赋范锥到赋范线性空间的嵌入定理与赋范锥上的Hahn-Banach定理

The Theorems of Embedding Normed Cones into Normed Linear Spaces and the Hahn-Banach Extension Theorems

研究赋范锥到赋范线性空间的嵌入问题与赋范锥上连续线性泛函的Hahn-Banach正延拓问题.第一部分采用几何方法直接证明赋范锥到赋范线性空间的嵌入定理.对于给定的赋范线性空间中的凸锥,通过引进凸锥的"锐性模".第二部分研究由锥范数导出的延拓范数与原范数的等价关系.第三部分给出赋范锥上连续线性泛函的Hahn-Banach正延拓定理.

The problems of embedding normed cones into normed linear spaces and the prob- lems of extending continuous linear functionals from normed cones to normed linear spaces are studied in this paper. In the first part, by geometric methods, the embedding theorems of normed cones into normed linear spaces are proved directly. In the second part, for a convex cone in a given normed linear space, via the SHARPNESS MODULUS of the convex cone, thc equivalent relation of the extension norm derived from the conical norm with the original norm is studied. The Hahn-Banach positive extension theorems of continuous linear functionals from normed cones to normed linear spaces are obtained at last.