泛函分析结课论文资料

泛函分析结课论文 Functional Analysis Course Paper 学号 姓名
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，进而建立了赋范线性空间和度量空间。 在线性空间中赋以“范数”，然后在范数的基础上导出距离，即赋范线性空间，完备的赋范线性空间称为巴拿赫空间。范数可以看出长度，赋范线性空间相当于定义了长度的空间，所有的赋范线性空间都是距离空间。 在距离空间中通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。赋范线性空间和内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。 赋范线性空间是其中每个向量赋予了范数的线性空间，而且由范数诱导出的拓扑结构与代数结构具有自然的联系。完备的赋范线性空间是Banach空间。赋范线性空间的性质类似于熟悉的nR，但相比于距离空间，赋范线性空间在结构上更接近于nR。 赋范线性空间就是在线性空间中，给向量赋予范数，即规定了向量的长度，而没有给出向量的夹角。 在内积空间中，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范线性空间与内积空间的本质区别。任何内积空间都赋范线性空间，但赋范线性空间未必是内积空间。 距离空间和赋范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上还具有向量的内积，利用内积可以定义向量的模和向量的正交。但是在一般的赋范线性空间中没有定义内积，因此不能定义向量的正交。内积空间实际上是定义了内积的线性空间。在内积空间上不仅可以利用内积导出一个范数，还可以利用内积定义向量的正交，从而讨论诸如正交投影、正交系等与正交相关的性质。Hilbert空间是完备的内积空间。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,xyX，都有唯一的一个元素zX与之对应，称为xy与的和，记作 zxy ,xXK，都会有唯一的一个元素uX与之对应，称为x与的积，记作 ux 且,,xyzX，,K，上述的加法与数乘运算，满足下列8条运算规律： 10 xyyx
20 ()()xyzxyz 30 在X中存在零元素，使得xX，有xx 40 xX，存在负元素xX，使得()xx 50 1xx 60 ()()xx 70 ()+xxx 80 ()xyxy 当KR时，称X为实线性空间；当KC时，称X为复线性空间 （2）维数： 10 设X为线性空间，12,,,nxxxX若不存在全为0的数12,,,nK

，使得 11220nnxxx 则称向量组12,,,nxxx是线性相关的，否则称为线性无关。 20 设xX，若12,,,nK，12,,,nxxxX使得 1122nnxxxx 则称x可由向量组12,,,nxxx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:dXXR，使得,,xyzX，下列距离公理成立： 10 非负性(,)0,(,)=0dxydxyxy 20 对称性(,)(,)dxydyx 30 三角不等式(,)(,(,)dxydxzdzy 则称(,)dxy为xy与的距离，X为以d的距离空间，记作(,)Xd。 3 赋范线性空间 设X称为数域上K上的线性空间，若xX，都有一个实数x与之对应，使得,,xyXK，下列范数公理成立： 10 正定性0,00xxx 20 绝对齐次性xx
30 三角不等式xyxy 则称x为x上的范数，X为K上的赋范空间。 已知完备的距离空间中任一Cauchy列均收敛，而赋范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范数诱导的距离。在范数的语言下，点列{}nx为Cauchy列的定义改写为 0,,NN使当m,n>时，有mnxx 完备的赋范线性空间称为Banach空间。 4 内积空间 设X称为数域上K上的线性空间，若存在映射,：XXK，使得,,xyzX，,K，，下列内积公理成立： 对第一变元的线性,,,xyzxzyz 共轭对称性,,xyyx 正定性,0xx且,00xxx 则称,为X上的内积，X为K上的内积空间。 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，一个内积空间称为Hilbert空间，若其按由内积导出的范数是完备的距离空间。 在由内积导出的范数下，内积空间X成为一个赋范空间，它具有一般赋范空间的所有性质。 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nE映射到另一个m维向量空间mE的运算，就是借助于m行n列的矩阵 111212122212nnmmmnaaaaaaAaaa 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，我们就得到一般赋范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] 由赋范线性空间X中的某子集D到赋范线性空间Y中的映射T称为算子，D称为算子T的定义域，记为DT，为称像集,yyTxxDT为算
子的值域，记作TD或TD。 若算子T满足： （1）,TxyTxTyxyDT （2）(),TxTxFxDT 称T为线性算子。对线性算子，我们自然要求TD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 是线性赋范空间，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) ax b 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 可以表示成为内积的形式: I（f）= 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的内积是积分的形式: ∫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，它的范数‖I‖=sup|I（f）|/‖f‖=1.在这个L^2的单位闭球面（所有范
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上范数的定义，‖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 线性完备内积空间称为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。 内积（inner product）：一个从H×H-->R 的双线性映射，记为。它满足： i）≥0，

