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Functional Analysis Functional Analysis Lecture Notes of Winter Semester 2017/18 These lecture notes are based on my course from winter semester 2017/18. I kept the results discussed in the lectures (except for minor corrections and improvements) and most of their numbering. Typi- cally, the proofs and calculations in the notes are a bit shorter than those given in class. The drawings and many additional oral remarks from the lectures are omitted here. On the other hand, the notes con- tain a couple of proofs (mostly for peripheral statements) and very few results not presented during the course. With `Analysis 1{4' I refer to the class notes of my lectures from 2015{17 which can be found on my webpage. Occasionally, I use concepts, notation and standard results of these courses without further notice. I am happy to thank Bernhard Konrad, J¨orgB¨auerle and Johannes Eilinghoff for their careful proof reading of my lecture notes from 2009 and 2011. Karlsruhe, May 25, 2020 Roland Schnaubelt Contents Chapter 1. Banach spaces2 1.1. Basic properties of Banach and metric spaces2 1.2. More examples of Banach spaces 20 1.3. Compactness and separability 28 Chapter 2. Continuous linear operators 38 2.1. Basic properties and examples of linear operators 38 2.2. Standard constructions 47 2.3. The interpolation theorem of Riesz and Thorin 54 Chapter 3. Hilbert spaces 59 3.1. Basic properties and orthogonality 59 3.2. Orthonormal bases 64 Chapter 4. Two main theorems on bounded linear operators 69 4.1. The principle of uniform boundedness and strong convergence 69 4.2. Sobolev spaces 75 4.3. The open mapping theorem and invertibility 81 Chapter 5. Duality 86 5.1. The duals of sequence and Lebesgue spaces 86 5.2. The extension theorem of Hahn-Banach 91 5.3. Reflexivity and weak convergence 101 5.4. Adjoint operators 109 Chapter 6. The spectral theorem for compact self-adjoint operators 117 6.1. Compact operators 117 6.2. The spectral theorem 121 Bibliography 126 1 CHAPTER 1 Banach spaces In these notes X 6= f0g and Y 6= f0g are always vector spaces over the field F 2 fR; Cg. 1.1. Basic properties of Banach and metric spaces We start with the fundamental definitions of this course which con- nect the linear structure with convergence. Definition 1.1. A seminorm on X is a map p : X ! R satisfying a) p(αx) = jαj p(x) (homogeneity), b) p(x + y) ≤ p(x) + p(y) (triangle inquality) for all x; y 2 X and α 2 F. If p fulfills in addition c) p(x) = 0 =) x = 0 (definiteness) for all x 2 X, then p is a norm. One mostly writes p(x) = kxk and p=k·k. The pair (X; k·k) (or just X) is called a normed vector space. In view of Example 1.4(a), we interpret kxk as the length of x and kx − yk as the distance between x and y. Seminorms will only occur as auxiliary objects, see e.g. Proposition 1.8. Definition 1.2. Let k · k be a seminorm on a vector space X.A sequence (xn)n2N = (xn)n = (xn) in X converges to a limit x 2 X if 8 " > 0 9 N" 2 N 8 n ≥ N" : kxn − xk ≤ ": We then write xn ! x as n ! 1 or x = limn!1 xn. Moreover, (xn) is a Cauchy sequence in X if 8 " > 0 9 N" 2 N 8 n; m ≥ N" : kxn − xmk ≤ ": A normed vector space (X; k · k) is a Banach space if each Cauchy sequence in (X; k · k) converges in X. Then one also calls (X; k · k) or k · k complete. In this section we discuss (and partly extend) various results from Analysis 2 whose proofs were mostly omitted in the lectures. We start with simple properties of norms and limits. Remark 1.3. Let k · k be a seminorm on a vector space X and (xn) be a sequence in X. The following facts are shown as in Analysis 2, see e.g. Satz 1.2. a) The vector 0 has the seminorm 0. 2 1.1. Basic properties of Banach and metric spaces 3 b) We have kxk − kyk ≤ kx − yk and kxk ≥ 0 for all x; y 2 X. c) If (xn) converges, then it is a Cauchy sequence. d) If (xn) converges or is Cauchy, it is bounded; i.e., supn2N kxnk<1: e) If xn ! x and yn ! y in X as n ! 