1、泛函分析結(jié)課論文Functional Analysis Course Paper 學號 姓名 一、 泛函分析空間理論泛函中四大空間的認識第一部分我們將討論線性空間，在線性空間的基礎(chǔ)上引入長度和距離的概念，進而建立了賦范線性空間和度量空間。在線性空間中賦以“范數(shù)”，然后在范數(shù)的基礎(chǔ)上導出距離，即賦范線性空間，完備的賦范線性空間稱為巴拿赫空間。范數(shù)可以看出長度，賦范線性空間相當于定義了長度的空間，所有的賦范線性空間都是距離空間。在距離空間中通過距離的概念引入了點列的極限，但是只有距離結(jié)構(gòu)、沒有代數(shù)結(jié)構(gòu)的空間，在應用過程中受到限制。賦范線性空間和內(nèi)積空間就是距離結(jié)構(gòu)與代數(shù)結(jié)構(gòu)相結(jié)合的產(chǎn)物，較距離空間

2、有很大的優(yōu)越性。賦范線性空間是其中每個向量賦予了范數(shù)的線性空間，而且由范數(shù)誘導出的拓撲結(jié)構(gòu)與代數(shù)結(jié)構(gòu)具有自然的聯(lián)系。完備的賦范線性空間是空間。賦范線性空間的性質(zhì)類似于熟悉的，但相比于距離空間，賦范線性空間在結(jié)構(gòu)上更接近于。賦范線性空間就是在線性空間中，給向量賦予范數(shù)，即規(guī)定了向量的長度，而沒有給出向量的夾角。在內(nèi)積空間中，向量不僅有長度，兩個向量之間還有夾角。特別是定義了正交的概念，有無正交性概念是賦范線性空間與內(nèi)積空間的本質(zhì)區(qū)別。任何內(nèi)積空間都賦范線性空間，但賦范線性空間未必是內(nèi)積空間。距離空間和賦范線性空間在不同程度上都具有類似于的空間結(jié)構(gòu)。事實上，上還具有向量的內(nèi)積，利用內(nèi)積可以定義向量

3、的模和向量的正交。但是在一般的賦范線性空間中沒有定義內(nèi)積，因此不能定義向量的正交。內(nèi)積空間實際上是定義了內(nèi)積的線性空間。在內(nèi)積空間上不僅可以利用內(nèi)積導出一個范數(shù)，還可以利用內(nèi)積定義向量的正交，從而討論諸如正交投影、正交系等與正交相關(guān)的性質(zhì)。空間是完備的內(nèi)積空間。與一般的空間相比較，空間上的理論更加豐富、更加細致。1 線性空間（1）定義：設(shè)是非空集合，是數(shù)域，稱為數(shù)域上上的線性空間，若，都有唯一的一個元素與之對應，稱為的和，記作，都會有唯一的一個元素與之對應，稱為的積，記作且，上述的加法與數(shù)乘運算，滿足下列8條運算規(guī)律：10 20 30 在中存在零元素，使得，有40 ，存在負元素，使得50 60

4、 70 80 當時，稱為實線性空間；當時，稱為復線性空間（2）維數(shù)：10 設(shè)為線性空間，若不存在全為0的數(shù)，使得則稱向量組是線性相關(guān)的，否則稱為線性無關(guān)。20 設(shè)，若，使得則稱可由向量組線性表示。30 設(shè)為線性空間，若在中存在個線性無關(guān)的向量，使得中任一向量可有個向量線性表示，則稱其為的一個基，稱為的維數(shù)。2 距離空間設(shè)是非空集合，若存在一個映射，使得，下列距離公理成立：10 非負性20 對稱性30 三角不等式則稱為的距離，為以的距離空間，記作。3 賦范線性空間設(shè)稱為數(shù)域上上的線性空間，若，都有一個實數(shù)與之對應，使得，下列范數(shù)公理成立：10 正定性20 絕對齊次性30 三角不等式則稱為上的范數(shù)

5、，為上的賦范空間。已知完備的距離空間中任一列均收斂，而賦范線性空間作為一類特殊的距離空間，同樣可以討論它的完備性。只是這里的距離是由范數(shù)誘導的距離。在范數(shù)的語言下，點列為列的定義改寫為完備的賦范線性空間稱為空間。4 內(nèi)積空間設(shè)稱為數(shù)域上上的線性空間，若存在映射：，使得，下列內(nèi)積公理成立：對第一變元的線性共軛對稱性正定性且則稱為上的內(nèi)積，為上的內(nèi)積空間。由于完備性的概念是建立在距離定義的基礎(chǔ)上的，故等價的說，一個內(nèi)積空間稱為空間，若其按由內(nèi)積導出的范數(shù)是完備的距離空間。在由內(nèi)積導出的范數(shù)下，內(nèi)積空間成為一個賦范空間，它具有一般賦范空間的所有性質(zhì)。二、 有界線性算子和連續(xù)線性泛函在線性代數(shù)中，我們

