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An Introduction to Functional Analysis Appalachian State University Department of Mathematics Ralph Chikhany Introduction to Functional Analysis c 2015 A Directed Research Paper in Partial Fulfillment of the Requirements for the Degree of Master of Arts May 2015 Approved: Dr William J Cook Member Dr William C Bauldry Member Contents Introduction 1 1 Metric Spaces 3 1.1 Definition and Examples . .3 1.2 Underlying Topology . .6 1.3 Convergence and Continuity . .9 1.4 Completeness and Completion . 11 1.5 Compactness . 15 2 Banach Spaces 18 2.1 Normed Vector Spaces . 18 2.1.1 Vector Spaces . 18 2.1.2 Normed Spaces . 19 2.1.3 Bounded Linear Operators . 22 2.2 Banach Spaces . 24 2.2.1 Introduction to Differential Equations in Banach Spaces . 25 2.3 Linear Functionals and Duality . 28 3 Hilbert Spaces 31 3.1 Inner Product Spaces and Orthogonality . 31 3.2 Hilbert Spaces . 35 3.3 Orthonormal Sequences and Bases . 38 3.4 Representation Theorems . 41 3.5 Special Operators . 44 3.6 Weak-Weak∗ Topologies and Convergence . 48 Bibliography 51 Introduction Functional Analysis was founded at the beginning of the 20th century to provide a general abstract domain for some problems, namely from physics, where the issue was finding functions verifying certain properties. With this incentive in mind, functional analysis can be perceived as the study of function spaces. To place this field in relation to differential and integral calculus, we can classify the study of functions in three levels, from the most elementary to the abstract: The first level consists of studying properties of individual functions, namely their domain, extremas, concavity, asymptotes... These properties can thus be represented and perceived graphically and geometrically. This is mainly the concern of beginner calculus courses. The second level consists of studying more general properties of collections of functions, such as continuity and convergence notions on sequences and series of functions. This is mainly the concern of real analysis courses. The third level consists of studying properties of function spaces. Those are sets of functions of given kind from one set to another. At this level, some abstract properties of collections of functions can be expressed as properties of certain function spaces. However, one should understand that moving from finite to infinite dimensions isn't the easiest jump for students, since we are losing the geometric intuition. Functional analysis is one of those fields that studies general concepts that might seem abstract and without practical use , but the results have turned out to be of utmost importance and efficacy for many fields, namely physics and economics, for the past century. To get familiar with the depth of some methods and key concepts, one must jump back and forth between definitions from one section to another, while relating theorems and trying to apply some previous examples to the concepts explained in a current section (for instance, one could go back and verify if the metric spaces discussed in the first chapter are normed, Banach, inner product or Hilbert, even if those are not clearly indicated in an exercise or example). And just like in any mathematics course, studying new structures often comes with the study of transformations and mappings between spaces. We will focus in this course on properties of linear transformations between spaces, namely functionals between normed spaces, which can be viewed as functions of functions. 1 The goal of this directed research project is to provide complete lecture notes with exercises for each section, for anyone willing to teach an introductory course to functional analysis at the senior undergraduate or graduate level. One could copy everything written in the next 50 or so pages verbatim, and the content would be more than enough for a semester's worth of material, provided that the strategy of lecturing for a couple of sessions then stopping and solving problems with students for a class meeting is adopted. This is essential since courses like this tend to get really boring and frustrating for students if the same routine lecturing method is adopted in every class meeting. With this strategy kept in mind, one could interchange the number of class meetings for lecturing or problem solving. A rough schedule