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Functional Analysis Functional Analysis Marco Di Francesco, Stefano Spirito 2 Contents I Introduction 5 1 Motivation 7 1.1 Someprerequisitesofsettheory. ............... 10 II Lebesgue integration 13 2 Introduction 15 2.1 Recalling Riemann integration . ............ 15 2.2 A new way to count rectangles: Lebesgue integration . .............. 17 3 An overview of Lebesgue measure theory 19 4 Lebesgue integration theory 23 4.1 Measurablefunctions................................. ............ 23 4.2 Simplefunctions ................................... ............ 24 4.3 Definition of the Lebesgue integral . ............ 25 5 Convergence properties of Lebesgue integral 31 5.1 Interchanging limits and integrals . ............... 31 5.2 Fubini’sTheorem................................... ............ 34 5.3 Egorov’s and Lusin’s Theorem. Convergence in measure . .............. 36 6 Exercises 39 III The basic principles of functional analysis 43 7 Metric spaces and normed spaces 45 7.1 Compactnessinmetricspaces. ............. 45 7.2 Introduction to Lp spaces .......................................... 48 7.3 Examples: some classical spaces of sequences . ............... 52 8 Linear operators 57 9 Hahn-Banach Theorems 59 9.1 The analytic form of the Hahn-Banach theorem . .......... 59 9.2 The geometric forms of the Hahn-Banach Theorem . .......... 61 10 The uniform boundedness principle and the closed graph theorem 65 10.1 The Baire category theorem . .......... 65 10.2 Theuniformboundednessprinciple . ................ 66 10.3 The open mapping theorem and the closed graph theorem . ............ 67 3 4 CONTENTS 11 Weak topologies 69 11.1 The inverse limit topology of a family of maps . ........... 69 11.2 The weak topology σ(E, E ∗) ........................................ 69 11.3 The weak ∗ topology σ(E∗, E )........................................ 73 11.4 Reflexive spaces and separable spaces . .............. 75 11.5 Uniformly Convex Space . .......... 78 12 Exercises 79 IV Lp spaces and Hilbert spaces 87 13 Lp spaces 89 p p 13.1 Density properties and separability of L spaces. Lloc spaces ...................... 89 13.2 Dual space of Lp and reflexivity of Lp spaces ............................... 92 14 Hilbert spaces 97 14.1 Definition of Hilbert spaces and geometrical properties . ................ 97 14.2 Dual of a Hilbert space and reflexivity of Hilbert spaces . 100 14.3 Hilbert sums. Orthonormal bases . 102 15 Exercises 109 V Operator theory 115 16 Introductory concepts 117 16.1 Basic concepts and examples . 117 16.2 Adjoint operators . 120 17 Compact operators 123 17.1 Definitions and basic properties . 123 17.2 TheRiesz-Fredholmtheory . 124 17.3 The spectrum of a compact operator . 128 18 Self-adjoint operators on Hilbert spaces 131 19 Exercises 133 Part I Introduction 5 Chapter 1 Motivation Why functional analysis? Recall a typical problem from linear algebra. Problem 1.0.1 . Let A be an n n matrix with real entries, and let y Rn be a given vector. Find all the vectors x Rn such that × ∈ ∈ Ax = y. (1.1) The vector x has to be determined. Borrowing geometrical terminology we call the vectors x, y points . The right to treat this problem is that of vector spaces (Rn) and linear mappings or operators (A). Now note that a point x = ( x ,...,x ) Rn can be viewed as a function x : 1,...,n R, i.e. 1 n ∈ { } → Rn = x : 1,...,n R . { { } → } So our points are actually functions on a finite set . Functional analysis is the right setting to treat problems (equations) where the sought-after object (unknown) is a function on an infinite set . Let us look at a more specific example. Problem 1.0.2 . Let α, y 0 R and assume we are given a continuous function g : [0 , + ) R. Find all the differentiable functions y :∈ [0 , + ) R such that ∞ → ∞ → y′(t) = αy (t) + g(t) for all t 0 ≥ (y(0) = 0 This is a Cauchy problem for a linear differential equation. Integrating the differential equation on the interval [0 , t ] for an arbitrary t > 0 we get t t y(t) α y(s)ds = y + g(s)ds. (1.2) − 0 Z0 Z0 The unknown ‘point’ is now a differentiable function y on the half line [0 , + ). Can we see the set of differentiable functions on [0 , + ) as a linear space? Can we see the mapping ∞ ∞ t y = y(t) A[y] = A[y]( t) A[y]( t) = y(t) α y(s)ds 7→ − Z0 as a linear operator acting on the ‘point’ y? If so, then the expression (1.2) could be written as . t A[y]( t) = h(t) , h (t) = y0 + g(s)ds, Z0 and the latter expression could be reminiscent of (1.1). 