6、曾遇到過把一個維向量空間映射到另一個維向量空間的運算，就是借助于行列的矩陣對中的向量起作用來達到的。同樣，在數(shù)學分析中，我們也遇到過一個函數(shù)變成另一個函數(shù)或者一個數(shù)的運算，即微分和積分的運算等。把上述的所有運算抽象化后，我們就得到一般賦范線性空間中的算子概念。撇開各類算子的具體屬性，我們可以將它們分成兩類：一類是線性算子；一類是非線性算子。本章介紹有界線性算子的基本知識，非線性算子的有關(guān)知識留在第5章介紹。定義3.1 由賦范線性空間中的某子集到賦范線性空間中的映射稱為算子，稱為算子的定義域，記為，為稱像集為算子的值域，記作或。若算子滿足：（1）（2）稱為線性算子。對線性算子，我們自然要求是的子

7、空間。特別地，如果是由到實數(shù)（復數(shù)）域的映射時，那么稱算子為泛函。我們已經(jīng)在第一章引入了線性算子與線性泛函的概念，同時也介紹了算子的連續(xù)性概念. 現(xiàn)在讓我們給出連續(xù)線性算子與連續(xù)線性泛函的一種形式上不同的定義，在基本空間是度量 空間的情況下，它們在實質(zhì)上是等價的.定義 1設(shè) X，Y 是線性賦范空間，T：XY 是線性算子.T 稱為是有界的，若對于 X中的任一有界集 A，T(A)是 Y 中的有界集. 注意應該把這一定義中的有界算子的概念與數(shù)學分析中有界函數(shù)的概念加以區(qū)別，后者是指在整個定義域中所取的值為有界的函數(shù). 同時要把線性算子與初等數(shù)學中所定義的線 性函數(shù)加以區(qū)別，后者是指形如 f (x)

8、= ax + b 的所有函數(shù). 但只有在 b=0 的情況，它才是我 們定義的線性算子. 三、 Hilbert空間主要結(jié)論 一個Hilbertspace的對偶空間（就是所有它的線性連續(xù)泛函組成的空間）等價于它自身，進一步，所有的線性連續(xù)泛函I（f）:H-R可以表示成為內(nèi)積的形式:I（f）=forsomeg*inH。（對了在這里再重新提一下，常用的平方可積函數(shù)空間L2的內(nèi)積是積分的形式:f*g，f，gL2，所以所有的線性連續(xù)泛函就都是帶一個因子g的積分了.）這個Hilbertspace上最根本的定理幾乎把Hilbertspace和Euclideanspace（歐幾里得空間）等同起來了，在那時大家都

9、很高興，畢竟Euclideanspace的性質(zhì)我們了解的最多，也最“好”。狄立克萊（Dirichlet）原理就是在這個背景下提出的：任何連續(xù)泛函在有界閉集上達到其極值。這個結(jié)論在Euclideanspace上是以公理的形式規(guī)定下來的（參見數(shù)學分析的實數(shù)基本定理部分），具體說來就叫做有界閉集上的連續(xù)函數(shù)必有極值，而且存在點使得這個函數(shù)達到它。在拓撲學上等價于局部緊性的這個東東，很可惜在一般的Hilbertspace上卻是不成立的：閉區(qū)間0，1上的L2空間有一個很自然的連續(xù)泛函：I（f）=|f（x）|dx。容易證明，它的范數(shù)I=sup|I（f）|/f=1.在這個L2的單位閉球面（所有范數(shù)等于1的f

10、）上存在這么一個子序列：f_n（x）=n，當x0，1/n2;f_n（x）=0，當x1/n2。按照L2上范數(shù)的定義，f_n=f2（x）dx=1，foralln。0I（f）=I在這個有界閉集上的最小值0，而且I（f_n）=1/n0。但是我們看到，當f_n弱收斂到常函數(shù)零時，它已經(jīng)不在單位閉球面上了（嚴格的證明可以在一些課本上找到）。一、定義線性完備內(nèi)積空間稱為Hilbertspace。線性（linearity）：對任意f，gH，a，bR，a*f+b*g仍然H。完備（completeness）：對H上的任意柯西序列必收斂于H上的某一點。相當于閉集的定義。內(nèi)積（innerproduct）：一個從HH-

11、R的雙線性映射，記為。它滿足：i）0，=0f=0；ii）=a*=foranyainR；iii）=+；iv）=在復內(nèi)積里是復數(shù)共軛關(guān)系四、 Banach空間主要結(jié)論Hahn-Banach 定理在理論上和應用上都是十分重要的，它往往提 供了某些學科或?qū)W科分支的理論基礎(chǔ). 這里介紹一些它們在逼近論方 面的應用.定義 3設(shè) X 是線性賦范空間， E 是 X 的子集合， x X ，稱 y E 是 x 關(guān)于 E 的最佳逼近元，若x - y= infzE x - z.(1)首先應該知道一般說來，最佳逼近元并不總是存在的.例 1設(shè) E C 0,1 ，E 是 0,1 上定義的任意階多項式全體構(gòu)成 的線性子空間，

