Bornological and ultrabornological C(X;E) spaces

Denote by C_s(X;E) the space of the continuous functions defined on the completely regular and Hausdorff space X, with values in the locally convex topological vector space E, when it is endowed with the simple or point-wise convergence topology. We give here some conditions on X and on E under which the space C_s(X;E) is bornological or ultrabornological and characterize in some cases the corresponding associated spaces. We give also a few results concerning the case of the compact connvergence topology.     

The bounded-open topology on function spaces

For any regular space Z It is shown, 1) that the bounded-open topology T on C(Y,Z) is splitting and it is also the smallest jointly continuous topology whenever Y is locally bounded, 2) if Y is locally bounded or if X × Y is a boundedly generated space, then there is a natural bijection on C(X × Y,Z) onto C(X,(C(Y,Z),T_eo) which is actually a homeomorphism with respect to the bounded-open topology on both function spaces, 3) The path components of (C(Y,Z),T_eo) are exactly its homotopy classes whenever Y is boundedly generated, 4) The bounded-open topology Teo induces contravariant and covariant Homotopy preserving function-space functors. Further, 5) T_eo reduces to the compact-open topology t_co whenever the domain Y is regular; but in general, T_eo is finer than T_co (assuming the domain is Hausdorff or the range is either Hausdorff or regular).     

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.     

Vectorial exceptional families of elements

Let Z{\mathcal{Z}} be an ordered Hausdorff topological vector space with a preorder defined by a pointed closed convex cone C ì Z{C \subset {\mathcal Z}} with a nonempty interior. In this paper, we introduce exceptional families of elements w.r.t. C for multivalued mappings defined on a closed convex cone of a normed space X with values in the set L(X, Z){L(X, {\mathcal Z})} of all continuous linear mappings from X into Z{\mathcal{Z}} . In Banach spaces, we prove a vectorial analogue of a theorem due to Bianchi, Hadjisavvas and Schaible. As an application, the C-EFE acceptability of C-pseudomonotone multivalued mappings is investigated.     

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

Weighted composition operators on nonlocally convex weighted spaces of continuous functions

LetV be a system of weights on a completely regular Hausdorff spaceX and letB(E) be the topological vector space of all continuous linear operators on a general topological vector spaceE. LetCV ₀(X, E) andCV _b (X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operatorsW _π,? onCV ₀(X, E) (orCV _b (X, E)) induced by the operator-valued mappings π:X→B(E) (or the vector-valued mappings π:X→E, whereE is a topological algebra) and the self-map ? ofX. Also, we characterize the mappings π:X→B(E) (or π:x→E) and ?:X→X which induce the compact weighted composition operators on these weighted spaces of continuous functions.     

Averaging operators on normed köthe spaces

Summary Under study is the existence of averaging operators determined by measurable maps φ from a measure space (S, Σ, μ) into an arbitrary Hausdorff topological space T. The map φ induces a continuous map φ^e from the space C_b(T) into the normed (Banach) function space L_ϱ = L_ϱ(S, Σ, μ) defined by φ^e(f)=foφ for all f ε C_b(T). An integral representation for such operators is first studied. The existence is then determined by the existence of an averaging operator U₁ for the restriction of φ to a certain measurable subset B₁ of S. Utilizing a representation of L_ϱ(S, Σ, μ) as a Banach function space over a compact extremally disconnected Hausdorff space Ŝ, we are able to give a definition for the concept of plural points and irreducible map. A significant upper bound is given for the operator U₁. Finally conditions are considered under which no bounded projection from L_ϱ onto the range of φ^e may exist. From a topological point of view the development is pursued in a general setting. Averaging operators have recently been used for the study of injective Banach spaces of the type C_b(T) and in non-linear prediction and approximation theory relative to Tshebyshev subspaces of L_ϱ. Entrata in Redazione l’ll settembre 1975.     

