Ghent University Faculty of Sciences Department of Mathematics Surjections in locally convex spaces by Lenny Neyt Supervisor: Prof. Dr. H. Vernaeve Master dissertation submitted to the Faculty of Sciences to obtain the academic degree of Master of Science in Mathematics: Pure Mathematics. Academic Year 2015{2016 Preface For me, in the past five years, mathematics metamorphosed from an interest into my greatest passion. I find it truly fascinating how the combination of logical thinking and creativity can produce such rich theories. Hopefully, this master thesis conveys this sentiment. I hereby wish to thank my promotor Hans Vernaeve for the support and advice he gave me, not only during this thesis, but over the last three years. Furthermore, I thank Jasson Vindas and Andreas Debrouwere for their help and suggestions. Last but not least, I wish to express my gratitude toward my parents for the support and stimulation they gave me throughout my entire education. 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Ghent, May 31, 2016 Lenny Neyt i Contents Preface i Introduction 1 1 Preliminaries 3 1.1 Topology . 3 1.2 Topological vector spaces . 7 1.3 Banach and Fr´echet spaces . 11 1.4 The Hahn-Banach theorem . 13 1.5 Baire categories . 15 1.6 Schauder's theorem . 16 2 Locally convex spaces 18 2.1 Definition . 18 2.2 Topological aspects of locally convex spaces . 24 2.2.1 Hausdorff . 24 2.2.2 Convergence . 25 2.2.3 Continuity of linear operators . 26 2.2.4 Boundedness . 30 2.3 Quotient spaces . 34 2.4 Hahn-Banach in locally convex spaces . 36 2.5 Application: Kakutani's fixed point theorem . 41 2.6 Completions . 43 3 Duality theory 46 3.1 Classification of admissible topologies . 46 3.2 Boundedness in duality theory . 54 3.3 Reflexivity . 56 3.4 Continuity and surjectivity of linear maps . 62 3.5 Mackey spaces . 68 3.6 Equicontinuous sets . 68 ii iii 4 Projective and inductive topologies 72 4.1 Definitions . 72 4.2 Bornological spaces . 77 4.3 Schwartz spaces . 80 4.4 The open mapping theorem . 85 4.5 Imbedding spectra and (LF)-spaces . 89 5 Fr´echet spaces and (DF)-spaces 93 5.1 Metrizable spaces . 93 5.2 (DF)-spaces . 98 5.3 Weakly compact imbedding spectra . 102 6 Surjections of Fr´echet Spaces 108 6.1 Characterizing surjectivity . 108 6.2 Surjectivity between Fr´echet spaces . 111 6.3 Applications . 117 6.3.1 Formal power series and Taylor expansions . 117 6.3.2 The existence of C8 solutions of a linear partial dif- ferential equation . 121 7 Surjectivity of dual mappings 124 7.1 Surjectivity of the dual map . 124 7.2 Acyclic (LF)-spaces . 126 7.3 Retractive (LF)-spaces . 133 7.4 Inductive limits of Fr´echet-Montel spaces . 137 Conclusion 142 Indication of the used sources 144 A Nederlandstalige samenvatting 145 B Distributions 147 B.1 The distribution space D1pΩq . 147 B.2 Differentiation . 148 B.3 Fourier transform . 149 B.4 Convolution . 150 Bibliography 152 Index 154 Introduction In mathematics, it is desirable to know whether or not a problem is generally solvable. For instance, consider the partial differential equation ¢ ¸ B k a φ ; (1) n Bx kPNn over some space of sufficiently differentiable functions. We wonder if (1) is solvable for each , i.e. if we put ¢ ¸ B k D : a ; n Bx kPNn we ask ourselves if for each there is a φ such that Dφ . Now, since D is a map, this will exactly be the case if D is a surjection. While this specific problem has been studied extensively (we refer to [2] for a nice overview), we may look at it in a more comprehensive framework. In general, suppose E and F are topological vector spaces, and A : E Ñ F is a continuous linear map, then the problem we will investigate throughout this master thesis is whether or not A is a surjection. This is exactly the case if the image of A is both closed and dense in F . For the latter condition we will find a general characterization, namely injectivity of the transpose map, but for closedness we will need more structure on the spaces to find verifiable requirements. This is why an extensive knowledge about the spaces used in functional analysis is vital, and we will therefore spend a great amount of time on it. In chapter 1, we discuss some preliminary analytic concepts and results we expect the reader to be familiar