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Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence theory of metric measure spaces. We investigate several topological properties on the space equipped with these distance functions toward a deep understanding of convergence theory. [ABSTRACT FROM AUTHOR]

The paper examines fixed point outcomes for the two-variable operator F acting in Fréchet algebras. These results can be viewed as modifications of the fixed point theorem for the C + A B operator within a wider context. Through our primary findings, we can establish the existence of solutions for nonlinear integral equations with deviations that are not bounded. The proofing technique employed also allows for an added description of the asymptotic stability of the solutions of the relevant equation. [ABSTRACT FROM AUTHOR]

Let A be a radical Banach algebra and A ′ its topological dual. We show that the multiplication in A is weakly continuous if, and only if, A, endowed with its weak topology σ (A , A ′) , is isomorphic to a subalgebra of a product of finite dimensional radical algebras. This answers an open question in the literature. We also show that such an algebra is either finite dimensional or possesses uncountably many closed ideals of finite codimension. We then obtain that a radical Banach algebra with weakly continuous multiplication admits at least one maximal closed ideal. We thus provide a further class of radical Banach algebras, where the answer to the long standing problem of existence of closed ideals is in the affirmative, as for compact ones. [ABSTRACT FROM AUTHOR]

We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior. [ABSTRACT FROM AUTHOR]

In this paper, we compare the Stone–Čech compactification of the space of Radon probability measures on a Tychonoff space , equipped with the weak topology, with the space of Radon probability measures on the Stone–Čech compactification of the space . It is shown that for any noncompact metric space , the compactification does not coincide with . We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds. [ABSTRACT FROM AUTHOR]

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

The paper is devoted to prove the existence of solutions of a nonlinear integral equation in a WC-Banach algebra. Making use of the measure of weak noncompactness and the weak topology, we prove some fixed point theorems for operators not necessarily weakly sequentially continuous acting in WC-Banach algebra involving weakly compact operators. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the continuity of positive periodic solutions for Lotka–Volterra equations in coefficient functions. It will be proven that these positive periodic solutions are continuous in coefficient functions with respect to weak topologies. [ABSTRACT FROM AUTHOR]

In this paper, we consider solvability of the Cauchy initial problem d x / dt = f (t , x) , x (0) = x 0 in a Banach space X. As a result, we generalize Peano's existence theorem in the following manner: For every Banach space X, the problem always has a solution x ∈ C 1 ([ 0 , a ] , X) for all a > 0 under the assumption that f : R + ⊕ X → X is weak-to-weak continuous on some bounded set with a relatively weakly compact range. We also show that for any infinite dimensional reflexive Banach space X with an unconditional basis, in particular, ℓ p and L p (Ω , ∑ , μ) ( 1 < p < ∞ and (Ω , ∑ , μ) is σ -finite), and for all a > 0 there is a bounded nowhere locally Lipschitz function f : R + ⊕ X → X which is weak-to-weak continuous on some bounded set so that the Cauchy initial problem has a solution x ∈ C 1 ([ 0 , a ] , X) . [ABSTRACT FROM AUTHOR]

We provide an exhaustive description of the class of locally convex spaces in which all Archimedean cones are closed. We introduce the notion of quasidense set and prove that the above class consists of all finite-dimensional and countable-dimensional spaces whose topological dual is quasidense in the algebraic dual of . [ABSTRACT FROM AUTHOR]

Let E and F be Banach spaces. We present some results about the topology induced on E by the strong operator topology on the space L (L (E , F) , F) of continuous linear operators from L (E , F) into F , and study several classes of homogeneous polynomials from E into F enjoying different types of w EF continuity. [ABSTRACT FROM AUTHOR]

A topology is said to be strongly useful if every weakly continuous preorder admits a continuous order-preserving function. A strongly useful topology is useful, in the sense that every continuous total preorder admits a continuous utility representation. In this paper, I study the structure of strongly useful topologies. The existence of a natural one-to-one correspondence is proved, between weakly continuous preorders and equivalence classes of families of complete separable systems. In some sense, this result completely clarifies the connections between order theory and topology. Then, I characterize strongly useful topologies and I present a property concerning subspace topologies of strongly useful topological spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. We introduce the concept of weak open and closed countable chain condition (WOCCC) relative to a topology, and we then show that a useful topology always satisfies this condition. The most important result in the paper shows that a completely regular topology is useful if and only if it is separable and it satisfies WFOCCC (a stricter version of WOCCC). In this way, we generalize all the previous results concerning useful topologies. We finish the paper by presenting a simple axiomatization of useful topologies under the well-known Souslin Hypothesis. [ABSTRACT FROM AUTHOR]

