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泛函中的重要定理之 Hahn-Banach 定理

个人学习笔记整理而来, 内容如有错漏欢迎电邮联系. a collection of personal math notes.

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.

[证明.]
Two steps to complete this proof.

Step (i). Add one element to Y and form a larger vector space, and proof for this space.

Assume YX, Given x0XY (Obviously, x00), define the subspace of X

Dom f:=(αx0+y)αR,yY,

then YDom f.

Claim 1. Exists linear functional f:Dom fR, such that

f(y)=l(y) yYandf(x)p(x) xDom f.

The problem can be reduced to find a real number λ:=f(x0), such that

f(αx0+y)=f(αx0)+f(y)=αλ+l(y)p(αx0+y),αR,yY.

This inequality holds for α=0, thus λ only need to satisfy:

λα1[p(αx0+y)l(y)]=p(x0+α1y)l(α1y),α>0,yY;λα1[p(αx0+y)l(y)]=p(x0+α1y)+l(α1y),α<0,yY;

By linearity of l and sublinearity of p, we have
p(x0+u)+l(u)p(x0+v)l(v),u,vY.

Let
a:=supuYp(x0+u)+l(u)b:=infvYp(x0+v)l(v),
and chose λ that satisfies a<λ<b will done.

Step (ii). functionals that satisfy previous property form a set with appropriate order, Zorn’s Lemma gives the existence of ˜l.

All the linear functionals f:Dom fR defined on subspaces of X which contains Y and satisfy the following properties form a set, we denote it by F:
f(y)=l(y) yYandf(x)p(x) xDom f.

F since lF. And F is partially ordered with relation :
Dom f1Dom f2,f1(x)=f2(x),xDom f1.

Given a totally ordered subset E of F, let
Dom g:=fEDom f,
then it is a subspace of X.

Claim 2. xDom g,
g(x):=f(x),fE s.t xDom f
clearly define a linear functional g:Dom gR, and xDom g, there is g(x)p(x).

Let xDom g such that xDom f1 and xDom f2, and f1,f2E, without loss of generality, assume f1f2. Then g(x)=f1(x)=f2(x)p(x).

If x1Dom f1,x2Dom f2, then f1f2 implies (x1+x2)Dom f2, thus g(x1+x2)=f2(x1+x2)=f2(x1)+f2(x2)=g(x1)+g(x2); and αR, g(αx1)=f1(αx1)=αf1(x)=αg(x1). [This shows that g(x) is a linear functional indeed!]

By construction of g, it is an upper bound of E. By Zorn’s Lemma, F has a maximum element ˜l, it is defined on a subspace Dom ˜lX.

Claim 3. Dom ˜l=X, and ˜lF satisfies every property that we required.

Prove this claim by contradiction. Suppose that ˜lX, then repeat step (i), a linear functional ˜f:Dom ˜fR would exist, and it satisfies all the related properties, and ˜l˜f. Hence the claim holds.

Theorem is now proved.

Note. Norm and Semi-norm are examples of sublinear functional, but the concept of sublinear functional is more general, since p(αx)=|α|p(x) is only required for α>0.

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

Let X be a complex vector space, and p:XR is a semi-norm in X. And let Y be a subspace of X, l:YC is a linear functional in Y which satisfies

|l(y)|q(y),yY.

Then there exists a linear functional ˜j:XC, such that

˜l(y)=l(y),yY, and |˜l(y)|p(x),xX.

Hahn-Banach Theorem in Normed Vector Space

Denote the dual space of a normed space X by X.

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

Let X be a normed vector space, Y is a subspace of X, and let l:YK is a continuous linear functional. Then there exists a continuous linear functional ˜l:XK that satisfies

˜l(y)=l(y)yY, and 

证明. [Here prove the complex case] Let X be a complex normed vector space.

Some notions need to be clarified first: \|\cdot\| is the norm in X; \|\cdot\| _{X^{*}} is the norm in X^{*}; so is \|\cdot\| _{Y^{*}}; and |\cdot| is the norm of complex numbers.

\forall x\in X, let p(x):=\|l\|_{Y^*}\|x\|. p:X\rightarrow\mathbb{C} is a norm (while l\neq 0), thus a sublinear functional. As l is continuous, then

|l(y)|\leqslant\|l\|_{Y^*}\|x\|\quad \forall y\in Y.

By Theorem\ref{THM:HB-Complex}, there exists a linear functional \widetilde{l}:X\rightarrow\mathbb{C} that satisfies

\widetilde{l}(y)=l(y),\quad \forall y\in Y,\quad \text{ and } \left|\widetilde{l}(x)\right|\leqslant p(x) = \|l\|_{Y^*}\|x\|,\quad \forall x\in X.

Hence \widetilde{l}(x) is continuous and

\|l\| _{Y ^*} = \sup _{y\in Y, y\neq 0}\frac{|l(y)|}{\|y\|}\leqslant \sup _{x\in X, x\neq 0}\frac{|l(x)|}{\|x\|} = \left\| \widetilde{l} \right\| _{X ^*}\leqslant \|l\| _{Y^*}.

The real normed space case can be proved in similar way.

Remark. (1) In the case of Hilbert Space, Hahn-Banach Theorem can be proved in a simpler way without using the Axiom of Choice, Furthermore, it the uniqueness of expension is given.

(2) In general, norm-preserving extension is not necessarily unique. several examples can be found in \cite{ref1}\cite{ref2}. But with certain conditions, the uniqueness can be guaranteed (Corollary\ref{COR:HB-TF}).

Geometric Form

Several preparatory knowledge to be mentioned first.

定义.[Hyperplane]

Let X be a normed vector space. A Hyperplane is a set defined as

H:=x\in X\mid f(x)=\alpha},

in which f is a linear functional that is not always zero, and \alpha\in \mathbb{R}. [f=\alpha] is called the equation of hyperplane H.

命题. Hyperplane [f=\alpha] is closed if and only if f is continuous.

定义. [Separation, Strictly Separation]
Let A,B\subset X. A, B are said to be separated by hyperplane [f=\alpha] if

f(x)\leqslant\alpha,\quad \forall x\in A,\quad \text{and } f(x)\geqslant \alpha,\quad \forall x\in B.

A, B are said to be strictly separated by hyperplane [f=\alpha] if \exists \varepsilon>0, such that
f(x)\leqslant\alpha-\varepsilon,\quad \forall x\in A,\quad \text{and } f(x)\geqslant \alpha+\varepsilon,\quad \forall x\in B.

fig1

fig1

定理. [1st Geometric Form]

Let A\subset X, B\subset X are two nonempty convex set. If A is an open set, then there exists a hyperplane which separates A and B.

定理. [2nd Geometric Form]

Let A\subset X, B\subset X are two nonempty convex set. If A is an closed set and B is a compact set, then there exists a hyperplane which strictly separates A and B.

Note. 更多关于 Hahn-Banach 型定理以及 “凸空间”的内容和结果将在另一份笔记中加以陈述.(这些内容非常重要且有意义.)

作者

Zengfk

发布于

2022-07-15

更新于

2022-09-15

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