泛函分析报告结课论文设计

泛函分析结课论文Functional Analysis Course Paper学号一、泛函分析空间理论泛函中四大空间的认识第一部分我们将讨论线性空间，在线性空间的基础上引入长度和距离的概念，进而建立了赋线性空间和度量空间。

在线性空间中赋以“数”，然后在数的基础上导出距离，即赋线性空间，完备的赋线性空间称为巴拿赫空间。

数可以看出长度，赋线性空间相当于定义了长度的空间，所有的赋线性空间都是距离空间。

在距离空间过距离的概念引入了点列的极限，但是只有距离结构、没有代数结构的空间，在应用过程中受到限制。

赋线性空间和积空间就是距离结构与代数结构相结合的产物，较距离空间有很大的优越性。

赋线性空间是其中每个向量赋予了数的线性空间，而且由数诱导出的拓扑结构与代数结构具有自然的联系。

完备的赋线性空间是Banach空间。

赋线性空间的性质类似于熟悉的n R，但相比于距离空间，赋线性空间在结构上更接近于n R。

赋线性空间就是在线性空间中，给向量赋予数，即规定了向量的长度，而没有给出向量的夹角。

在积空间中，向量不仅有长度，两个向量之间还有夹角。

特别是定义了正交的概念，有无正交性概念是赋线性空间与积空间的本质区别。

任何积空间都赋线性空间，但赋线性空间未必是积空间。

距离空间和赋线性空间在不同程度上都具有类似于nR 的空间结构。

事实上，nR 上还具有向量的积，利用积可以定义向量的模和向量的正交。

但是在一般的赋线性空间中没有定义积，因此不能定义向量的正交。

积空间实际上是定义了积的线性空间。

在积空间上不仅可以利用积导出一个数，还可以利用积定义向量的正交，从而讨论诸如正交投影、正交系等与正交相关的性质。

Hilbert 空间是完备的积空间。

与一般的Banach 空间相比较，Hilbert 空间上的理论更加丰富、更加细致。

1 线性空间（1）定义：设X 是非空集合，K 是数域，X 称为数域上K 上的线性空间，若,x y X ∀∈，都有唯一的一个元素z X ∈与之对应，称为x y 与的和，记作z x y =+,x X K α∀∈∈，都会有唯一的一个元素u X ∈与之对应，称为x α与的积，记作u x α=且,,x y z X ∀∈，,K αβ∈，上述的加法与数乘运算，满足下列8条运算规律： 10 x y y x +=+20 ()()x y z x y z ++=++30 在X 中存在零元素θ，使得x X ∀∈，有x x θ+= 40 x X ∀∈，存在负元素x X ∀-∈，使得()x x θ+-= 50 1x x ⋅=60 ()()x x αβαβ= 70 ()+x x x αβαβ+= 80 ()x y x y ααα+=+当K R =时，称X 为实线性空间；当K C =时，称X 为复线性空间 （2）维数：10 设X 为线性空间，12,,,n x x x X∈若不存在全为0的数12,,,n Kααα∈，使得11220n n x x x ααα+++=则称向量组12,,,nx x x 是线性相关的，否则称为线性无关。

20 设x X ∀∈，若12,,,n K ααα∈，12,,,n x x x X ∈使得1122n n x x x x ααα=+++则称x 可由向量组12,,,nx x x 线性表示。

30 设X 为线性空间，若在X 中存在X 个线性无关的向量，使得X 中任一向量可有n 个向量线性表示，则称其为X 的一个基，称n 为X 的维数。

2 距离空间设X 是非空集合，若存在一个映射:d X X R ⨯→，使得,,x y z X ∀∈，下列距离公理成立：10 非负性(,)0,(,)=0d x y d x y x y ≥⇔= 20 对称性(,)(,)d x y d y x =30 三角不等式(,)(,(,)d x y d x z d z y ≤+则称(,)d x y 为x y 与的距离，X 为以d 的距离空间，记作(,)X d 。

3 赋线性空间设X 称为数域上K 上的线性空间，若x X ∀∈，都有一个实数x与之对应，使得,,x y X K α∀∈∈，下列数公理成立： 10 正定性0,00x x x ≥=⇔=20 绝对齐次性x xαα=30 三角不等式x y x y+≤+则称x为x 上的数，X 为K 上的赋空间。

已知完备的距离空间中任一Cauchy 列均收敛，而赋线性空间作为一类特殊的距离空间，同样可以讨论它的完备性。

只是这里的距离是由数诱导的距离。

在数的语言下，点列{}n x 为Cauchy 列的定义改写为0,,N N ε∀>∃∈N 使当m,n>时，有m n x x ε-<完备的赋线性空间称为Banach 空间。