f>=0 <==> f=0； ii）=a*= for any a in R； iii）=+； iv）= ——在复内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 是线性赋范空间， 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 y inf zE x z . (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 et C 0,1 ，尽管 d x, E inf zE x z 0 ， 但不存在 y E 使得 x y 0 ，因为 et 不是多项式. 这说明不存在 et
n 0 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 是线性赋范空间， E X 是有限维子空间，则对于 每个 x X ， x 关于 E 的最佳逼近元存在. 证 明 任取 y0 E ，考虑集合 F z E; x z x y0 . 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 d x, F d x, E . 取 zn F 使得 x zn d x, F ，此时存在子列 z z F ，于是 k x z0 lim nx zn d x, F d x, E . z0 即是 x 关于 E 的最佳逼近元.
英文翻译部分 First, the functional analysis space theory Understanding of the four major functional space The 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. Normed linear space is a linear space, to give the vector norm that specifies the length of the vector, but did not give the angle of the vector. Inner product spaces, there is not only a vector length angle between two v

ectors. In particular, the definition of the concept of orthogonal, with or without the concept of orthogonality is the essential difference between normed linear space with 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 within 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 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 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 as
And ,, above-mentioned 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 that Called vectors are linearly related, otherwise known linearly independent. 20 is provided, if so Called 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 Space Set up a non-empty set, if there is a map, so that, from the following axiom holds: 10 non-negative 20 Symmetry 30 triangle inequality Distance is called for in order of distance space, denoted by. 3 normed linear space Let called linear spatial domain on, if, there is a corresponding real number, such that, following the establishment of the norm axioms: 10 positive definiteness 20 Absolute Homogeneity 30 triangle inequality Norm 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 completeness. But here is the distance induced by the norm of the distance. In the language norm, defined as the point column is rewritten Complete normed linear space is called a space. 4 inner product space Called a linear space provided on the upper number field, if there is a mapping: so ,,, the following inner product axioms Founded: Linear first ARGUMENTS Conjugate symmetry Positive qualitat

ive and Inner 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 inner 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 functionals Linear algebra, we have encountered a put-dimensional vector space is mapped into another dimensional vector space operations, that is, by means of the ranks of the matrix Of the vector act to achieve. Similarly, in mathematical analysis, we also encountered a function into another function or a number of operations, that operations such 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 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 set 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 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 latter Refers 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 conclusions A 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) = 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. 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 extreme value, and makes this function reaches the point of presence it. Topologically equivalent to the local compactness of this stuff, it is a pity that in general the Hilbert space it is not true: a closed interval L ^ 2 space is a natural continuous functional [0,1]: I (f) = ∫ | f (x) | dx. Easy to prove that its norm ‖I‖ = sup | I (f) | / ‖f‖ = 1 unit in the L ^ 2 closed spherical (all norm equal to 1 f) the presence of such a sequence: f_n (x) = n, when x∈ [0,1 / n ^ 2]; f_n (x) = 0, when x> 1 / n ^ 2. L is defined according to the norm of 2 ^, ‖f_n‖ = ∫f ^ 2 (x) dx = 1, for all n. 0≤I (f) ==> I minimum ≤0 closed set bounded on this, and I (f_n) = 1 / n → 0. But we see that when f_n often weak convergence to zero function, it has not closed the unit sphere of the (strict proof can be found in some textbooks). I. Definitions The linear complete product space called Hilbert space. Linear (linearity): For any f, g∈H, a, b∈R, a * f + b * g still ∈H. Complete (completeness): H on an arbitrary Cauchy sequence must converge to a point H on. - Corresponds to the definition of closed set. The inner product (inner product): from a H × H - R bilinear mapping>, denoted by . It satisfied: i) ≥0, = 0 <==> f = 0; ii) = a * = for any a in R; iii) = + ; iv) = - in the complex inner product is inside the complex conjugate relationship Four, Banach spaces main conclusions