Topology Of Uniform Convergence Research Articles

A note on the space of all Toeplitz operators

Abstract We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

The uniform convergence topology on separable subsets

The uniform convergence topology on separable subsets

Relative weak compactness in infinite-dimensional Fefferman-Meyer duality

Let E be a Banach space such that E′ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the E-valued martingale Hardy space H1(μ,E) to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in L1(μ,E). In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for H1(μ,E)-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of H1(μ,E)-boundedness.

Lipschitz-free spaces over properly metrizable spaces and approximation properties

Let T T be a topological space admitting a compatible proper metric, that is, a locally compact, separable and metrizable space. Let M T \mathcal {M}^T be the non-empty set of all proper metrics d d on T T compatible with its topology, and equip M T \mathcal {M}^T with the topology of uniform convergence, where the metrics are regarded as functions on T 2 T^2 . We prove that the set A T , 1 \mathcal {A}^{T,1} of metrics d ∈ M T d\in \mathcal {M}^T for which the Lipschitz-free space F ( T , d ) \mathcal {F}(T,d) has the metric approximation property is a dense set in M T \mathcal {M}^T , and is furthermore residual in M T \mathcal {M}^T when T T is zero-dimensional. We also prove that if T T is uncountable then the set A f T \mathcal {A}^T_f of metrics d ∈ M T d\in \mathcal {M}^T for which F ( T , d ) \mathcal {F}(T,d) fails the approximation property is dense in M T \mathcal {M}^T . Combining the last statement with a result of Dalet, we conclude that for any ‘properly metrizable’ space T T , A f T \mathcal {A}^T_f is either empty or dense in M T \mathcal {M}^T .

Baire 1 Functions and the Topology of Uniform Convergence on Compacta

Let X be a Tychonoff topological space, B1(X,R) be the space of real-valued Baire 1 functions on X and τUC be the topology of uniform convergence on compacta. The main purpose of this paper is to study cardinal invariants of (B1(X,R),τUC). We prove that the following conditions are equivalent: (1) (B1(X,R),τUC) is metrizable; (2) (B1(X,R),τUC) is completely metrizable; (3) (B1(X,R),τUC) is Čech-complete; and (4) X is hemicompact. It is also proven that if X is a separable metric space with a non isolated point, then the topology of uniform convergence on compacta on B1(X,R) is seen to behave like a metric topology in the sense that the weight, netweight, density, Lindelof number and cellularity are all equal for this topology and they are equal to c= |B1(X,R)|. We find further conditions on X under which these cardinal invariants coincide on B1(X,R).

Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded hyperbolic metric space X and a concave, nonzero and nondecreasing function ω:[0,+∞)→[0,+∞) with ω(0)=0 and study the space Cω(X) of uniformly continuous self-mappings on X whose modulus of continuity is bounded from above by ω. We endow Cω(X) with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of a generic mapping in Cω(X), in the sense of Baire categories, is precisely ω. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A.J. Zaslavski, C. Bargetz and D. Thimm.

Compact subsets of C λ,u (X)

Abstract The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in C λ,u (X) for a locally-λ space X when λ ⊇ 𝓕(X), for a hemi- λ λf -space X when λ ⊆ 𝓟 𝓢(X), and for a k-space X when λ ⊇ 𝓚(X). This paper also studies that every bounded subset of C λ,u (X) has compact closure for some classes of topological spaces X.

On Grothendieck-type duality for spaces of holomorphic functions of several variables

We describe the strong dual space $({\mathcal O} (D))^*$ of the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables in a bounded domain $D$ with Lipschitz boundary and connected complement (as usual, ${\mathcal O} (D)$ is endowed with the topology of local uniform convergence in $D$). We identify the dual space with the closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, whose elements vanish at the point at infinity and satisfy the Cauchy-Riemann tangential conditions on $\partial D$. In particular, we generalize classical Grothendieck-Köthe-Sebastião e Silva duality for holomorphic functions of one variable to the multivariate situation. We prove that the duality we produce holds if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of Sobolev-class holomorphic functions in $D$ is dense in ${\mathcal O} (D)$. Bibliography: 35 titles.

Lipschitz-Free Spaces over Cantor Sets and Approximation Properties

Let K=2^mathbb {N} be the Cantor set, let mathcal {M} be the set of all metrics d on K that give its usual (product) topology, and equip mathcal {M} with the topology of uniform convergence, where the metrics are regarded as functions on K^2. We prove that the set of metrics din mathcal {M} for which the Lipschitz-free space mathcal {F}(K,d) has the metric approximation property is a residual F_{sigma delta } set in mathcal {M}, and that the set of metrics din mathcal {M} for which mathcal {F}(K,d) fails the approximation property is a dense meager set in mathcal {M}. This answers a question posed by G. Godefroy.

Multipliers on Spaces of Holomorphic Functions

We consider multipliers on the space of holomorphic functions of one variable H(Omega ),,Omega subset mathbb {C} open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to Omega and Omega is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on H(Omega ). We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when Omega is convex.

Generalization of the Grothendieck's theorem

In this paper, we obtain a generalization of Grothendieck's theorem for the space of continuous mappings Cλ,μ(X,Y) where Y is a complete uniform space with the uniformity μ endowed with the topology of uniform convergence on the family λ of subsets of X. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type.The developed technique is used to generalize the Grothendieck's theorem for the space of continuous mappings endowed with the set-open topology.

