Introduction to Functional Analysis
Introduction to Functional Analysis
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DOI：
10.1007/0-387-22620-6_7
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发表时间：
1997-10
期刊：
影响因子：
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通讯作者：
R. Meise;D. Vogt
中科院分区：
文献类型：
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作者：
R. Meise;D. Vogt
在这一章中，我们首先讨论了Hahn-Banach定理，其中最著名的情形给出了赋范空间X的子空间上的有界线性泛函可以在保持范数不变的情况下扩张为X上的有界线性泛函的条件。然后，我们给出了这个定理的几个应用，其中一些应用说明了赋范空间和它的对偶之间的相互作用。在第二节中，我们利用Hahn-Banach定理得到关于凸集的超平面分离的结果。本章的最后一节介绍了Baire范畴定理，并介绍了它在经典分析和泛函分析中的一些应用。
In this chapter we first discuss the Hahn–Banach Theorem, the most famous case of which provides conditions under which a bounded linear functional on a subspace of a normed space X can be extended, with preservation of its norm, to a bounded linear functional on the whole of X. We then present several applications of this theorem, some of which illustrate the interplay between a normed space and its dual. In Section 2 we use the Hahn–Banach Theorem to obtain results about the separation of convex sets by hyperplanes. The last section of the chapter introduces the Baire Category Theorem, and includes some of its many applications in classical and functional analysis.