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:X→R is a function satisfies the following properties:
p(αx)=αp(x),∀α>0 and x∈X,p(x+y)⩽p(x)+p(y)∀x,y∈X.
And let Y be a subspace of X, l:Y→R is a linear functional in Y which satisfies
l(y)⩽q(y),∀y∈Y.
Then there exists a linear functional ˜j:X→R, such that
˜l(y)=l(y),∀y∈Y. and ˜l(y)⩽p(x),∀x∈X.
[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 Y⫋X, Given x0∈X−Y (Obviously, x0≠0), define the subspace of X
Dom f:=(αx0+y)∣α∈R,y∈Y,
then Y⊂Dom f.
Claim 1. Exists linear functional f:Dom f→R, such that
f(y)=l(y) ∀y∈Yandf(x)⩽p(x) ∀x∈Dom 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),∀α∈R,y∈Y.
This inequality holds for α=0, thus λ only need to satisfy:
λ⩽α−1[p(αx0+y)−l(y)]=p(x0+α−1y)−l(α−1y),∀α>0,y∈Y;λ⩾α−1[p(αx0+y)−l(y)]=−p(−x0+α−1y)+l(−α−1y),∀α<0,y∈Y;
By linearity of l and sublinearity of p, we have
−p(−x0+u)+l(u)⩽p(x0+v)−l(v),∀u,v∈Y.
Let
a:=supu∈Y−p(−x0+u)+l(u)⩽b:=infv∈Yp(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 f→R 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) ∀y∈Yandf(x)⩽p(x) ∀x∈Dom f.
F≠∅ since l∈F. And F is partially ordered with relation ≼:
Dom f1⊂Dom f2,f1(x)=f2(x),∀x∈Dom f1.
Given a totally ordered subset E of F, let
Dom g:=⋃f∈EDom f,
then it is a subspace of X.
Claim 2. ∀x∈Dom g,
g(x):=f(x),∀f∈E s.t x∈Dom f
clearly define a linear functional g:Dom g→R, and ∀x∈Dom g, there is g(x)⩽p(x).
Let x∈Dom g such that x∈Dom f1 and x∈Dom f2, and f1,f2∈E, without loss of generality, assume f1≼f2. Then g(x)=f1(x)=f2(x)⩽p(x).
If x1∈Dom f1,x2∈Dom f2, then f1≼f2 implies (x1+x2)∈Dom f2, thus g(x1+x2)=f2(x1+x2)=f2(x1)+f2(x2)=g(x1)+g(x2); and ∀α∈R, 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 ˜l⊂X.
Claim 3. Dom ˜l=X, and ˜l∈F satisfies every property that we required.
Prove this claim by contradiction. Suppose that ˜l⫋X, then repeat step (i), a linear functional ˜f:Dom ˜f→R 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:X→R is a semi-norm in X. And let Y be a subspace of X, l:Y→C is a linear functional in Y which satisfies
|l(y)|⩽q(y),∀y∈Y.
Then there exists a linear functional ˜j:X→C, such that
˜l(y)=l(y),∀y∈Y, and |˜l(y)|⩽p(x),∀x∈X.
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:Y→K is a continuous linear functional. Then there exists a continuous linear functional ˜l:X→K that satisfies
˜l(y)=l(y)∀y∈Y, 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.
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.)
泛函中的重要定理之 Hahn-Banach 定理
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