Hahn-Banach延拓定理的另一形式

A Version of Hahn-Banach Extension Theorem

运用Zorn引理,研究了算子在延拓过程中是否保持序关系,解决了在次线性算子的控制下正保序算子的延拓问题,得到了如下的结论:设X和Y是Banach格,且X是可分的,Y具有Cantor性质.P:X→Y+是绝对且连续的次线性算子,T:X→Y是正线性算子.如果X0是X的一个线性子空间,V是从X0到Y的连续线性算子,满足在X0上V≥T且对于任意x∈X0有V(x)≤P(x),则V在P的控制下可连续延拓到整个空间,且延拓算子仍满足原有的序关系.

Based on Zorn’s lemma, the problem whether the operator is order-preserving in the extension or not was studied and was studied the extension problem of positive operators preserving order under the domination of a sublinear operator was solved . X and Y are Banach lattices, such that X is separable and Y has the Cantor property. Let P:X→Y^+ be a continuous sublinear operator such that P is absolute and T:X→Y be a positive linear operator. The main result in this paper was that if X_0 is a vector subspace of X and V:X_0→Y is a continuous operator with V≥T｜_(X_0) and V(x)≤P(x) for all x∈X_0, then there exists a continuous extension V^ of V to all of X also satisfying V^≥T and V^(x)≤P(x) for all x∈X.