2 CONTENTS Index 123 Contents 1 Banach spaces 5 1.1 Metric spaces ............................... 5 1.2 Normed spaces .............................. 9 1.3 H¨older and Minkowski inequality .................... 18 2 Bounded maps; the dual space 21 2.1 Bounded linear maps ........................... 21 2.2 The dual space and the Hahn-Banach theorem ............ 26 2.3 Examples of dual spaces ......................... 30 2.4 The Banach space adjoint and the bidual ............... 32 3 Linear operators in Banach spaces 37 3.1 Baire’s theorem .............................. 37 3.2 Uniform boundedness principle ..................... 38 3.3 The open mapping theorem ....................... 45 3.4 The closed graph theorem ........................ 47 3.5 Projections in Banach spaces ...................... 50 3.6 Weak convergence ............................ 52 4 Hilbert spaces 55 4.1 Hilbert spaces ............................... 55 4.2 Orthogonality ............................... 58 4.3 Orthonormal systems ........................... 61 4.4 Linear operators in Hilbert spaces ................... 65 4.5 Projections in Hilbert spaces ...................... 69 4.6 The adjointDRAFT of an unbounded operator ................. 71 DRAFT 5 Spectrum of linear operators 79 5.1 The spectrum of a linear operator ................... 79 5.2 The resolvent ............................... 80 5.3 The spectrum of the adjoint operator .................. 87 5.4 Compact operators ............................ 89 5.5 Hilbert-Schmidt operators ........................ 99 5.6 Polar decomposition ........................... 102 A Exercises 105 References 116 Problem Sheets 119 Problem Sheet 1. Espacios m´etricos y normados. .............. 120 Problem Sheet 2. Operadores lineales. .................... 121 Problem Sheet 3. Hahn Banach, Espacios duales. .............. 122 1 Last Change: Tue 5 Feb 11:22:41 COT 2013 CONTENTS 3 4 CONTENTS The notes are written and modified while I am teaching an introductory course Notation on functional analysis at the Universidad de los Andes, Bogot´a, Colombia. They are by no means a new presentation of basic functional analysis, they are rather a The letter K usually denotes either the real field R or the complex field C. The positive real numbers are denoted by R := (0, ). collection of excerpts of the books of the list of references I used to prepare classes. + ∞ Many thanks to the students from my 2009 class who found a lot of mistakes in the first draft of the notes. Prerequisites for the course are a solid knowledge in analysis, linear algebra and very basic notions of topology. Having attended a lecture in measure theory or an advanced course in analysis is of advantage but not necessary. An important part of any mathematics lecture are exercises. For each week there is a problem sheet with exercises (stolen from various books) which hopefully help to understand the material presented in the lecture. Bogot´a, February 2013, M.W. DRAFT DRAFT These lecture notes are work in progress. They may be abandoned or changed radically at any moment. Very likely they contain a lot of mistakes and ambiguities. If you find mistakes or have suggestions how to improve the lecture notes, please let me know. Last Change: Tue 5 Feb 11:22:41 COT 2013 Last Change: Thu 7 Feb 17:21:44 COT 2013 Chapter 1. Banach spaces 5 6 1.1. Metric spaces Let (M, d) be a metric space. Recall that the metric d induces a topology on M: a set U M is open if and only if for every p U there exists an ε > 0 such ⊆ ∈ that Bε(p) U. In particular, the open balls are open and closed balls are closed subsets of M⊆. Let x M. A subset U M is called a neighbourhood of x if there ∈ ⊆ exists an open set Ux such that x Ux U. It is easy to see that the topology∈ generated⊆ by d has the Hausdorff property, that is, for every x = y M there exist neighbourhoods U of x and U of y with Chapter 1 6 ∈ x y Ux Uy = . Recall∩ that∅ a set N M is called dense in M if N = M, where N denotes the ⊆ Banach spaces closure of N. Definition 1.2. A sequence (x ) N M converges to x M if and only if n n∈ ⊆ ∈ lim d(xn, x) = 0, that is, n→∞ 23 Jan 2012 ε> 0 N N : n N = d(x , x) < ε. 1.1 Metric