Functional Analysis Notes (2011) Mr. Andrew Pinchuck

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107页的经典泛函分析，入门经典，比国内老是讲述不知道好多少倍
2011 FUNCTIONAL ANALYSIS ALP 6 Baire's Category Theorem and its Applications 99 99 6.2 Uniform Boundedness principle 10l 6.3 The Open Mapping Theorem 6.4 Closed Graph Theorem 104 2011 FUNCTIONAL ANALYSIS ALP Introduction These course notes are adapted from the original course notes written by prof. Sizwe mabizela when he last gave this course in 2006 to whom I am indebted. I thus make no claims of originality but have made several changes throughout. In particular, I have attempted to motivate these results in terms of applications n science and in other important branches of mathematics Functional analysis is the branch of mathematics, specifically of analysis, concerned with the study of vector spaces and operators acting on them. It is essentially where linear algebra meets analysis. That is an important part of functional analysis is the study of vector spaces endowed with topological structure. Functional analysis arose in the study of tansformation of functions, such as the Fourier transforM, and in the study of differential and integral equations. The founding and early development of functional analysis is largely due to a group of Polish mathematicians around Stefan Banach in the first half of the 20th century but continues to be an area of intensive research to this day. Functional analysis has its main applications in differential equations, probability theory, quantum mechanics and measure theory amongst other areas and can best be viewed as a powerful collection of tools that have far reaching consequences As a prerequisite for this course, the reader must be familiar with linear algebra up to the level of a standard second year university course and be familiar with real analysis. The aim of this course is to introduce the student to the key ideas of functional analysis. It should be remembered however that we only scratch the surface of this vast area in this course. We examine normed linear spaces, Hilbert spaces, bounded linear operators, dual spaces and the most famous and important results in functional analysis such as the Hahn-Banach theorem, Baires category theorem, the uniform boundedness principle, the open mapping theorem and the closed graph theorem. We attempt to give justifications and motivations for the ideas developed as we go along Throughout the notes, you will notice that there are exercises and it is up to the student to work through these. In certain cases, there are statements made without justification and once again it is up to the student to rigourously verify these results. For further reading on these topics the reader is referred to the following . G. BACHMAN. L. NARICI, Functional Analysis, Academic Press, NY1966 . E KREYSZIG, Introductory Functional Analysis, John Wiley sons, New York-Chichester-Brisbane Toronto. 1978 .G. F. SIMMONS, Introduction to topology and modern analysis, McGraw-Hill Book Company, Sin galore. 1963 AE. TAYLOR. In on to Functional Analysis, John Wiley sons, N.Y1958 have also found wikipedia to be quite useful as a general Chapter 1 Linear Spaces 1.1 Introducton In this first chapter we review the important notions associated with vector spaces. We also state and prove some well known inequalities that will have important consequences in the following chapter. Unless otherwise stated, we shall denote by R the field of real numbers and by C the field of complex numbers. Let f denote either r or c 1.1.1 Definition A linear space over a field F is a nonempty set X with two operations + X×X→X( called addition)and F×X→X( called multiplication) satisfying the following propertics 1Jx+y∈ X whenever x,y∈X y+ x for all x,y∈X; [3/ There exists a unique element in X, denoted by 0, such that x +0=0+x=x for allx E X 14 Associated with each x e X is a unique element in X, denoted by -x, such that x +(x)= x+x=0 [5/(x +y)+z=x+(+z) for allx, y,zE X; 6c.x∈ X for all x∈ X and for all a∈; 7a(x+y)=:x+a: y for all x,y∈Xand叫ll∈]; 8J(α+B)·x=α·x+B· x for all x∈Xand叫ll,β∈I; 19(aB)·x=a·(B·x) for all x∈ X and all a,B∈Ⅳ l10/1·x= x for all x∈X. We emphasize that a linear space is a quadruple(X, F,+, where X is the underlying set, F a field,+ addition, and- multiplication. When no confusion can arise we shall identify the linear space (x, F,+, with the underlying set X. To show that X is a linear space, it suffices to show that it is closed under ddition and scalar multiplication operations Once this has been shown it is easy to show that all the other axioms hold 2011 FUNCTIONAL ANALYSIS ALP 1.1.2 Definition A real(resp. complex) linear space is a linear space over the real(resp. complex) field A linear space is also called a vector space and its elements are called vectors 1. 1.3 Examples [1 For a fixed positive integer n, let X n=x=(x1, x2, ):x;∈R, 3-the set of all n-tuples of real or complex numbers. Define the operations of addition and scalar multiplication pointwise as follows: For all x=(x1, x2,...,n) Vn) in en and a∈F, y (x1+y1,x2+y2,,xn+yn) (ax1, ax Then n is a linear space over F [2] Let X= Cla. b]=x: [a, b]>FIx is continuous ). Define the operations of addition and scalar multiplication pointwise: For all x, ye X and all a E R, define x + y)(t) )+y(t) and ax(t) fora!t∈[,b] Then Cla, b] is a real vector space Sequence Spaces: Informally, a sequence in X is a list of numbers indexed by n. equivalently, a sequence in X is a function x: n->X given by n b>x(n)= xn. We shall denote a sequence x1, x2,... by x=(x1,x2 3 The sequence space s. Let s denote the set of all sequences x =(n)i of real or complex numbers. Define the operations of addition and scalar multiplication pointwise: For all x (x1,x2,…),y=(y1,12,…)∈ s and all o∈F, define xty (ax1,ax2,) Then s is a linear space over F 4] The sequence space loo. Let loc= exo( n denote the set of all bounded sequences of real or complex numbers. That is, all sequences x=(n) such that sup xil< Define the operations of addition and scalar multiplication pointwise as in example (3). Then loo is a linear space over f [5] The sequence space p lp(n), 1< p<o. Let lp denote the set of all sequences x=(n)i of real or complex numbers satistying the condition ∑ i=1 Define the operations of addition and scalar multiplication pointwise: For all x =(xn),y (n)in Ep and all a E F, define +y 1,a2,…) 2011 FUNCTIONAL ANALYSIS ALP Then tp is a linear space over F Proof Let x=(x1,x2,,y=(y1,y2,…)∈p. We must show that x+y∈en. Since, for each i∈N, xz+y;12≤[2max{|x;l.y;}≤2Pmax{lx;P.lyv;}≤2p(xP+1y2P), it follows that ∑+y≤2(∑N2+∑ Thus,x+y∈Cp.Also,x=(xn)∈ p and a∈E,then a ∑ That is 6 The sequence space c=c(N). Let c denote the set of all convergent sequences x =(xn)i of real or complex numbers. That is, c is the set of all sequences x =(xn)i such that lim xn n→ exists. Define the operations of addition and scalar multiplication pointwise as in example (3). Then c is a linear space over F. [7 The sequence space co Co(N). Let co denote the set of all sequences x OX of real or complex numbers which converge to zero. That iS, co is the space of all sequences x=(n i such that lim xn=0. Define the operations of addition and scalar multiplication n→ pointwise as in example ( 3). Then co is a linear space over F 8 The sequence space lo =lo(N). Let lo denote the set of all sequences x =(xn)i of real or complex numbers such that xi =0 for all but finitely many indices i. Define the operations of addition and scalar multiplication pointwise as in example( 3). then lo is a linear space over F 2011 FUNCTIONAL ANALYSIS ALP 1.2 Subsets of a linear space Let X be a linear space over F, x E X and A and B subsets of X and 2 E F. we shall denote by x+ A {x+a:a∈ A+B:={+b:a∈A,b∈B} 入A {.a:a∈A} 1.3 Subspaces and convex sets 1.3.1 Definition A subset M of a linear space X is called a linear subspace of X if (a)x+y∈ M for all x,y∈M,and (b)λx∈ M for allx∈ m and for all1∈. Clearly, a subset M of a linear space X is a linear subspace if and only if M+M cm andam cM for all入∈P 1.3.2E Xamples [1 Every linear space X has at least two distinguished subspaces: M=0 and M= X These are called the improper subspaces of X. All other subspaces of X are called the proper subspaces [2] Let