4 积空间设X 称为数域上K 上的线性空间，若存在映射,<⋅⋅>：X X K ⨯→，使得,,x y z X ∀∈，,K αβ∈，，下列积公理成立： 对第一变元的线性,,,x y z x z y z αβαβ<+>=<>+<> 共轭对称性,,x y y x <>=<>正定性,0x x <>≥且,00x x x <>=⇔= 则称,<⋅⋅>为X 上的积，X 为K 上的积空间。

由于完备性的概念是建立在距离定义的基础上的，故等价的说，一个积空间称为Hilbert 空间，若其按由积导出的数是完备的距离空间。

在由积导出的数下，积空间X 成为一个赋空间，它具有一般赋空间的所有性质。

二、 有界线性算子和连续线性泛函在线性代数中，我们曾遇到过把一个n 维向量空间n E 映射到另一个m 维向量空间m E 的运算，就是借助于m 行n 列的矩阵111212122212n n m m mn a a a a a a A a a a ⎛⎫⎪ ⎪= ⎪⎪⎝⎭对nE 中的向量起作用来达到的。

同样，在数学分析中，我们也遇到过一个函数变成另一个函数或者一个数的运算，即微分和积分的运算等。

把上述的所有运算抽象化后，我们就得到一般赋线性空间中的算子概念。

撇开各类算子的具体属性，我们可以将它们分成两类：一类是线性算子；一类是非线性算子。

本章介绍有界线性算子的基本知识，非线性算子的有关知识留在第5章介绍。

[定义3.1] 由赋线性空间X 中的某子集D 到赋线性空间Y 中的映射T 称为算子，D 称为算子T 的定义域，记为()D T ，为称像集(){},y y Tx x D T =∈为算子的值域，记作()T D 或TD 。

若算子T 满足：（1）()()(),T x y Tx Ty x y D T +=+∀∈ （2）()()(),T x TxF x D T ααα=∀∈∈称T 为线性算子。

对线性算子，我们自然要求()T D 是X 的子空间。

特别地，如果T 是由X 到实数（复数）域F 的映射时，那么称算子T 为泛函。

我们已经在第一章引入了线性算子与线性泛函的概念，同时也介绍了算子的连续性概念. 现在让我们给出连续线性算子与连续线性泛函的一种形式上不同的定义，在基本空间是度量 空间的情况下，它们在实质上是等价的.定义 1 设 X ，Y 是线性赋空间，T ：X →Y 是线性算子. T 称为是有界的，若对于 X中的任一有界集 A ，T(A)是 Y 中的有界集. 注意应该把这一定义中的有界算子的概念与数学分析中有界函数的概念加以区别，后者是指在整个定义域中所取的值为有界的函数. 同时要把线性算子与初等数学中所定义的线 性函数加以区别，后者是指形如 f (x)axb 的所有函数. 但只有在 b=0 的情况，它才是我 们定义的线性算子.三、 Hilbert 空间主要结论一个Hilbert space 的对偶空间（就是所有它的线性连续泛函组成的空间）等价于它自身，进一步，所有的线性连续泛函 I （f ）: H---> R 可以表示成为积的形式: I （f ）=<f ，g*> for some g* in H 。

（对了在这里再重新提一下，常用的平方可积函数空间L^2的积是积分的形式: ∫f*g ，f ，g ∈L^2，所以所有的线性连续泛函就都是带一个因子g 的积分了.） 这个Hilbert space 上最根本的定理几乎把Hilbert space 和Euclidean space （欧几里得空间）等同起来了，在那时大家都很高兴，毕竟Euclidean space 的性质我们了解的最多，也最“好”。

狄立克莱（Dirichlet ）原理就是在这个背景下提出的：任何连续泛函在有界闭集上达到其极值。

这个结论在Euclidean space上是以公理的形式规定下来的（参见数学分析的实数基本定理部分），具体说来就叫做有界闭集上的连续函数必有极值，而且存在点使得这个函数达到它。

在拓扑学上等价于局部紧性的这个东东，很可惜在一般的Hilbert space上却是不成立的：闭区间[0，1]上的L^2空间有一个很自然的连续泛函：I（f）=∫|f（x）|dx。

容易证明，它的数‖I‖=sup|I（f）|/‖f‖=1.在这个L^2的单位闭球面（所有数等于1 的f）上存在这么一个子序列：f_n（x）=n，当x∈[0，1/n^2]; f_n（x）=0，当x>1/n^2。