Approximation by sums of shifts and dilations of a single function and neural networks

We find sufficient conditions on a function f to ensure that sums of functions of the form f(αx−θ), where α∈A⊂R and θ∈Θ⊂R, are dense in the real spaces C0 and Lp on the real line or its compact subsets. That is, we consider linear combinations in which all coefficients are 1. As a corollary we deduce results on density of sums of functions f(w⋅x−θ), w∈W⊂Rd, θ∈Θ⊂R in C(Rd) in the topology of uniform convergence on compact subsets.

Topological properties of quasiconformal automorphism groups

Let G subsetneq mathbb {C} be a bounded, simply connected domain in mathbb {C}, and denote by Q(G):=f:G⟶G|fis a quasiconformal mapping ofGontoG\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Q(G):= \\left\\{ f: G \\longrightarrow G \\ |\\ f \ ext { is a quasiconformal mapping of } G \ ext { onto } G \\ \\right\\} \\end{aligned}$$\\end{document}the quasiconformal automorphism group of G. In a canonical manner, the set Q(G) carries the structure of a (non–abelian) group with respect to composition of mappings. Moreover, we endow the set Q(G) with the topology of uniform convergence by the supremum metric dsup(f,g):=supz∈Gf(z)-g(z),f,g∈Q(G).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} d_{\\sup }(f, g):= \\sup \\limits _{z \\in G} \\left| f(z) - g(z) \\right| , f, g \\in Q(G). \\end{aligned}$$\\end{document}In this paper, we present results concerning topological properties of Q(G) such as completeness, separability, path–connectedness, discreteness and compactness.

Embedding theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space C(X,Y) of all continuous functions from a topological space X into a uniform space Y with the topology of uniform convergence on a family of subsets of X and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space C(X,Y) is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on C(X,Y).

Convolution on distribution spaces characterized by regularization

AbstractLocally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function‐valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex lattices of measures.

Tightness of discrete Gibbsian line ensembles

A discrete Gibbsian line ensemble L=(L1,…,LN) consists of N independent random walks on the integers conditioned not to cross one another, i.e., L1≥⋯≥LN. In this paper we provide sufficient conditions for convergence of a sequence of suitably scaled discrete Gibbsian line ensembles fN=(f1N,…,fNN) as the number of curves N tends to infinity. Assuming log-concavity and a KMT-type coupling for the random walk jump distribution, we prove that under mild control of the one-point marginals of the top curves with a global parabolic shift, the full sequence (fN) is tight in the topology of uniform convergence over compact sets, and moreover any weak subsequential limit possesses the Brownian Gibbs property. If in addition the top curves converge in finite-dimensional distributions to the parabolic Airy2 process, then a result of Dimitrov (2021) implies that (fN) converges to the parabolically shifted Airy line ensemble. These results apply to a broad class of discrete jump distributions, including geometric as well as any log-concave distribution whose support forms a compact integer interval.

Minimal cusco maps and the topology of uniform convergence on compacta

Minimal cusco maps have applications in functional analysis, in optimization, in the study of weak Asplund spaces, in the study of differentiability of functions, etc. It is important to know their topological properties. Let X be a Hausdorff topological space, MC(X) be the space of minimal cusco maps with values in R and ?UC be the topology of uniform convergence on compacta. We study completemetrizability and cardinal invariants of (MC(X), ?UC). We prove that for two nondiscrete locally compact second countable spaces X and Y, (MC(X), ?UC) and (MC(Y), ?UC) are homeomorphic and they are homeomorphic to the space C(Ic) of continuous real-valued functions on Ic with the topology of uniform convergence.

More on generalizations of topology of uniform convergence and m-topology on C(X)

This paper conglomerates our findings on the space C(X) of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the m-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of Z-ideals of C(X) induced by the UI and the mI-topologies on C(X) [5]. Motivated by the definition of the mItopology, another generalization of the topology of uniform convergence, called UI-topology, is introduced here. Among several other results, it is established that for a convex ideal I in C(X), a necessary and sufficient condition for UI-topology to coincide with mI-topology on C(X) is the boundedness of X ? T Z[I] in X. As opposed to the case of the UI-topologies (and mI-topologies) on C(X), it is proved that each UI-topology (respectively, mI-topology) on C(X) is uniquely determined by the ideal I. In the last section, the denseness of the set of units of C(X) in CU(X) (= C(X) with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space X. Also, the space X turns out to be a weakly P-space if and only if the set of zero divisors (including 0) in C(X) is closed in CU(X). Computing the closure of CP(X) (={ f ? C(X) : the support of f ? P} where P is an ideal of closed sets in X) in CU(X) and Cm(X) (= C(X) with the m-topology), the results clUCP(X) = CP? (X) (= { f ? C(X) : ?n ? N, {x ? X : | f (x)| ? 1n } ? P}) and clmCP(X) = { f ? C(X) : f.1 ? CP? (X) for each 1 ? C(X)} are achieved.

Fréchet subspaces of minimal usco and minimal cusco maps

Minimal usco/cusco maps are very important in functional analysis, optimization, selection theorems, the study of differentiability of Lipschitz functions, etc. We study topologies of uniform convergence on bornologies on the space of minimal usco and minimal cusco maps. We find sufficient conditions for metrizability and complete metrizability of minimal usco and minimal cusco maps equipped with the topology of uniform convergence on bornologies. We also investigate Fréchet locally convex subspaces of these spaces.

StatisticalLY τ-Bounded Operators on Ordered Topological Vector Spaces

In this paper, we introduce statistical bounded sets on topological vector space. Also, we give the three notions of bounded operators from topological vector spaces to ordered topological vector spaces. Moreover, we show some relations between these operators and order bounded operators. Also, we give some algebraic properties of these operators with respect to the uniform convergence topology.