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LECTURES on FUNCTIONAL ANALYSIS Contents 1. Elements Of LECTURES ON FUNCTIONAL ANALYSIS R. SHVYDKOY Contents 1. Elements of topology2 1.1. Metric spaces3 1.2. General topological spaces. Nets6 1.3. Ultrafilters and Tychonoff's compactness theorem7 1.4. Exercises8 2. Banach Spaces9 2.1. Basic concepts and notation9 2.2. Geometric objects9 2.3. Banach Spaces 10 2.4. Classical examples 11 2.5. Subspaces, direct products, quotients 14 2.6. Norm comparison and equivalence 15 2.7. Compactness in finite dimensional spaces 16 2.8. Convex sets 17 2.9. Linear bounded operators 18 2.10. Invertibility 20 2.11. Complemented subspaces 22 2.12. Completion 23 2.13. Extensions, restrictions, reductions 23 2.14. Exercises 24 3. Fundamental Principles 24 3.1. The dual space 24 3.2. Structure of linear functionals 25 3.3. The Hahn-Banach extension theorem 26 3.4. Adjoint operators 28 3.5. Minkowski's functionals 28 3.6. Separation theorems 29 3.7. Baire Category Theorem 30 3.8. Open mapping and Closed graph theorem 31 3.9. Exercises 33 4. Hilbert Spaces 34 4.1. Inner product 34 4.2. Orthogonality 36 4.3. Orthogonal and Orthonormal sequences 38 4.4. Existence of a basis and Gram-Schmidt orthogonalization 40 4.5. Riesz Representation Theorem 42 4.6. Hilbert-adjoint operator 43 1 2 R. SHVYDKOY 4.7. Exercises 45 5. Weak topologies 46 5.1. Weak topology 46 5.2. Weak∗ topology 49 5.3. Exercises 52 6. Compact sets in Banach spaces 52 6.1. Compactness in sequence spaces 54 6.2. Arzel`a-AscoliTheorem 56 6.3. Compactness in Lp(Ω) 57 6.4. Extreme points and Krein-Milman Theorem 59 6.5. Compact maps 60 6.6. Exercises 60 7. Fixed Point Theorems 60 7.1. Contraction Principles 60 7.2. Brouwer Fixed Point theorem and its relatives 61 8. Spectral theory of bounded operators 61 8.1. Spectral Mapping Theorem and the Gelfand formula 65 8.2. On the spectrum of self-adjoint operators 67 8.3. On the spectrum of unitary operators 70 8.4. Exercises 70 1. Elements of topology We start our lectures with a crush course in elementary topology. We will not need the full extent of this section till after we start discussing weak and weak-star topologies on Banach spaces. However, everything related to metric spaces and compactness will become useful right away. Definition 1.1. A set X is called a topological space if has a designated family of subsets τ, called topology whose elements are called open sets and which satisfy the following axioms (i) ;;X 2 τ, S (ii) If Uα 2 τ for α 2 A is any collection of open sets, then α2A Uα 2 τ, Tn (iii) If Ui, i = 1; : : : ; n is a finite collection of open sets, then i=1 Ui 2 τ. We call F ⊂ X a closed set if F c = XnF 2 τ. Closed sets verify a complementary set of axioms: (i) ;;X are closed, T (ii) If Fα for α 2 A is any collection of closed sets, then α2A Fα is closed, Sn (iii) If Fi, i = 1; : : : ; n is a finite collection of closed sets, then i=1 Fi is closed. For any subset A ⊂ X, we define the closure of A, denoted A¯, to the minimal closed set containing A. In other words, if F is a closed set containing A, then A¯ ⊂ F . In view of property (ii) of closed sets, we can define the closure equivalently by \ A¯ = F: F is closed;A⊂F LECTURES ON FUNCTIONAL ANALYSIS 3 One of the most important uses of topological spaces is that one can define the notion of a continuous map between them. Definition 1.2. Let (X; τX ) and (Y; τY ) be two topological spaces. A map f : X ! Y −1 is called continuous if f (U) 2 τX for every U 2 τY . In other words the preimage of every open set is an open set. One can also define "small sets" that in a sense possess properties of a finite dimensional object. Definition 1.3. A subset K ⊂ X is called compact if every open cover of K contains a finite subcover, in other words if one can embed K into a collection of open sets [ K ⊂ Uα; α2A then one can find only finitely many of them Uα1 ;:::;Uαn which still cover the entire set Sn K ⊂ i=1 Uαi . Lemma 1.4. Let f : X ! Y be a continuous map. If K ⊂ X is compact, then f(K) is compact in Y . Proof. Consider