7 8 CHAPTER 1. MOTIVATION Functional analysis provides the best setting to extend notions such as linear space and linear operator to functions on infinite sets . In the example of problem 1.0.2, the candidate linear space to work with is the space of differentiable functions on the half line [0 , + ). This is actually a subset of a wider set, the set of continuous functions on [0 , + ). Let us focus on this set for∞ a moment, and change the half line [0 , + ) to [0 , 1] for simplicity. ∞ ∞ Fact 1.0.1 . The set of continuous functions on [0 , 1] is a linear space . In order to see this we must equip said set (that we name C([0 , 1])) with two operations, namely sum between points and product of a point by a real number. Given f, g C([0 , 1]), we introduce f + g C([0 , 1]) defined by ∈ ∈ (f + g)( x) = f(x) + g(x). Given λ R and f C([0 , 1]), we introduce λf C([0 , 1]) defined by ∈ ∈ ∈ (λf )( x) = λf (x). Note that in the above two definitions we are using two well known operations (sum and product between real numbers) to define operations between functions. Every linear space needs its zero point. In our case, the zero point is obviously the function f(t) = 0 for all t [0 , 1]. ∈ Fact 1.0.2 . The space C([0 , 1]) has infinite dimension . To see this, recall from linear algebra that the dimension of a linear space is the largest possible integer number N of linearly independent vectors, i.e. the largest possible number of vectors v1,...,v N for which α1v1 + . + αN vN = 0 implies α1 = . = αN = 0, i.e. the largest possible number of vectors in which none of them can be written as a linear combination of the others. Now, assume that the dimension of our linear space C([0 , 1]) is given by some (possibly very large) integer N N. Consider the set of points ∈ . 2 . N v0(t) = 1 , v 1(t) = t, v 2(t) = t , ...,v N (t) = t . We claim that the above N + 1 points are linearly independent. Assume N i αit = 0 (1.3) i=0 X for some α , α ,...,α R. Note that (1.3) must hold for all t [0 , 1], so in particular for t = 0. This implies 0 1 N ∈ ∈ α0 = 0. Now divide (1.3) by t > 0, which implies N i 1 α1t − = 0 i=1 X for all t > 0. Now t can be clearly sent to zero in the above expression, and this easily implies α1 = 0. Iterating the procedure we prove that α2 = . = αN = 0. Hence, the N + 1 monomials are linearly independent. But that contradicts the fact that the dimension of the space was N, and N was an arbitrary integer. Hence, the dimension is infinite. So, here is the big issue with functional analysis. Linear spaces of functions such as C([0 , 1]) do not fit the classical background of linear algebra that we learned in our bachelor studies, because there linear spaces were always finite dimensional . This had a lot of implications, most importantly linear mappings could be expressed via matrices with a finite number of entries. Functional analysis is the extension of linear algebra to linear spaces with infinite dimension . Such spaces are typically spaces of functions on infinite sets, often functions on subsets of R or subsets of the Euclidean space Rn. It is known that one can measure distances in a finite dimensional linear space. In the most intuitive case, n namely the Euclidean space R , the distance between two points x = ( x1,...,x n) and y = ( y1,...,y n) is given by the canonical formula d(x, y ) = (x y )2 + . + ( x y )2. 1 − 1 n − n p 9 Other non-canonical ways could be d (x, y ) = x y + . x y , 1 | 1 − 1| | n − n| or d (x, y ) = max xi yi . ∞ i=1 ,...,n | − | We could measure distances between points using all of those distances, and of course they might (except in the case n = 1) give rise to different concepts of distance. However, here is an important fact. + n n Fact 1.0.3 . Given a sequence of points xk k=1∞ R and a point x R . Then d1(xk, x ) 0 as k + if and { } ⊂ ∈ → → ∞ n only if the same holds with d2 or d replacing d1. In other words, the notion of convergence of a sequence in R ∞ is not affected by the choice of the ‘distance’ ( d1, d, or d ). Another way to see it: ‘being close’ in the d1 sense is equivalent to being close in the d sense of in the d sense.∞ Proving this fact is not too difficult, but we shall omit ∞ the details at this stage. The main idea is the following: think of x = ( x1,...,x n) and xk = (( xk)i,..., (xk)n) as n vectors in R ; having xk and x very close in any of the three distances under consideration is equivalent to having the i-th components ( xk)i and xi very close (as real numbers!) for all i = 1 ,...,n .
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