with and may be bypassed at first, since all used results are referred to. On top of these results, we also expect the reader to be familiar with some basic notions in the field of algebra, such as vector spaces and groups. In chapter 2, we introduce the locally convex spaces, which are the gen- eral spaces we will be working with. These spaces can be seen as a vector spaces equipped with a family of continuous seminorms. The chapter mainly serves to establish the structure of these spaces, in order for it to be easy applicable in the sequel. Nonetheless, we will already utilize these results to 1 2 give an application in the form of a fixed point theorem due to S. Kakutani [5]. In chapter 3, we will be looking at the duals of locally convex spaces and study the possible ways to topologize them. Via this, one can discover a remarkable interplay between a space and its dual, which will greatly increase our understanding of the locally convex spaces. Our first general characterization of the surjectivity problem will also be made here. In chapter 4, we will construct new spaces out of other locally convex spaces, using projective and inductive topologies. This will pave the way for many new categories of spaces, such as the bornological spaces. The main result of this chapter is an extension of the open mapping theorem, a tool that is critical in our search for surjections, to more general spaces, due to de Wilde [18]. In chapter 5, we will focus on Fr´echet spaces and their duals. Since Fr´echet spaces are both complete and metrizable, they will have some strong properties, which we will apply in the chapter following it to characterize surjectivity between Fr´echet spaces. In chapter 6, an equivalent condition for the surjectivity of a map be- tween a Mackey webbed space and an ultra-bornological space, under certain conditions, is shown. This will be an extension of a result due to S. Banach. In particular, the characterization will work for a map between two Fr´echet spaces. Said conditions will be used to discuss two applications: an easy proof of a classical theorem by E. Borel and a characterization of C8 solu- tions of linear partial differential equations. In chapter 7, our perspective will shift by looking at the surjectivity problem for the transpose map. As it will turn out, this problem is linked with the notion of acyclicness, following the definition of Palamodov [8]. Because of this, we will investigate equivalent definitions for this notion, which were discovered by Wengenroth [17]. Chapter 1 Preliminaries This chapter serves as a quick reference to a list of important notions and results that will be used in the sequel. We expect the reader to be familiar with most of what will be discussed, hence not all results shall be given a proof. In order to confirm that all that is reviewed is fact, we refer to the books of Meise-Vogt [7] and Tr`eves [15]. 1.1 Topology Definition 1.1.1. Given a set E, a filter F is a family of subsets of E submitted to the following three conditions: (F1) the empty set H does not belong to the family F; (F2) if A1 and A2 belong to F, then A1 X A2 P F; (F3) any set which contains a set belonging to F should also belong to F. Definition 1.1.2. Let E be a set and suppose that for each x P E, there is a filter Fx of E such that: (a) for each U P Fx, we have x P U; (b) if U P Fx, then there is a V P Fx contained in U with the property that for each y P V , we have V P Fy. This family of filters is called a topology on E, and a set equipped with a topology is called a topological space. Each U P Fx is called a x- neighborhood. Notation 1.1.3. For a topological space E and a point x P E, the family of x-neighborhoods will also be denoted by UxpEq. We may now introduce the following list of notions. 3 CHAPTER 1. PRELIMINARIES 4 Definition 1.1.4. Let E be a topological space. 1. A subset U E is called open if U P UxpEq for each x P U. 2. A subset V E is called closed if EzV is open. 3. A point x of a subset W E is called an interior point of W if there is a U P UxpEq such that U W . The set of all interior points of W is called the interior of W and is denoted by W ¥. 4. The closure W of a subset W E is the intersection of all closed sets that contain W .