局部β-凸空间中β-次半范的Hahn-Banach延拓定理及其应用

The Hahn-Banach Theorems about β-Subseminorms In Locally β-convex spaces and Their Applications

研究了β-次半范的Hahn-Banach延拓问题,得到线性空间中β-次半范的控制延拓定理,连续β-次半范在局部β-凸空间中的连续延拓定理及在赋β-范空间中的保范延拓定理等.作为Hahn-Banach延拓定理的应用,最后证明了局部β-凸空间的共轭锥对空间本身的分离定理.

This paper deals with the extension problems about β- subseminorms, the control extension theorem of β - subseminorms in linear spaces, the Hahn - Banach continuous extension theorem of continous β- subseminorms in locally β- convex spaces and the norm - preserving extension theorem of continous β- subseminorms in β- normed spaces are obtained. As the application of Hahn - Banach extension theorems, the theorem of Xβ distinguishing X is obtained at the end of this paper.