一致空间完备性的几种等价形式与空间(B_β(X,Y),‖‖_γ)的完备性定理

Some Equivalent Forms of Completeness of Uniform Spaces and the Completeness Theorem of (B_β(X,Y),‖‖_γ)

第一部分给出一致空间聚点完备、滤了基完备的两种完备新定义 ,进而证明了这两种完备与通常完备等价 .据此在第二部分中使用聚点方法证明了当空间 (Y,‖‖y)完备时算子空间(Bβ(X ,Y) ,‖‖y)

In the first part of this paper,two new forms of completeness,i.e.accumulation complete and filter-base complete about uniform spaces are defined,then it is proved that the two new forms of completeness are equivalent to the ordinary completion.As an application of the equivalence theorem,by means of accumulation method it is proved that the space (B β(X,Y),‖‖ γ) is complete whenever (Y,‖‖ γ) is complete.