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Analysis of Banach Spaces

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Analysis of Banach Spaces Analysis of Banach Spaces Andrew Kobin Spring 2014 Contents Contents Contents 0 Introduction 1 1 Normed Linear Spaces and Banach Spaces 3 1.1 Normed Linear Spaces . .3 1.2 Generalizing the Reals . .5 1.3 Sequences Spaces . 10 2 Function Spaces 16 2.1 Norms on Function Spaces . 16 2.2 The Arzela-Ascoli Theorem . 21 2.3 Approximation . 25 2.4 Contraction Mapping . 31 2.5 Differentiable Function Spaces . 34 2.6 Completing a Metric Space . 35 2.7 Sobolev Space . 38 3 Calculus on Normed Linear Spaces 41 3.1 Differentiability . 41 3.2 Linear Operators . 44 3.3 Rules of Differentiation . 48 3.4 The Mean Value Theorem . 50 3.5 Mixed Partials . 53 3.6 Directional Derivatives . 54 3.7 Sard's Theorem . 55 3.8 Inverse Function Theorem . 61 i 0 Introduction 0 Introduction These notes follow a course on Banach spaces taught by Dr. Stephen Robinson at Wake Forest University during the spring of 2014. The primary reference for the semester was a set of course notes compiled by Dr. Robinson. Our motivation for studying functional analysis is the following question which describes phase transitions in physics (e.g. water $ ice). Question. Given a system whose energy is described by the integral equation Z 1 Z 1 E(u) = ju0j2 du + F (u) du 0 0 for some function F , what are the values for which the system's energy is minimized? In general, this can be thought of as a critical point search. For example, consider F (u) ice state water state The trick is the domain of E(u) is some set of functions u, whereby E is a function from this set of functions to the real numbers R. For this reason we want to be able to perform calculus on such sets of functions, called function spaces. They are a special type of vector space called a normed linear space. The main topics covered in these notes are Normed linear spaces, Banach spaces and their properties The Arzela-Ascoli Theorem Polynomial approximation and the Weierstrass Approximation Theorem Contraction mapping Completion of a metric space Calculus on normed linear spaces, including the Fr´echet derivative, the Mean Value Theorem, Sard's Theorem and the Inverse Function Theorem 1 0 Introduction We briefly review some basic definitions from real analysis that will be relevant through- out the course of these notes. Definitions containing the notation j · j require that X be a metric space, a concept which will be reviewed at the start of the first chapter. Definition. A sequence (sn) in a space X converges to L 2 X, denoted (sn) ! L or lim sn = L, if for every " > 0 there is an N 2 such that for all n > N, jsn − Lj < ". n!1 N Definition. A sequence (sn) is Cauchy if, given any " > 0, there is some N 2 N such that for all n; m > N, jsn − smj < ". Definition. For a set A ⊂ X, a is a limit point of A if there is a sequence (sn) ⊂ A converging to a. Definition. A set A ⊂ X is closed if A contains all of its limit points. Definition. The complement of a closed set is called an open set. Equivalently, A ⊂ X is open if and only if for every a 2 A, there is some neighborhood U (a metric ball if X is a metric space, or a set in a topological basis otherwise) containing a and contained within A. Definition. A set D ⊂ X is dense if X is the smallest closed set containing D. The next three concepts are sometimes referred to as the \Three C's" of analysis, as they represent some of the most fundamental concepts in analytical mathematics. Definition. A space X is said to be complete if every Cauchy sequence in X converges to a point in X. Definition. A set A ⊂ X is compact if every sequence (an) ⊂ A has a subsequence that converges in A. Definition. A function f : X ! X is continuous at a point x0 2 X if for all " > 0, there is some δ > 0 such that jx − x0j < δ implies jf(x) − f(x0)j < ". 2 1 Normed Linear Spaces and Banach Spaces 1 Normed Linear Spaces and Banach Spaces 1.1 Normed Linear Spaces We begin with the fundamental object of study in analysis: a metric space. Definition. A space X is a metric space if there is a function d : X × X ! R satisfying 1) (Positivity) For all x; y 2 X, d(x; y) ≥ 0 and equality holds if and only if x = y. 2) (Symmetry) For all x; y 2 X, d(x; y) = d(y; x). 