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake a systematic study of those lcs E such that (E, w) is (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). The following results obtained in our paper play a key role: for every barrelled lcs E, the space (E, w) is Eberlein–Grothendieck (linearly Eberlein–Grothendieck) if and only if E is metrizable (E is normable, respectively). The main applications concern to the space of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology C k (X) . We prove that (C k (X) , w) is Eberlein–Grothendieck (linearly Eberlein-Grothen—dieck) if and only if X is hemicompact (X is compact, respectively). Besides this, we show that the class of E for which (E, w) is linearly Eberlein–Grothendieck preserves linear continuous quotients. Various illustrating examples are provided. [ABSTRACT FROM AUTHOR]

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of N-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of , we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set A in Riesz spaces lp (1 < p < ∞) is relatively sequential weakly compact if and only if it is normed bounded, that says sup . The result again confirms the conclusion of the Banach-Alaoglu theorem. [ABSTRACT FROM AUTHOR]

In this paper, we present a new simple axiomatization of useful topologies, i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific application to optimization theory, we characterize the continuous representability of all continuous total preorders, which admit both a maximal and a minimal element. [ABSTRACT FROM AUTHOR]

We present several random fixed point theorems for random operators with deterministic or stochastic domains. The main assumptions of our results are formulated in terms of the weak topology and an axiomatic definition of the measure of weak noncompactness. The results herein extend in a broad sense some new and classical results in the literature. As an application, we discuss the solvability of a random Hammerstein integral equation with lack of compactness. [ABSTRACT FROM AUTHOR]

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time. [ABSTRACT FROM AUTHOR]

We compare the finite central truncations of a given matrix with respect to two non-equivalent Hilbert space norms. While the limit sets of the finite sections spectra are merely located via numerical range bounds, the weak $$*$$ -limits of the counting measures of these spectra are proven in general to be gravi-equivalent with respect to the logarithmic potential in the complex plane. Classical methods of factorization of Volterra type or Wiener-Hopf type operators lead to a series of effective criteria of asymptotic equivalence, or uniform boundedness of the two sequences of truncations. Examples from function theory, integral equations and potential theory complement the theoretical results. [ABSTRACT FROM AUTHOR]

Graham, Hamada, Kohr and Kohr studied the normalized time $$T$$ reachable families $$\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )$$ of the Loewner differential equation, which are generated by the Carathéodory mappings with values in a subfamily $$\Omega $$ of the Carathéodory family $${\mathcal {N}}_A$$ for the Euclidean unit ball $${\mathbb {B}}^n$$ , where $$A$$ is a linear operator with $$k_+(A)<2m(A)$$ ( $$k_+(A)$$ is the Lyapunov index of $$A$$ and $$m(A)=\min \{\mathfrak {R}\left\langle Az,z\right\rangle \big |z\in {\mathbb {C}}^n,\Vert z\Vert =1\}$$ ). They obtained some compactness and density results, as generalizations of related results due to Roth, and conjectured that if $$\Omega $$ is compact and convex, then $$\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )$$ is compact and $$\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},ex\,\Omega )$$ is dense in $$\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )$$ , where $$ex\,\Omega $$ denotes the corresponding set of extreme points and $$T\in [0,\infty ]$$ . We confirm this, by embedding the Carathéodory mappings in a suitable Bochner space. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a new family of measures of noncompactness in the space $${L^1_{\text{loc}}(\mathbb{R}_+)}$$ consisting of all real functions locally integrable on $${\mathbb{R}_+}$$ , equipped with a suitable topology. As an example of applications of the technique associated with that family of measures of noncompactness, we study the existence of solutions of a nonlinear Volterra integral equation in the space $${L^1_{\text{loc}}(\mathbb{R}_+)}$$ . The obtained result generalizes several ones obtained earlier with help of other methods. [ABSTRACT FROM AUTHOR]

In this paper, some infinite alternative theorems in locally convex vector spaces are proved and some applications of these theorems in optimization theory are addressed. Extensions of constraint qualification conditions, in the absence of differentiability using Clarke's generalized gradient and Mordukhovich's subdifferential, in Banach and Asplund spaces are introduced; and relationships between them are established. Some of these relations, are proved using the provided alternative theorems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate functional large deviation behaviors of some nonparametric function estimates. As a first step, we define a a vector process W and study its large deviation behavior in the space L× L× L with respect to the weak convergence topology. As by-products, we derive large deviation principles in the L-space equipped with the weak convergence topology simultaneously for several density and regression estimators built up using the delta-sequence estimation method. [ABSTRACT FROM AUTHOR]

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L p topologies, but also with respect to the weak topologies of the L p spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way. [ABSTRACT FROM AUTHOR]

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces Lγ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in Lγ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence. [ABSTRACT FROM AUTHOR]