X=R2. Then the nontrivial linear subspaces of X are straight lines through the origin 3 M=x=(0, x2, 3, .. xn): x R, i=2, 3, .. n is a subspace of r [4]M=x: [1, 1]-R, x continuous and x(0)=0 is a subspace of C[-1,I 5]M=x: [1,1]->R, x continuous and x(0)=l is not a subspace of cl-1,1 6 Show that co is a subspace of c. 1.3.3 Definition Let k be a subset of a lincar space X. The linear hull of K, denoted by lin(k)or span(K), is the intersection of all linear subspaces of X that contain K The linear hull of K is also called the linear subspace of X spanned (or generated) by K It is easy to check that the intersection of a collection of linear subspaces of X is a linear subspace of X. It therefore follows that the linear hull of a subset K of a linear space X is again a linear subspace of x. In fact, the linear hull of a subset k of a linear space X is the smallest linear subspace of X which contains K 1.3. 4 Proposition Let k be a subset of a linear space X. Then the linear hull of k is the set of all finite linear combinations of elements of K. that is (K)={∑1x,x2 ∈K,入1, F,n∈N 2011 FUNCTIONAL ANALYSIS ALP Proof. exercise 1.3.5 Definition [1 A subset x1, x2,..., xn of a linear space X is said to be linearly independent if the equation 0 only has the trivial solution aI =C2= 0. Otherwise, the set x1, x2,..., xn is linearly dependent 12/ A subset K of a linear space X is said to be linearly independent if every finite subset( x1: x2,..., xng of K is linearly independent 1.3.6 Definition If a is a linearly independent subset of X and X=linxi, x2,..., xn, then X is said to have dimension n. In this case we say that (x1, x2,..., xni is a basis for the linear space X. If a linear space X does not have a finite basis, we say that it is infinite dimensional 1.3.7 Examples [1] The space rn has dimension n. Its standard basis is el, e2,..., eni, where, for each ple of real numbers with I in the j-th position and zeroes elsewhere: i.e ej=(0、0,,1,0,……,0), where 1 occurs in the j- th position [2 The space IPn of polynomials of degree at most n has dimension n +1. Its standard basis is [3 The function space Cla, b is infinite-dimensional 4] The spaces e,Wthl≤p≤∞, are infinite-dimensional 1. 3. 8 Definition Let k be a subset of a linear space X. We say that (a) k is convex ifλx+(1-λ)y∈ K whenever x,y∈Kandλ∈[0.1] (b) K is balanced ifλx∈ K whenever x∈Kand||≤1; (c)K is absolutely convex if K is convex and balanced 1.3.9 Remark [1 It is easy to verify that K is absolutely convex if and only if nx +uy e K whenever, yE K and|A+|u|≤1 [2 Every linear subspace is absolutely convex 1.3.10 Definition Let s be a subset of the linear space X. the convex hull of S, denoted co(S), is the intersection of all convex sets in X which contain s Since the intersection of convex sets is convex, it follows that co(S)is the smallest convex set which contains S. The following result is an alternate characterization of co(S) 2011 FUNCTIONAL ANALYSIS ALP 1.3.11 Proposition Let s be a nonempty subset of a linear space X. Then co(s) is the set of all convex combinations of elements of s. That is CO(S) 1,V2 xn∈S,入≥0V=12,…,n,∑ j=1,n∈N Proof. let c denote the set of all convex combinations of elements of s. that is 心下x1,x2,…,xn∈S,;≥0Vj=1,2 n,∑入=1.n∈N Letx,y∈Cand0≤λ≤. Then x=∑入1,y=∑u1n, where ai,H10,∑ ∑似=1,andx;,n∈S.Thn 入x+(1-)=∑入;x1+∑(1-厘 is a linear combination of elements of s, with nonnegative coefficients, such that ∑λ入;+∑(-)=2∑入;+(1-人)∑;=λ+(1-入) That is, hx+(1-h)y eC and C is convex. Clearly S CC. Hence CO(S)CC We now prove the inclusion CC CO(S). Note that, by definition, S C Co(S). Let x1, x2 E S, λ1≥0,λ2≥0andλ1+λ2=1.Then, by convexity of co(S),λ1x1+入2x2∈CO(S). Assume that λ;x∈co(S) whenever x;x2,…,xn1∈S,≥0,j=1,2,…,n-1and∑入=1. c1, ,xn∈Sand1,λ2,…,λ n be such that n;≥0,j=1,2 n and ∑ 1. If j=1 ∑=0. then an=1.Henc∑x=nxn∈co(S) Assume that B=∑>0.Then务≥0 for all j=1, 2 n-1 and 1. By the induction assumption x∈co(S). Hence λ;x;= (2 x;+Anxn∈co(S) Thus CC CO(S) 1.4 Quotient Space Let m be a linear subspace of a linear space X over F. For all x,yE X, define x= y(mod M)<>x-y∈M. 【实例截图】
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