3) For all x; y; z 2 X, d(x; z) ≤ d(x; y) + d(y; z). Examples. 1 R is a metric space under the usual distance function d(x; y) = jx − yj. 2 For any n ≥ 2, Rn is a metric space under the Euclidean norm n !1=2 X 2 d(x; y) = jjx − yjj := jxi − yij : i=1 3 There is a generalization of Euclidean n-space called `2 which will be important in subsequent sections. One can think of `2 as the space of sequences of real numbers 2 (x1; x2; x3;:::). Then ` is a metric space under the 2-norm: 1 !1=2 X 2 d(x; y) = jxi − yij i=1 4 Let C[0; 1] denote the space of continuous functions on the interval [0; 1]. Then C[0; 1] is a metric space with several compatible metrics. The two most important are: d(f; g) := sup jf(x) − g(x)j x2[0;1] Z 1 1=2 2 and d2(f; g) := jf(x) − g(x)j dx : 0 All of the spaces in these examples have a vector space structure. In addition, the notion of distance is derived from the distance between vectors: d(x; y) = jjx − yjj. In the study of functional analysis, we want to restrict our consideration to a certain type of metric space. Definition. Suppose that X is a real vector space and there is a function jj · jj : X ! R satisfying 1) jjxjj ≥ 0 for all x 2 X, and jjxjj = 0 if and only if x is the zero vector. 2) jjx + yjj ≤ jjxjj + jjyjj for all x; y 2 X. 3) jjαxjj = jαj jjxjj for any α 2 R. Then X is a normed linear space. 3 1.1 Normed Linear Spaces 1 Normed Linear Spaces and Banach Spaces Example 1.1.1. Not every metric space is a normed linear space. For instance, let X = R and define ( 0 x = y d(x; y) = 1 x 6= y which is called the discrete metric. By virtue of d being a metric, (1) and (2) above are satisfied. However, if (3) held we would have 1 = d(2; 0) = jj2 − 0jj = jj2jj = 2jj1jj = 2jj1 − 0jj = 2 · d(1; 0) = 2 · 1 = 2 clearly a contradiction. Therefore (R; d) is not a normed linear space. Example 1.1.2. Consider the unit circle in R2: X = f(x; y) 2 R2 j x2 + y2 = 1g X is not a vector space, so it is certainly not a normed linear space. While not every metric space is a normed linear space, the reverse implication is true. Theorem 1.1.3. If (X; jj · jj) is a normed linear space, then (X; d) is a metric space under the function d(x; y) := jjx − yjj. Proof. Norms are nonnegative, so d ≥ 0. Likewise, d(x; y) = 0 () x = y follows directly from (1) in the definition of a normed linear space. For the triangle inequality (2), consider d(x; z) = jjx − zjj = jjx − y + y − zjj ≤ jjx − yjj + jjy − zjj = d(x; y) + d(y; z): Finally, to show symmetry consider d(y; x) = jjy − xjj = jj − (x − y)jj = j − 1j jjx − yjj = jjx − yjj = d(x; y): Hence d is a metric so (X; d) is a metric space. 4 1.2 Generalizing the Reals 1 Normed Linear Spaces and Banach Spaces 1.2 Generalizing the Reals It is common to prove the some of the following results in a course on real analysis. Lemma 1.2.1. (R; j · j) is a normed linear space. Theorem 1.2.2 (Cauchy Convergence). Every Cauchy sequence in R converges, that is, R is complete. Theorem 1.2.3 (Heine-Borel). A set D ⊂ R is compact () D is closed and bounded. Theorem 1.2.4. The set of rationals Q is dense in R. In this section we will focus on generalizing these properties to Euclidean n-space Rn for finite n. For notational convenience, we will actually restrict our focus to the proofs in R2, but 2 p 2 2 moving up in dimension doesn't alter the proofs much. Consider R with jjxjj = x1 + x2 2 for x = (x1; x2) 2 R . We first prove Proposition 1.2.5. R2 is a normed linear space. The main ingredients in this proof are three inequalities, which we state and prove sep- arately before returning to the main proof. 1 2 1 2 Lemma 1.2.6 (Young's Inequality). For any nonnegative a; b 2 R, ab ≤ 2 a + 2 b . Proof. Consider 1 2 1 2 1 2 1 2 ln(ab) = ln(a) + ln(b) = 2 ln(a ) + 2 ln(b ) ≤ ln 2 a + 2 b where the inequality comes from the concavity of ln(x) { see the diagram below. ln(x) ln(b2) 1 2 1 2 ln a + b 1 2 1 2 2 2 2 ln(a ) + 2 ln(b ) ln(a2) 2 1 2 2 2 a 2 (a + b ) b 1 b 1 2 Taking the exponential of both sides of the inequality ln(ab) ≤ ln 2 a + 2 b proves Young's Inequality.
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