Riesz product spaces and representation theory

Let {E _i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏ _i∈I E_i. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff spaceX such that ∏ _i∈I E_i is Riesz isomorphic toC(X) if and only if for everyi∈I there exists a completely regular Hausdorff spaceX _i such thatE _i is Riesz isomorphic toC(X _i). Supported by the National Natural Science Foundation of China     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

The <Emphasis Type="Italic">(S)</Emphasis><Stack><Subscript>+</Subscript><Superscript>1</Superscript></Stack>Condition for Generalized Vector Variational Inequalities

In this paper, we extend the (S) ₊ ¹ condition to multivalued mappings in an ordered Hausdorff topological vector space and we derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S) ₊ ¹ condition. We generalize also an existence result of Cubiotti and Yao for generalized variational inequalities of class (S) ₊ ¹ to barreled normed spaces. As consequences, some existence results for vector variational inequalities are established.This work was partially supported by grants from the National Science Council of the Republic of China. Communicated by H. P. Benson     

Some topologies on the spaces of USCO maps and densely continuous forms

We study the completeness of three (metrizable) uniformities on the sets D(X, Y) and U(X, Y) of densely continuous forms and USCO maps from X to Y: the uniformity of uniform convergence on bounded sets, the Hausdorff metric uniformity and the uniformity U _B . We also prove that if X is a nondiscrete space, then the Hausdorff metric on real-valued densely continuous forms D(X, ?) (identified with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y) equipped with the Hausdorff metric is dense equicontinuity introduced by Hammer and McCoy in [7].     

On the duals of Lp spaces with 0<p<1

Given a measure space < Ω,m,μ >, a locally bounded, Hausdorff topological linear space < X, τ > and a real number 0<p<1, one can define the space L_p(Ω,m,μ,X), which is, under certain assumptions, a Fréchet space if endowed with a suitable topology. M.M. Day [1] has given a necessary and sufficient condition, in terms of the properties of the measure space < Ω,m,μ >, for the dual of L_p(Ω,m,μ,C) to be trivial. In this paper a different proof along with a slight generalization is given for this result, using standard and elementary measure theoretic arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

Existence and stability of fixed point set of Suzuki-type contractive multivalued operators in <Emphasis Type="Italic">b</Emphasis>-metric spaces with applications in delay differential equations

In this paper, we introduce a class of Suzuki-type contractive multivalued mappings and establish the existence of fixed points of such mappings in the setup of b-metric spaces. Some examples are presented to support the results proved herein. An estimate of Hausdorff distance between the fixed point sets of two Suzuki-type contractive multivalued mappings is obtained. As an application of results thus obtained, existence and uniqueness of periodic solution of delay differential equations are shown.     

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P ₀(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P ₀(X) if and only if d ^–1 is hereditarily precompact.     

Products of resolvents and multivalued hybrid mappings in CAT(0) spaces

In this article, we introduce and investigate the concept of multivalued hybrid mappings in CAT(0) spaces by using the concept of quasilinearization. Also, we present a new iterative algorithm involving products of Moreau-Yosida resolvents for finding a common element of the set of minimizers of a finite family of convex functions and a common fixed point of two multivalued hybrid mappings in CAT(0) spaces.     

Weak selections and weak orderability of function spaces

It is proved that for a zero-dimensional space X, the function space C _p (X, 2) has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if X is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space E, the function space C _p (X, E) is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial positive answer to a question posed by van Mill and Wattel.     

Operads,configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, [`(C)]\bar C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A <sub>∞</sub>-algebras and their homotopy morphisms, (ii) the 2-coloured operad of L <sub>∞</sub>-algebras and their homotopy morphisms, and (iii) the 4-coloured operad of openclosed homotopy algebras and their homotopy morphisms. Two gadgets — a (coloured) operad of Feynman graphs and a de Rham field theory on [`(C)]\bar C — are introduced and used to construct quantized representations of the (fundamental) chain operad of [`(C)]\bar C which are given by Feynman type sums over graphs and depend on choices of propagators.     

Splittings of spaces CX

The n-fold free loop space ⁿSⁿX is for connected spaces X weakly equivalent to a simpler space C_nX, which has a natural filtration F_inr C_nX. It is well known that there is a splitting S^tF_r(C_nX) V _m=1 ^p S^t(F_m(C_nX)¦F_m–1(C_nX) inducing a stable splitting of C_nX. We give a simple construction for such a splitting with comparatively low estimates for the number t of necessary suspension coordinates.