for the number of meetings for each section is provided below: Section Lecture days Problems days Total 1.1 1.5 1.5 3 1.2 1 1 2 1.3 2 1 3 1.4 2.5 1.5 4 1.5 1 1 2 2.1 2 2 4 2.2 2 1 3 2.3 1 2 3 3.1 2 1 3 3.2 1.5 1.5 3 3.3 2 1 3 3.4 2 1 3 3.5 1 2 3 3.6 2 1 3 Not all exercises should be done by the professor. Most of them should be turned in for credit, but problem sessions should be aimed at helping students get started with the tricky proofs or problems. If this class is dual listed (audience is a mix of senior undergraduates and graduate students), some problems in each section (starred) should be additionally assigned to graduate students for their portion of the grade. A midterm and a final are necessary, and preferably given as a take home since a lot of definitions, theorems and proofs are being covered, and 50 minute exams are not enough to test student on all concepts. With a total of 14 sections, a suggested grade distribution could be the following: 65% for homework (5% each, drop the lowest), 15 % for the midterm and 20% for the final. If anyone decides to teach or take this course as an independent study at Appstate, or anywhere else, and decides to use my notes and exercises, I hope to get constructive feedback and criticism, since I was learning the material while working on the next 50 pages. Happy learning/teaching! 2 Chapter 1 Metric Spaces 1.1 Definition and Examples Definition. A metric space (M; d) is an ordered pair consisting of a set M and a distance + function d : M × M ! R such that for all x; y; z 2 M: (a) d(x; y) = 0 () x = y, (b) d(x; y) = d(y; x), (c) d(x; z) ≤ d(x; y) + d(y; z) (triangle inequality). Definition. A subspace (M 0; d0) of (M; d) is a metric space on M 0 ⊆ M with the metric d 0 0 0 0 restricted to M × M . The map d = djM 0×M 0 is called the metric induced on M by d. Example. The real line equipped with the usual metric d(x; y) = jx − yj is a metric space. In n general, the set F where F = Z; Q; R or C equipped with the metric v u n uX 2 d(~x;~y) = t jξi − ηij i=1 where ~x = ξ1; ξ2; ...ξn and ~y = η1; η2; ...ηn. This is also known as the Euclidean metric. When 2 n = 2 and F = R, we obtain the regular distance function in R pictured next. 2 Example. Sets can be equipped with multiple distinct metrics. Another metric on R is d = jξ1 − η1j + jξ2 − η2j and is known as the taxicab metric, since its the distance one would travel by taxi on a rectangular grid of streets. It is also pictured below. Proof. We will prove that taxicab metric is indeed a metric. We need to verify the three conditions listed in the definition. Let ~x = (x1; x2), ~y = (y1; y2) and ~z = (z1; z2). (a) d(x; y) = 0 () jx1 − y1j + jx2 − y2j = 0 () jx1 − y1j = 0 and jx2 − y2j = 0 () x = y 3 2 Figure 1.1: Euclidean and Taxicab Metrics in R (b) d(x; y) = jx1−y1j+jx2−y2j = jy1−x1j+jy2−x2j = d(y; x) (c) d(x; y) + d(y; z) = jx1 − y1j + jx2 − y2j + jy1 − z1j + jy2 − z2j = (jx1 − y1j + jy1 − z1j) + (jx2 − y2j + jy2 − z2j) ≥ (jx1 − y1 + y1 − z1j) + (jx2 − y2 + y2 − z2j) = (jx1 − z1j) + (jx2 − z2j) = d(x; z) ( 0 x = y Example. Define d(x; y) = . Then (M; d) for any set M is called a discrete 1 x 6= y metric space equipped with the discrete metric. Example. Consider the function space C[a; b] = ff :[a; b] ! C j f continuousg, i.e. the set of all continuous functions on a given closed interval, and define 8 1 Z b p > p > jf(x) − g(x)j dx 1 ≤ p < 1 <> a dp(f; g) = > > :> max jf(x) − g(x)j p = 1 a≤x≤b Then (C[a; b]; dp) is a metric space. 1 p X p Example. We define the space ` of sequences ~x = (x1; x2; :::) such that jxij converges for i=1 1 1 ! p X p a fixed p ≥ 1, and the metric d(x; y) = jxi − yij . Then (`p; d) is a metric space. When i=1 p = 2 we obtain the Hilbert sequence space, the earliest example of Hilbert spaces studied later in this course. 4 Proof. Clearly, d satisfies the first two conditions of being a metric. Proving that d satisfies the triangle inequality is not trivial. One needs to recall a couple of results. First, define q to be the conjugate exponent of p, meaning 1 1 + = 1 or (p − 1)(q − 1) = 1 p q This form helps in deriving the first of these necessary inequalities. 1 1 1=p 1 1=q X X p X q • Holder's inequality for sequence spaces: jxk ykj ≤ jxkj jykj k=1 k=1 k=1 Note that if p = 2 then q = 2 and this yields the Cauchy-Schwarz inequality.
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