Topologies on function spaces

In the present paper we introduce notions of A-splitting and A-jointly continuous topology on the set C(Y,Z) of all continuous maps of a topological space Y into a topological space Z, where A is any family of spaces. These notions satisfy the basic properties of splitting and jointly continuous topologies on C(Y,Z). In particular, for every A, the greatest A-splitting topology on C(Y,Z) (denoted by τ(A) always exists. We indicate some families A of spaces for which the topology τ(A) coincides with the greatest splitting topology on C(X,Y). We give a notion of equivalent families of spaces and try to find a “simple” family which is equivalent to a given family. In particular, we prove that every family is equivalent to a family consisting of one space, and the family of all spaces is equivalent to a family of all T₁-spaces containing at most one nonisolated point. We compare the topologies τ({X}) for distinct compact metrizable spaces X and give some examples. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 208, 1993, pp. 82–97. Translated by A. A. Ivanov.     

An Elementary Approach to Exponential Spaces

In 1970, Day and Kelly characterized exponential spaces by a condition (C). Eight years later, Hofmann and Lawson pointed out that this is equivalent to quasi-local compactness, i.e. every neighborhood V of a point contains a smaller one W such that any open cover of V admits a finite subcover of W. These characterizations work with topologies on topologies and may be felt to be not really elementary. This note instead offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.     

λ-Topologies on Function Spaces

This paper is devoted to the spaces C _λ(X) of all continuous real-valued functions on X endowed with arbitrary λ-topologies. This is a fairly complete survey of the results obtained by the author in the following domains of the theory of λ-topologies: cardinal functions; locally convex properties; weak and strong topologies; dual spaces; lattices of λ-topologies; completeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 3–56, 2003.     

Dependence of separately continuous functions on n coordinates on products of compact sets

We investigate the problem of the dependence of separately continuous functions on n coordinates on products of spaces each of which is a topological product. In the case where X and Y are products of completely regular countably compact spaces, we establish necessary and sufficient conditions for such a dependence. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 822–829, June, 1998.     

Some properties of set-open topologies

The space of continuous real-valued functions with set-open topology is studied. The coincidence of the set-open topologies for various families λ is considered and conditions for a set-open topology to coincide with the uniform topology are investigated. The separation axioms in the spaces C ^λ(X) are studied. Criteria for the local compactness and σ-compactness of C ^λ(X) are given. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.     

Polynomials in topological vector spaces over valued fields

We consider the vector space of continuousm-homogeneous polynomials between topological vector spaces over a non-trivially valued field of characteristic zero and certain natural vector topologies on such spaces, and we prove polynomial versions of certain well known theorems of the linear theory of locally convex spaces. Partially supported by CNPq.     

Characteristic functions and their factorizations

A new definition of the characteristic function is introduced for contractions on Hilbert spaces. The relationship with other definitions is established. A factorization formula corresponding to an invariant subspace is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 71–78. Translated by V. V. Kapustin.     

Diagonal sequence property in Banach spaces with weaker topologies

We investigate the diagonal sequence property in Banach spaces with weaker topologies. In particular, we present examples of Banach spaces with weaker locally convex topologies which have the diagonal sequence property but are not Fréchet–Urysohn. The examples answer negatively a question of Averbukh and Smolyanov. We give also a very simple proof of the fact that each Banach space contains a subset A whose weak closure includes 0, but 0 is not contained in the weak closure of any bounded subset of A.     

Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies

We provide a representation for strong-weak continuous dynamic risk measures from L^p into L^p_t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.     

Exponential homeomorphisms in the category of topological spaces with base point

Topologies on base point preserving function spaces are studied making use of the Brown-Booth-Tillotson C-smash product and the topologies on function spaces defined by the classes C of exponentiable spaces. Some conditions on the classes C are obtained for exponential bijections and exponential homeomorphisms in the category of topological spaces with base point. Conditions for exponential homeomorphisms are obtained for CW-complexes whose cellular structure implies precise results.     

Harmonic analysis in (UMD)-spaces: Applications to the theory of bases

In the paper, a general method for the construction of bases and unconditional finite-dimensional basis decompositions for spaces with the property of unconditional martingale differences is proposed. The construction makes use of a certain strongly continuous representation of Cantor's group in these spaces. The results are applied to vector function spaces and symmetric spaces of measurable operators associated with factors of type II.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 890–905, December, 1995.     

A study concerning splitting and jointly continuous topologies on C(Y,Z)

Let Y and Z be two fixed topological spaces and C(Y, Z) the set of all continuous maps from Y into Z. We construct and study topologies on C(Y, Z) that w^{e call F}n(τn)-family-open topologies. Furthermore, we find necessary and su?cient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.     

Compact sets of additive measures

In this note one gives compactifications, having a general character, for the family of all probability measures and for convex, bounded sets in linear spaces, considered in weak topologies.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 14–18, 1982.     

Maximally mobile Davies metric spaces

The spaces whose linear elements are given by a homogeneous function of degree two with respect to the direction arguments are studied. The maximal order of the group of motionsG _r (r=n(n–1)/2+2) in Davies spaces is determined. A tensor characterization of maximally mobile spaces is given.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 47–54.     

Covering dimension of topological products

The first part of the paper is concerned with conditions under which the inequality dim X × Y ≤ dim X + dim Y and similar inequalities for infinite topological products hold. The second part contains examples of spaces such that the sums of their dimensions are smaller than the dimensions of their products. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 34, General Topology, 2005.     

Functorial Duality for Ortholattices and De Morgan Lattices

Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.     

Domain函数空间上的Isbell拓扑和Scott拓扑的一致性

本文主要讨论了Domain函数空间上Isbell拓扑和Scott拓扑的一致性.利用Domain函数空间给出了一个例子: Scott拓扑有开滤子基的非连续的DCPO.     

Hardy's inequalities in function spaces containing derivatives of noninteger order

A theorem on Hardy's inequality in function spaces containing derivatives of noninteger order is proved. Translated fromMatematichcskie Zametki, Vol. 63, No. 5, pp. 673–678, May, 1998. The author wishes to thank Professor V. A. Kondrat'ev for his attention to this work.     

Almost isometric operators: Functional model,invariant subspaces,commutant

A new function model for an arbitrary bounded operator on a Hilbert space is constructed. This model generalizes the model of Sz.-Nagy and Foiaş for contractions and seems to be useful for operators close to an isometry (in a sense). All the model spaces are Hilbert spaces, but instead of dilation a generalization of it is used. The model admits a symmetry with respect to the map z→1/z of the complex plane. In terms of the model the question of lifting the commutant is investigated, a relationship between invariant subspaces of a unitary operator is established, and the characteristic function of the model operator is calculated. Some other problems are solved as well. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 95–116. Translated by V. Vasyunin.     

Resolution of the uniform lower bound problem in constructive analysis

In a previous paper we constructed a full and faithful functor ?? from the category of locally compact metric spaces to the category of formal topologies (representations of locales). Here we show that for a real‐valued continuous function f, ??(f) factors through the localic positive reals if, and only if, f has a uniform positive lower bound on each ball in the locally compact space. We work within the framework of Bishop constructive mathematics, where the latter notion is strictly stronger than point‐wise positivity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)