We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows—provided the sequence is tight—from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the Λ-coalescents. We show that the Λ-coalescent defines an infinite (random) metric measure space if and only if the so-called “dust-free”-property holds. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the generalization ability of the empirical risk minimization technique in the framework of agnostic learning, and consider the support vector regression method as a special case. We give a set of analytic conditions that characterize the empirical risk minimization methods and their approximations that are distribution-free consistent. Then utilizing the weak topology of the feature space, we show that the support vector regression, possibly with a discontinuous kernel, is distribution-free consistent. Moreover, a tighter generalization error bound is shown to be achieved in certain cases if the value of the regularization parameter grows as the sample size increases. The results carry over to the ν-support vector regression. [ABSTRACT FROM AUTHOR]

We will introduce a type of Fredholm operators which are shown to have a certain continuity in weak topologies. From this, we will prove that the fundamental matrix solutions of k-th, k ⩾ 2, order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies. Consequently, Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies. Moreover, for the scalar Hill’s equations, Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues, and rotation numbers are all continuous in potentials with weak topologies. These results will lead to many interesting variational problems. [ABSTRACT FROM AUTHOR]

We study hypercyclicity and supercyclicity of weighted shifts on ℓ∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c0 or ℓ p (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ∞ is norm nowhere dense. Finally, we show that ℓ∞ supports an isometry that is weak * sequentially supercyclic. [ABSTRACT FROM AUTHOR]

Ansari and Bourdon showed that an isometry on an infinite dimensional Banach space cannot be norm supercyclic. However, in this paper, we show there does exist a weakly supercyclic isometry on the Banach space $$c_0 (\mathbb{Z}).$$ [ABSTRACT FROM AUTHOR]

We discuss Gossez's 'type (D)' maximal monotone multifunctions and the newer 'type (ED)' subfamily (for which an analog of the Brøndsted-Rockafellar property holds). We then discuss the 'locally maximal monotone' (= type (FP)) and 'maximal monotone locally' (= type (FPV)) multifunctions of Fitzpatrick-Phelps and Fitzpatrick-Phelps-Verona-Verona. Finally, we discuss the strongly maximal monotone multifunctions. We prove that every maximal monotone multifunction of type (D) is locally maximal monotone, and every maximal monotone multifunction of type (ED) is both maximal monotone locally and strongly maximal monotone. [ABSTRACT FROM AUTHOR]

An analog of the finite-dimensional theorem about the upper semicontinuity of the geometric difference of continuous multivalued maps for separable Banach spaces is obtained. Sufficient conditions for the continuity of the geometric difference of multivalued maps in finite-dimensional spaces without the “nonempty interior” condition are obtained. Examples that demonstrate the unimprovability of these results are given. [ABSTRACT FROM AUTHOR]

To solve a statistical decision problem from a Bayesian viewpoint, the decision maker must specify a probability distribution on the parameter space, his prior distribution. In order to analyze the influence of this prior distribution on the solution of the problem, Bernardo (1981) proposed to compare the results with those that one would obtain by using that prior distribution which maximizes the useful experimental information, thus introducing the concept of reference decision. This definition is too involved for most of the problems usually found in practice. Here we analyze situations in which it is possible to simplify the definition of the reference decision, and we provide conditions guaranteeing the existence of such decisions. [ABSTRACT FROM AUTHOR]

The subject of this paper is to study the problem of the minimum distance to the complement of a convex set. Nirenberg has stated a duality theorem treating the minimum norm problem for a convex set. We state a duality result which presents some analogy with the Nirenberg theorem, and we apply this result to polyhedral convex sets. First, we assume that the polyhedral set is expressed as the intersection of some finite collection of m given half-spaces. We show that a global solution is determined by solving m convex programs. If the polyhedral set is expressed as the convex hull of a given finite set of extreme points, we show that a global minimum for a polyhedral norm is obtained by solving a finite number of linear programs. [ABSTRACT FROM AUTHOR]

The article focuses on the properties of invertible transformations, T : M → M preserving a finite measure μ on M, known as automorphisms of the measure space(M, Σ, μ). It assigns meanings to the generic and nongeneric properties of the transformations' asymptotic or algebraic properties. It also describes construction of transformations with "unusual" properties in ergodic theory using Rudolph's counterexample machine.

The purpose of this paper is to study the following system of nonlinear operator equations for N-variable pseudo-contractive mappings T 1 , T 2 , ... , T N in a Banach space: T 1 (x 1 , x 2 , ... , x N) = x 1 , T 2 (x 1 , x 2 , ... , x N) = x 2 , T 3 (x 1 , x 2 , ... , x N) = x 3 , ⋯ T N (x 1 , x 2 , ... , x N) = x N. The existence theorems of solutions are proved using a comprehensive analytical method. To get the expected results, the reflexivity of the product of Banach space and Opial's condition of the product of Banach spaces are discussed. Meanwhile, the weak topology of the product of Banach spaces and normal structure of the product of Banach spaces are also discussed. The results of this article improve and extend the previous classical results given in the literature. [ABSTRACT FROM AUTHOR]