spaces ∀ ∃ ∈ ≥ ⇒ n The limit x is unique. A sequence (xn)n∈N is a Cauchy sequence in M if and only We repeat the definition of a metric space. if Definition 1.1. A metric space (M, d) is a non-empty set M together with a map ε> 0 N N : m,n N = d(x , x ) < ε. ∀ ∃ ∈ ≥ ⇒ n m d : M M R × → Definition 1.3. A metric space in which every Cauchy sequence is convergent, is such that for all x,y,z M: called a complete metric space. ∈ (i) d(x, y)=0 x = y, Definition 1.4. Let (X, ) and (Y, ) be topological spaces. ⇐⇒ X Y (ii) d(x, y)= d(y, x), O O (i) A function f : X Y is called continuous if and only if f −1(U) is open in X (iii) d(x, y) d(x, z)+ d(z,y). → ≤ for every U open in Y . The last inequality is called triangle inequality. Usually the metric space (M, d) is denoted simply by M. (ii) An bijective function f : X Y is called a homeomorphism if and only if f and f −1 are contiunous. → Note that the triangle inequality together with the symmetry of d implies The following lemma is often useful. d(x, y) 0, x,y M, ≥ ∈ Lemma 1.5. Let (M, d) be a complete metric space and N M. Then N is closed since 0 = d(x, x) d(x, y)+ d(y, x)=2d(x, y). in M if and only if (N, d ) is complete. ⊆ ≤ |M It is easy to check that DRAFT Remarks. DRAFTEvery convergent sequence is a Cauchy sequence. d(x, y) d(y,z) d(x, z), x,y,z M. • | − |≤ ∈ Every Cauchy sequence is bounded. Recall that a sequence (xn)n∈N is bounded • if the set x : n N is bounded. { n ∈ } A subset N M is called bounded if ⊆ Not every metric space is complete, but every metric space can be completed in the diam N := sup d(x, y): x, y N < . following sense. { ∈ } ∞ Let r> 0 and x M. Then Definition 1.6. Let (M, d ) and (N, d ) be metric spaces. A map f : M N is ∈ M N → called an isometry if and only if dN (f(x),f(y)) = dM (x, y) for all x, y M. The B (x) := y M : d(x, y) < r =: open ball with centre x and radius r, ∈ r { ∈ } spaces M and N are called isometric if there exists a bijective isometry f : M N. K (x) := y M : d(x, y) r =: closed ball with centre x and radius r, → r { ∈ ≤ } Sr(x) := y M : d(x, y)= r =: sphere with centre x and radius r. Note that an isometry is necessarily injective since x = y implies f(x) = f(y) { ∈ } because d(f(x),f(y)) = d(x, y) = 0, and that every isometry6 is continuous. 6 6 Examples. R with the d(x, y)= x y is a metric space. • | − | Let X be a set and define d : X X R by d(x, y) = 0 for x = y and Theorem 1.7. Let (M, d) be a metric space. Then there exists a complete metric • d(x, y)= 1 for x = y. Then (X, d)× is a metric→ space. d is called the discrete space (M, d) and an isometry ϕ : M M such that ϕ(M) = M. M is called → metric on X. 6 completion of M; it is unique up to isometry. c b c c c Last Change: Thu 7 Feb 17:21:44 COT 2013 Last Change: Thu 7 Feb 17:21:44 COT 2013 Chapter 1. Banach spaces 7 8 1.1. Metric spaces Proof. Let Next we show that (M, d) is complete. Let (ˆxn)n∈N be a Cauchy sequence in M. Since ϕ(M) is dense in M there exists a sequence z = (zn)n∈N M such that := (x ) N M : (x ) N is a Cauchy sequence in M ⊆ CM { n n∈ ⊆ n n∈ } c b c c 1 be the set of all Cauchy sequences in M. We define the equivalence relation on d(xn,zn) < , n N. n ∈ by ∼ CM The sequence z is a Cauchy sequenceb b in M because x y d(x ,y ) 0, n ∼ ⇐⇒ n n → → ∞ d(zn,zm)= d(ϕ(zn), ϕ(zm)) d(ϕ(zn), xˆn)+ d(ˆxn, xˆm)+ d(ˆxm, ϕ(zm)) for all x = (x ) N, y = (y ) N . It is easy to check that is indeed a ≤ n n∈ n n∈ M 1 1 equivalence relation (reflexivity and∈ symmetry C follow directly from∼ properties (i) b < b + d(ˆxn, xˆm)+b 0,b m,n . and (ii) of the definition of a metric and transitivity of is a consequence of the n m → → ∞ triangle inequality). ∼ The sequence (ˆxn)n∈N converges to [z] becauseb Let M := M / the set of all equivalence classes. The equivalence class containing C ∼ 1 x = (xn)n∈N is denoted by [x]. On M we define d(ˆxn,z) d(ˆxn, ϕ(zn)) + d(ϕ(zn),z) < + lim d(zn,zm) 0, n . c ≤ n m→∞ → → ∞ d : M M R, c d([x], [y]) = lim d(xn,yn). (1.1) × → n→∞ Web have shownb that ϕ(M) isb a dense subset of the complete metric space (M, d) and that ϕ is an isometry. We have to showb thatcd isc well-defined.b c b Finally, we have to show that M is unique (up to isometry).