点态性质与一致性质的一个新关系

A New Relation Between the Pointwise and Uniform Properties

为了研究点态性质和一致性质之间的关系,应用紧性性质在拓扑意义下得到逐点有界性质蕴含一致有界性质的结论,并将其推广到连续函数族的情况.基于Baire定理,得到完备度量空间中的逐点收敛性质可导出一致有界性质的结论.

In order to study the relations between the pointwise and uniform properties,we apply the compactness to obtain the result that the pointwise boundedness implies the uniform boundedness under the sense of topology,and then we extend it to the case of a family of continuous functions under certain conditions.Based on the famous Baire＇s theorem,we also verify the assertion that pointwise convergence can deduce the pointwise boundedness in complete metric spaces.