赋范空间中最小范数问题的研究  

On Minimum Morm Problem in Normed Space

作　　者：王立柱[1] 

机构地区：[1]沈阳师范大学数学与系统科学学院,沈阳110034

出　　处：《沈阳师范大学学报（自然科学版）》2010年第4期473-476,共4页Journal of Shenyang Normal University:Natural Science Edition

基　　金：国家自然科学基金资助项目(10471096);沈阳师范大学教改立项(JG2009-YAD15)

摘　　要：Hahn-Banach定理的延拓形式是在赋范线性空间中研究最优化理论的重要基础之一,同时也是研究最小范数问题的有力工具。在Hilbert空间中最小范数问题可以利用投影定理使问题得到完满的解决,但在一般的赋范空间及凸集上情况就变得复杂的多。在一般的赋范空间中解决最小范数问题的办法是利用对偶定理来解决,它同时涉及两个不同的赋范空间对问题的研究带来一定的麻烦。这里从另外一个角度,利用Hahn-Banach定理的几何形式来研究最小范数问题,它的优点在于摆脱了同时设想在两个不同的赋范空间中考虑问题带来的烦杂。另外,把原赋范空间与其对偶空间的有关对象很好的结合在了一起,对最小范数问题给出了直观的几何解释。Hahn-Banach continuation is important for us to research optimization theory on normed space.At the same time,it is a powerful tools to research minimum norm.Minimum norm problem in Hilbert can be excellently solved through projection theorem.However,it is complex on the ordinary normed space and convex set.In ordinary normed space,minimum norm problem can work out by Duality theorem,but involving two different normed spaces.From another point of view,this paper discuss the minimum norm through formal geometry of Hahn-Banach theorem.It is easy to get out of trouble in considering two different normed spaces.In addition, formal geometry of Hahn-Banach theorem is associated with related objects in original space with objects in dual space,so geometric interpretation is given for the problem of minimum norm.

关 键 词：HAHN-BANACH定理 最小范数 超平面 凸集 

分 类 号：O177[理学—数学]