按照L^2上数的定义，‖f_n‖=∫f^2（x）dx =1，for all n。

0≤I （f）==>I在这个有界闭集上的最小值≤0，而且I（f_n）=1/n→0。

但是我们看到，当f_n弱收敛到常函数零时，它已经不在单位闭球面上了（严格的证明可以在一些课本上找到）。

一、定义线性完备积空间称为Hilbert space。

线性（linearity）：对任意f，g∈H，a，b∈R，a*f+b*g仍然∈H。

完备（completeness）：对H上的任意柯西序列必收敛于H上的某一点。

——相当于闭集的定义。

积（inner product）：一个从H×H-->R 的双线性映射，记为<f，g>。

它满足：i）<f，f>≥0，<f，f>=0 <==> f=0；ii）<a*f，g>=a*<f，g>=<f，a*g> for any a in R； iii）<f+g，h>=<f，h>+<g，h>；iv）<f，g>=<g，f> ——在复积里是复数共轭关系四、Banach空间主要结论Hahn-Banach 定理在理论上和应用上都是十分重要的，它往往提供了某些学科或学科分支的理论基础. 这里介绍一些它们在逼近论方面的应用.定义 3 设X 是线性赋空间，E是X 的子集合，x X ，称y E 是x 关于E的最佳逼近元，若x yinfz E xz. (1)首先应该知道一般说来，最佳逼近元并不总是存在的.例 1 设E C 0,1，E 是0,1上定义的任意阶多项式全体构成的线性子空间，取x t e t C 0,1，尽管d x, E infz Exz0 ，但不存在y E 使得xy0 ，因为e t 不是多项式. 这说明不存在e tn关于 E 的最佳逼近元.定理 1 实际上是最佳逼近元的判定定理. 下面定理可以看成最佳 逼近元的存在定理.定理 2 设 X 是线性赋空间， E X 是有限维子空间，则对于 每个 xX ， x 关于 E 的最佳逼近元存在.证 明 任取 y 0E ，考虑集合F zE ; xzxy 0.容易验证 F 是 E 中的有界 闭 集，是 E 有限 维的，从 而 是紧集并 且dx , Fdx , E . 取 z nF 使得 xz ndx , F ，此时存在子列 z z F ，于是kxz 0limnxz ndx , Fdx , E .z 0 即是 x 关于 E 的最佳逼近元.英文翻译部分First, the functional analysis space theoryUnderstanding of the four major functional spaceThe first part we will discuss the linear space, linear space is introduced based on the concept of length and distance, thereby establishing a normed linear spaces and metric spaces.Linear space assigned to the "norm", and then on the basis of export norm distance, ie normed linear space, complete normed space is called a Banach space. Norm can be seen that the length of normed linear space is equivalent to define the length of the space, all the spaces are normed linear distance space.In the distance space introduced by the concept of distance limit point of the column, but only from the structure, there is no room algebraic structure is limited in the application process. Normed linear spaces and inner product space is the distance between the structure and the algebraic structure of the combination product, the more distance space has a big advantage.Normed linear space is given to each of the linear vector space norm, and the norm induced by the topology of algebraic structure has a natural link. Complete normed linear space is space. Nature normed linear space is similar to the familiar, but compared to the distance space, normed linear space is closer in structure. 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In particular, if it is the real number (plural) field mapping, then the operator is called functional.In the first chapter we have introduced the concept of linear operators and linear functional, but also introduces the concept of continuity of the operator. Now let us give a continuous linear operator with one form of continuous linear functionals different definition, in the case of basic space metric space, they are essentially equivalent.Definition 1. Let X, Y be normed linear space, T: X →Y is called a linear operator T is bounded, if for X.Either a bounded set A, T (A) Y is a bounded set. Note that this definition should be a distinction between the bounded operator concepts and mathematical analysis of the concept of bounded functions, the latterRefers to the definition of the entire field is taken bounded functions. At the same time make a linear operator and a linear function of elementary mathematics as defined distinction between the latter refers to the form f (x) ax b of All function, but only if b = 0, it is our definition of a linear operator.Three, Hilbert space of the main conclusionsA Hilbert space dual space (that is, all of its linear continuous functional spatial composition) equivalent to its own, further, all linear continuous functionals I (f): H ---> R can be expressed as an inner product form: I (f) = <f, g *> for some g * in H.(On the re-mention here, the commonly used square integrable function space L ^ 2 is the inner product integral form: ∫f * g, f, g∈L ^ 2, so all linear continuous functional on all is with a factor g of calculus.) on the Hilbert space of the most fundamental theorems almost Hilbert space and Euclidean space (Euclidean space) equated, at that time we are very happy, after all, the nature of Euclidean space we Learn the most, but also the most "good."Di Li Clay (Dirichlet) principle is proposed in this context: any continuous functional on a bounded closed set reaches its extreme value. 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