泛函中的重要定理之 Hahn-Banach 定理

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Hahn-Banach Theorem is one of the core theorems of linear functional analysis. The Hahn-Banach Theorem in Vector Space is also called Analytic Form of Hahn-Banach Theorem. Two corollaries are especially important: Hahn-Banach Theorem in Normed Vector Space, and Geometric Form of Hahn-Banach Theorem.

Analytic Form

定理.[Hahn-Banach Theorem in Real Vector Space]

Let X be a real vector space, and p is a sublinear functional in X, that is, p:XR is a function satisfies the following properties:

p(αx)=αp(x),α>0 and xX,p(x+y)p(x)+p(y)x,yX.
And let Y be a subspace of X, l:YR is a linear functional in Y which satisfies
l(y)q(y),yY.
Then there exists a linear functional j~:XR, such that
l~(y)=l(y),yY. and l~(y)p(x),xX.

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为函数空间赋予拓扑结构 - 一些例子

对于函数空间, 在需要研究其某些特定的性质时, 对于这个空间赋予适当的拓扑结构, 是非常有用的一种方法. 一个常见的方面就是: 通过其拓扑在给定的义意下讨论函数族收敛的概念.

在线性泛函分析学习中学过的赋范线性空间, 就是通过对空间赋予范数来引入拓扑结构–由范数可以自然的诱导相应的度量, 度量则给出了空间的拓扑结构. 当然, 并非所有的函数空间的拓扑结构都可以由范数给出, 于是我们有更一般的关联于所谓半范数族的拓扑线性空间.

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