On Functions of Finite Baire Index

Functions of Baire class one

Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $β$ and the convergence index $γ$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$ for some countable ordinals $ξ_1$ and $ξ_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise su…

具有任意有限Baire阶的Banach空间

Banach spaces with arbitrary finite Baire order

本文构造了可分Banach空间，解决了关于内在Baire阶的开放问题，并推广了Lindenstrauss的结果，表明任何具有基的Banach空间可作为对偶空间中特定Baire类的泛函空间。

On Baire measurability in spaces of continuous functions

Let C(K) be the Banach space of all continuous functions on a given compact space K. We investigate the w*-sequential closure in C(K)* of the set of all finitely supported probabilities on K. We discuss the coincidence of the Baire sigma-algebras on C(K) associated to the weak and pointwise convergence topologies.

Extension of Functions with Small Oscillation

A classical theorem of Kuratowski says that every Baire one function on a G_δsubspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that β_{Y}(f)<ω^α, α< ω_1, then f has an extension F onto X so that β_X(F)is no…

Index of symmetry and topological classification of asymmetric normed spaces

Let X, Y be asymmetric normed spaces and Lc(X, Y) the convex cone of all linear continuous operators from X to Y. It is known that in general, Lc(X, Y) is not a vector space. The aim of this note is to prove, using the Baire category theorem, that if Lc(X, Y) is a vector space for some asymmetric normed space Y , then X is isomorphic to its associated normed space (the converse is true for every asymmetric normed space Y and is easy to establish). For this, we introduce an index of symmetry of the space X denoted c(X) $\in$ [0, 1] and we give the link between the index c(X) and the fact that L…

拓扑阿贝尔群中的对偶性与类贝尔性质

Duality and Baire like properties in topological abelian groups

本文研究阿贝尔拓扑群的对偶性，引入$\omega$-桶状群并证明其蕴含$\aleph_0$-桶状性质，解决开放问题并刻画完全有界群紧子集有限性与对偶群类贝尔性质的关系。