstein spaces. We obtain a tensor characteristic of spaces which are not conformally flat and have r = n − 1, which is the maximum possible value. Thus, we have found maximum mobile nonconformally flat spaces with r = n − 1.     

Collectionwise Hausdorff versus collectionwise normal with respect to compact sets

Collectionwise normal (CWN) and collectionwise Hausdorff (CWH) spaces have played an increasingly important role in topology since the introduction of these concepts by R.H. Bing in 1951 [3]. It has remained an open and frequently raised question as to whether CWH T₃-spaces are CWN with respect to compact sets. Recently, a counterexample requiring the existence of measurable cardinals and having little additional topological structure was constructed by W.G. Fleissner and the author. In this paper, the author gives a simple example in ZFC of a CWH, first countable, perfect T₃-space that is not CWN with respect to compact, metrizable sets, and, under Martin's Axiom, such an example that is also a Moore space. In addition, the author considers the analogous question for strongly collectionwise Hausdorff (SCWH) T₃-spaces and characterizes the existence of SCWH T₃-spaces that are not CWN with respect to compact sets in set-theoretic and box product formulations. The constructions utilized throughout the paper are of a general nature and several apparently new set-theoretic techniques for interchanging ‘points’ and ‘sets’ are introduced.     

Multiplication operators on spaces of integrable functions with respect to a vector measure

We study continuity and other properties related to some kind of compactness of multiplication operators between different spaces of pth power integrable scalar functions with respect to a vector measure.