1 and α; β 2 F, then the linear combinations αxn + βyn tend to αx + βy in X. f) Limits are unique in the norm case: Let k·k be a norm. If xn ! x and xn ! y in X as n ! 1 for some x; y 2 X, then x = y. ♦ For our basic examples below and later use, we introduce some no- tation. Let X be a vector space and S 6= ; be a set. For maps f; g : S ! X and numbers α 2 F one defines the functions f + g : S ! X;(f + g)(s) = f(s) + g(s); αf : S ! X;(αf)(s) = αf(s): It is easily seen that the set ff : S ! Xg becomes a vector space en- dowed with the above operations. Function spaces are always equipped with this sum and scalar multiplication. Let X = F. Here one puts fg : S ! F;(fg)(s) = f(s)g(s): Let α; β 2 R. We then write f ≥ α (f > α, respectively) if f(s) ≥ α (f(s) > α, respectively) for all s 2 S. Similarly one defines α ≤ f ≤ β, f ≤ g, and so on. Example 1.4. a) X = Fm is a Banach space for the norms 8 1 < Pm p p k=1 jxkj ; 1 ≤ p < 1; jxjp = :maxfjxkj j k = 1; : : : ; mg; p = 1; m m where x = (x1; : : : ; xm) 2 F . Moreover, vectors vn converge to x in F as n ! 1 for each of these norms if and only if all components (vn)k tend to xk in F as n ! 1. See Satz 1.4, 1.9 and 1.13 in Analysis 2. We always equip X = F with the absolute value j · j which coincides with each of the above norms. b) Let (X; k · k) be a Banach space and K a compact metric space. Then the set E = C(K; X) = ff : K ! X j f is continuousg endowed with the supremum norm kfk1 = sup kf(s)k s2K is a Banach space. We equip E with this norm, unless something else is specified. Before proving the claim, we note that the above supremum is a maximum and thus finite, cf. Theorem 1.45 or Analysis 2, and that convergence in k·k1 is just uniform convergence from Analysis 1. In the special case X = R, the norm kf −gk1 is the maximal vertical distance 1.1. Basic properties of Banach and metric spaces 4 between the graphs of f; g 2 E. If f ≥ 0 describes the temperature in K, for instance, then kfk1 is the maximal temperature. Moreover, the closed ball BE(f; ") around f of radius " > 0 consists of all functions g in E whose graph belongs to an `"{tube' around f. Proof. It is clear that E is a vector space. Let f; g 2 E and α 2 F. Since k · k is a norm, we obtain kfk1 = 0 =) 8 s 2 K : f(s) = 0 =) f = 0; kαfk1 = sup kαf(s)k = sup jαj kf(s)k = jαj sup kf(s)k = jαj kfk1; s2K s2K s2K kf + gk1 = sup kf(s)+g(s)k ≤ sup(kf(s)k+kg(s)k) ≤ kfk1 + kgk1; s2K s2K so that E is a normed vector space. Take a Cauchy sequence (fn) in E. For each " > 0 there is an index N" 2 N with kfn(s) − fm(s)k ≤ kfn − fmk1 ≤ " for all n; m ≥ N" and s 2 K. By this estimate, (fn(s))n is a Cauchy sequence in X. Since X is complete, there exists the limit f(s) := limn!1 fn(s) in X for each s 2 K. Let s 2 K and " > 0 be given. Take the index N" from above and n ≥ N". We then estimate kf(s) − fn(s)k = lim kfm(s) − fn(s)k ≤ lim sup kfm − fnk1 ≤ ": m!1 m!1 Because N" does not depend on s, we can take the supremum over s 2 K and derive the inequality kf − fnk1 ≤ " for all n ≥ N". To see that f belongs to E, fix an integer N with kf−fN k1 ≤ ". The continuity of fN yields a radius δ > 0 such that kfN (s) − fN (t)k ≤ " for all s 2 BK (t; δ). For such s we deduce kf(s)−f(t)k ≤ kf(s)−fN (s)k+kfN (s)−fN (t)k+kfN (t)−f(t)k ≤ 3"; i.e., f 2 E. Summing up, (fn) converges to f in E as required. 2 c) Let X = C([0; 1]). We set Z 1 kfk1 = jf(s)j ds 0 for f 2 X. The number kf − gk1 yields the area between the graphs of f; g 2 X (if F = R), and kfn − fk1 ! 0 is called convergence in the mean. For a mass density f ≥ 0 of a substance, the integral kfk1 is the total mass. We claim that k · k1 is an incomplete norm on X. Proof. Let α 2 F and f; g 2 X.
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