赋β-范线性空间上的齐性算子性质初探

Some properties of homogeneous operator in β-normal linear space

证明了赋β-范空间上的有界齐性算子与在零点连续的齐性算子等价 ;对两个赋β-范空间 X和 Y之间的有界性算子全体 B(X,Y) ,按引入的算子范数及线性运算 ,在 X具有共轭分离性时 ,B(X,Y)为赋 β-范线性空间 ;指出 B(X,Y)完备与 Y完备是等价的 ,只要 X具有共轭分离性 .这些推广了赋范空间上的关于有界线性算子已有的结论 .

The equivalent relation of boundedness and continuity at zero for homogeneous operator in β normal linear space is proved.Let X and Y be two β normal linear spaces and B(X,Y) the set of all bounded homogeneous operators between X and Y.According to the operator norm and linear operation introduced,it is proved that B(X,Y) is a β normal linear space if X is dual separated.It is pointed out that B(X,Y) is complete if and only if the space Y is complete if X is dual separated of bound linear operator in normal linear space.