R Haydon, A Molto, J Orihuela
Let $Γ$ be a Polish space and let $K$ be a separable and pointwise compact set of real-valued functions on $Γ$. It is shown that if each function in $K$ has only countably many discontinuities then $C(K)$ may be equipped with a $T_p$-lower semicontinuous and locally uniformly convex norm, equivalent to the supremum norm.
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We use lexicographic products to give examples of compact spaces of first Baire class functions on a compact metric space that cannot be represented as spaces of functions with countably many discontinuities.
By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe \textit{rigorously} the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} Φ\circ | ΔX_s |$, where $X : Ω\times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $Φ: \R_+ \longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf. P…
We show that if K is Rosenthal compact which can be represented by functions with countably many discontinuities then every Radon measure on K is countably determined. We also present an alternative proof of the result stating that every Radon measure on an arbitrary Rosenthal compactum is of countable type. Our approach is based on some caliber-type properties of measures, parameterized by separable metrizable spaces.
We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^ω$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes of ``large'' sets and Kuczma--Ger classes in the topological vector spaces $\mathbb R^n$ for $n\le ω$.
Statistical stability for systems semi-conjugate to pre-piecewise \textit{convex or expanding} maps with countably many branches
本文研究半 conjugate 于预分段凸或扩张映射的系统统计稳定性，提出一般条件保证未扰动测度在适当拓扑下连续，并给出测度连续模的定量估计。
In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich's transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.
We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any…
Let $E(\mathscr{A})$ denote the shift-invariant space associated with a countable family $\mathscr{A}$ of functions in $L^{2}(\mathbb{H}^{n})$ with mutually orthogonal generators, where $\mathbb{H}^{n}$ denotes the Heisenberg group. The characterizations for the collection $E(\mathscr{A})$ to be orthonormal, Bessel sequence, Parseval frame and so on are obtained in terms of the group Fourier transform of the Heisenberg group. These results are derived using such type of results which were proved for twisted shift-invariant spaces and characterized in terms of Weyl transform. In the last sectio…
Here we present an overview of countably-normed spaces. We discuss the main topologies--weak, strong, and inductive--placed on the dual of a countably-normed space and discuss the sigma-fields generated by these topologies. In particular, we show that under certain conditions the strong and inductive topologies coincide and the sigma-fields generated by the weak, strong, and inductive topologies are equal. With these sigma-fields, we develop a Gaussian measure on the dual of a nuclear space. The purpose in mind is to provide the background material for many of the results used is White Noise A…
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternati…
We construct two Frechet algebras admitting countably many mutually inequivalent Frechet algebra topologies. The second example is a modification of the maiden (and first) example of a non-Banach Frechet algebra with two inequivalent Frechet algebra topologies, considered by Read to show that the famous Singer-Wermer conjecture does not hold in the Frechet case. However the conjecture does hold by the second example which also admits countably many mutually equivalent Frechet algebra topologies.
Convergence of Random Products of Countably Infinitely Many Projections
本文结合伪周期函数概念研究可数无穷多个投影的随机乘积的收敛性，对M. Sakai提出的相关问题给出了肯定回答，推广了投影乘积收敛性的已有结果。
We give examples of real Banach spaces with exactly infinite countably many complex structures and with $ω_1$ many complex structures.
Given a Banach space we consider the $σ$-ideal of all of its subsets which are covered by countably many hyperplanes and investigate its standard cardinal characteristics as the additivity, the covering number, the uniformity, the cofinality. We determine their values for separable Banach spaces, and approximate them for nonseparable Banach spaces. The remaining questions reduce to deciding if the following can be proved in ZFC for every nonseparable Banach space $X$: (1) $X$ can be covered by $ω_1$-many of its hyperplanes; (2) All subsets of $X$ of cardinalities less than ${\rm cf}([{\rm dens…
A hybrid method for countable equilibrium, variational inequality and maximal monotone inclusion problems with fixed point constraints
提出一种混合投影方法，在一致光滑和一致凸的实Banach空间中，逼近可数族广义非扩张型映射的不动点集、可数多个均衡问题解集、可数多个变分不等式问题解集和可数多个极大单调算子零点集的公共元，并证明强收敛性。
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