an open cover [ f(K) ⊂ Vα: α2A −1 Then f (Vα) = Uα form an open over of K. By the compactness of K there exists a finite Sn Sn subcover K ⊂ i=1 Uαi . But then f(K) ⊂ i=1 Vαi . One can define a limit of a sequence x = limn!1 xn by declaring that for every open set U containing x there exists an N 2 N such that for all n > N, xn 2 U. However, in general the definitions of continuity and compactness are not equivalent to their sequential counterparts n as, say, in R . Thus, the continuity in the sense that \if xn ! x, then f(xn) ! f(x)" in not equivalent to Definition 1.2. Likewise, sequential compactness in the sense that "every sequence fxng ⊂ K has a convergent subsequence to an element of K" is not equivalent to Definition 1.3. These sequential counterparts appeal to the situation when one can choose a \sequence of balls" shrinking towards x, and respectively towards f(x) in Y , which may not be possible if the topology does not have a \countable base". Before we venture into the general settings let us narrow our discussion down to the more special case of a metric space where most topological concepts do in fact have sequential analogues. 1.1. Metric spaces. A metric on a set X, or distance, is a map d : X × X ! R+ such that (i) d(x; y) = 0 if and only if x = y, (ii) d(x; y) = d(y; x), (iii) d(x; z) ≤ d(x; y) + d(y; z). For any point x there is a family of balls Br(x) = fy : d(x; y) < rg around x which has a countable base, namely B1=n(x). We define metric topology on X by declaring that a set U is open if for every x 2 U there is a ball Br(x) ⊂ U. Every metric space separates points, namely for every x1; x2 2 X there are open neigh- borhoods U(xi) such that xi 2 U(xi) and U(x1) \ U(x2) = ;. Indeed, pick U(xi) to be the 4 R. SHVYDKOY ball centered at xi of radius r = d(x1; x2)=3. Such topologies are called Hausdorff. We say that x = limn!1 xn if d(x; xn) ! 0. Because of the Hausdorff property all limits are unique. We call a sequence Cauchy if 8" 9N such that 8n; m > N, d(xn; xm) < ". Definition 1.5. The space X is called complete if every Cauchy sequence in X has a limit. Proposition 1.6 (Sequential definitions). Let (X; d) be a metric space. (a) A subset F ⊂ X of a metric space is closed iff the limit of a convergent sequence xn 2 F , belongs to F . (b) A function f : X ! Y , where Y is any topological space, is continuous iff for every sequence xn ! x one has f(xn) ! f(x) (sequentially continuous). (c) A subset K ⊂ X is compact iff every sequence fxngn ⊂ K contains a subsequence which converges to a point in K (sequentially compact). Consequently, K is complete as a metric space. Proof. (a): Suppose F is closed. If x2 = F , then there is a neighborhood U(x) \ F = ;. By definition of the limit there exists n such that xn 2 U(x), so not in F , a contradiction. Conversely, if F is sequentially closed but F c is not open, then there exists a point x 2 F c such that B1=n(x) \ F 6= ;. Pick a sequence xn 2 B1=n(x) \ F . Then xn ! x and all xn 2 F , yet x2 = F , a contradiction. (b): Suppose f is continuous, but there exists xn ! x such that f(xn) does not converge to f(x). Then thee exists an open neighborhood U of f(x) and a subsequence xnk such that −1 f(xnk ) 2= U. Since f (U) is open and contains x, some elements of xnk will eventually get −1 in f (U) which means f(xnk ) 2 U, a contradiction. Conversely, suppose f is sequentially continuous but for some open U ⊂ Y , f −1(U) is not −1 −1 open. Then there exists a point x 2 f (U) and a sequence xn 2 B1=n(x)nf (U). But then xn ! x and so f(xn) ! f(x), which means that from some n, f(xn) 2 U, a contradiction. (c): Let K be compact and let fxng ⊂ K. We call point y 2 K a cluster point if every ball Br(y) contains a subsequence fxnk g of the given sequence. We show that there exists at least one cluster point. Suppose not, then for every y 2 K we can find a ball Br(y)(y) such that starting from n > N(y), all xn 2= Br(y)(y). This defines a cover, hence there exists a finite subcover Br(y1)(y1);:::;Br(ym)(ym).
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