12、取 x (t ) = et C 0,1 ，盡管d ( x, E ) = infzE x - z= 0 ，但不存在 y E 使得x - y= 0 ，因為 et 不是多項式. 這說明不存在 et關(guān)于 E 的最佳逼近元.定理 1 實際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設(shè) X 是線性賦范空間， E X 是有限維子空間，則對于 每個 x X ， x 關(guān)于 E 的最佳逼近元存在.證 明任取 y0 E ，考慮集合F = z E;x - z x - y0 .容易驗證 F 是 E 中的有界 閉 集，是 E 有限 維的，從 而 是緊集并 且d ( x, F ) = d (

13、 x, E ) . 取 zn F 使得 x - zn d ( x, F ) ，此時存在子列 n0z z F ，于是kx - z0= limnx - zn= d ( x, F ) = d ( x, E ) .z0 即是 x 關(guān)于 E 的最佳逼近元.英文翻譯部分First, the functional analysis space theoryUnderstanding of the four major functional spaceThe first part we will discuss the linear space, linear space is introduced base

14、d 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

15、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

16、. 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 algeb

17、raic 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.Normed linear space is a linear space, to give the vector norm that specifies the length of

18、the vector, but did not give the angle of the vector.Inner product spaces, there is not only a vector length angle between two vectors. In particular, the definition of the concept of orthogonal, with or without the concept of orthogonality is the essential difference between normed linear space wit

19、h an inner product space. Any inner product space Ode normed linear spaces, but not necessarily a normed linear space inner product space.Distance and space normed linear space in varying degrees have a structure similar to the space. In fact, in addition with vector inner product, use the product w

20、ithin the mold may define vectors and orthogonal vectors. But not within the definition of product in general normed linear space, and therefore can not be defined orthogonal vectors. Inner product space actually defines the inner product of linear space. On the inner product space can not only use

21、the inner product export a norm, you can also use the product within the definition of orthogonal vectors, which discussed the quadrature-related properties such as orthogonal projection, orthogonal system and so on. Space is complete inner product space. Compared with ordinary space, the theory of

22、space richer, more detailed.1 linear space(1) Definition: Let a non-empty set is the number of domains, called linear spatial domain on, if, there is only one corresponding element, and called, denoted, There will be only one element corresponding, called the product, referred to asAnd , above-menti

23、oned number of addition and multiplication, the following eight arithmetic rules:At the time, it called real linear space; at that time, known as complex linear space(2) dimension:10 is set to linear space, if the presence of the whole number of 0 is such thatCalled vectors are linearly related, oth

24、erwise known linearly independent.20 is provided, if soCalled linear representation by vectors.30 is set to linear space, if there is a linearly independent vectors such that the vector can have any one of a linear vector representation, called it a group is, the dimension known as.2 distance SpaceS

25、et up a non-empty set, if there is a map, so that, from the following axiom holds:10 non-negative20 Symmetry30 triangle inequalityDistance is called for in order of distance space, denoted by.3 normed linear spaceLet called linear spatial domain on, if, there is a corresponding real number, such tha

26、t, following the establishment of the norm axioms:10 positive definiteness20 Absolute Homogeneity30 triangle inequalityNorm is called on, on a normed space.Known complete metric space in any one converge, and normed linear space as a special kind of distance space, the same can discuss its completen

27、ess. But here is the distance induced by the norm of the distance. In the language norm, defined as the point column is rewrittenComplete normed linear space is called a space.4 inner product spaceCalled a linear space provided on the upper number field, if there is a mapping: so , the following inn

28、er product axioms Founded:Linear first ARGUMENTSConjugate symmetryPositive qualitative andInner product is called on for the inner product space.Since the concept of completeness is based on a defined distance, it is equivalent to that inner product space is called a space, by the press if their inn

29、er product derived from the norm is a complete space.In the inner product derived from the norm, inner product space becomes a normed space, it has all the properties of the general normed spaces.Second, bounded linear operator and a continuous linear functionalsLinear algebra, we have encountered a

30、 put-dimensional vector space is mapped into another dimensional vector space operations, that is, by means of the ranks of the matrixOf the vector act to achieve. Similarly, in mathematical analysis, we also encountered a function into another function or a number of operations, that operations suc

31、h as differentiation and integration. After all the above-mentioned operation of abstraction, we get the general normed spaces operator concept. Leaving aside the specific properties of various types of operators, we can divide them into two categories: one is a linear operator; a class of nonlinear

32、 operator. This chapter describes the basic knowledge of the boundaries of the operator, nonlinear knowledge about the child to remain in Chapter 5.Definition 3.1 by the normed linear space to a subset of the domain space in normed linear mapping called operator, called the operator, denoted as a se

33、t is referred to as the range operator, Hutchison or make.If the operator is satisfied:(1)(2)Called a linear operator. Linear operator, we are naturally requires subspace. In particular, if it is the real number (plural) field mapping, then the operator is called functional.In the first chapter we h

34、ave 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 ess

35、entially 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 conc

36、ept 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 i

37、f 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

38、 product form: I (f) = 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, gL 2, so all linear continuous functional on all is with a factor g of calculus.) on the Hilbert space of the most fundamental theor

39、ems 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. This conclusion is based on the form of the axioms of Euclidean space set down (see the mathematical analysis of the real part of the fundamental theorem), specifically called closed bounded continuous function must be set on the e