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Extension of Lipschitz Functions University of Innsbruck Faculty of Mathematics, Computer Science and Physics Department of Mathematics Extension of Lipschitz Functions Lorenz Oberhammer Master’s Thesis Submitted to the Director of Studies of the University of Innsbruck in partial fulfilment of the requirements for the degree of Diplom-Ingenieur Supervisor: Univ.-Prof. Dr. Eva Kopecká Innsbruck, 2016 Abstract In this master’s thesis two recent results related to the problem of the extension of Lipschitz functions are presented. The first one deals with functions defined on subsets of metric spaces with values in the real numbers that are both Lipschitz and continuous with respect to some given topology on the metric space (not necessarily the topology induced by the metric). A condition is given when such functions admit extensions to the whole space that preserve both the Lipschitz condition and continuity with respect to the topology. The second one examines the possibility of “continuous” selections of extensions for Lipschitz functions between Hilbert spaces. iii Acknowledgments First and foremost, I would like to thank my supervisor, Univ.-Prof. Dr. Eva Kopecká, for giving me the opportunity to write about such beautiful mathemat- ics, for her continuous support, and for her patience during the sometimes lengthy process of finishing this master’s thesis. Many thanks also to my colleagues and friends for some interesting discussions we had, in particular Noema Nicolussi, who is so incredibly talented at finding counterexamples. Parts of the initial work on this master’s thesis have been carried out during my stay at Aristotle University of Thessaloniki from September 2014 to June 2015. I would like to thank all people who contributed to making this stay such a rewarding experience. Furthermore, I am grateful for the financial support I recieved during that period through the Erasmus+ Programme of the European Commission. Last but not least, I would also like to thank my family, for being always there for me. v Contents 1 Introduction 1 2 Extension Theorems for Lipschitz Functions 5 2.1 Basic definitions . 5 2.2 A word of caution . 5 2.3 The Kirszbraun intersection property . 7 2.4 An extension theorem of McShane . 8 2.5 Coordinate-wise extension . 10 2.6 Injective metric spaces . 11 2.7 Kirszbraun’s Theorem . 12 2.8 When the nearest point mapping is a retraction . 16 2.9 Preserving the closed convex hull of the image . 17 2.10 What about Banach spaces? . 17 3 Urysohn’s Lemma and Tietze’s Extension Theorem 19 3.1 Normal topological spaces . 19 3.2 A generalized variant of recursion . 21 3.3 Urysohn’s Lemma . 21 3.4 Tietze’s Extension Theorem . 24 4 Double Extensions 27 4.1 A counterexample . 27 4.2 Matoušková’s positive result . 28 4.3 Compact Hausdorff spaces . 41 4.4 Duals of Banach spaces . 42 4.5 Reflexive Banach spaces . 43 4.6 Closed points . 44 4.7 Unbounded functions . 45 4.8 A second proof for Theorem 4.2 . 47 4.9 But is it really weaker? . 54 5 Michael’s Selection Theorem 57 5.1 Paracompact topological spaces . 57 5.2 The relation of paracompactness to other properties . 57 vii Contents 5.3 Partitions of unity . 62 5.4 Michael’s Selection Theorem . 62 6 Continuous Selections 67 6.1 Closed and convex sets . 67 6.2 Compact sets in Euclidean space . 68 6.3 Continuous selections . 76 List of Symbols 81 Bibliography 83 Index 85 viii List of Figures 2.1 Illustration for Example 2.1. 6 2.2 The Kirszbraun intersection property. 7 2.3 Injective metric spaces. 11 3.1 Illustration for the proof of Urysohn’s Lemma. 23 4.1 Illustration for Lemma 4.3. 29 4.2 Illustration for Lemma 4.4. 30 4.3 Illustration for Lemma 4.5. 31 5.1 The relation of paracompactness to other properties. 61 5.2 Lower semi-continuity and upper semi-continuity. 64 6.1 Extending gA when Lip(gA) is small. 77 6.2 Extending gA when Lip(gA) is large. 78 ix Chapter 1 Introduction The aim of this master’s thesis is to present some recent results of research on two aspects of the following general question. Problem. Let (M, dM ) and (N, dN ) be metric spaces. Assume that S ⊆ M and that f : S → N is L-Lipschitz, for some L > 0. Does there exist an extension of f to ˜ ˜ M, i.e. a function f : M → N such that f|S = f, that is again L-Lipschitz? Not surprisingly, this is not always possible. However, there are two classic theorems which answer the above question pos- itively for all subsets S and all functions f if the spaces (M, dM ) and (N, dN ) are members of certain classes of metric spaces. The following result, which is due to E. J. McShane [13], implies that the above question can always be answered positively when (N, dN ) is the real line with the usual metric, regardless of what the space (M, dM ) is. Theorem (McShane 1934). Let (X, d) be a metric space, S ⊆ X and f : S → R a function that satisfies the Lipschitz condition |f(x) − f(y)| ≤ Ld(x, y) on S, for some L > 0. Then the function Φ(x) := sup(f(¯x) − Ld(¯x, x)) (x ∈ X) x¯∈S extends f and |Φ(x) − Φ(y)| ≤ Ld(x, y) for all x, y ∈ X. Likewise, by Kirszbraun’s Theorem [6] it is possible to find an extension as above when both M and N are Hilbert spaces and dM and dN are the metrics induced by the inner products of M and N, respectively. Theorem (Kirszbraun 1934). Let H1 and H2 be Hilbert spaces, S ⊆ H1 and g : S → H2 a function with the property that kg(x) − g(y)kH2 ≤ Lkx − ykH1 for all x, y ∈ S, for some L > 0. Then there exists a function f : H1 → H2 such that kf(x) − f(y)kH2 ≤ Lkx − ykH1 for all x, y ∈ H1, and such that f|S = g. We begin our discussion of the subject with those two theorems, which will be presented with full proof in Chapter 2. Occasionally we will also make some remarks on possible generalizations. Other extension theorems, for the special case where the range space is R, are Urysohn’s Lemma [20] and Tietze’s Extension Theorem [19]. 1 1. Introduction Theorem (Urysohn 1925). Let X be a normal topological space. Then for each pair of closed and disjoint subsets A and B of X, there exists a continuous function f : X → [0, 1] such that A ⊆ f −1({0}) and B ⊆ f −1({1}). Theorem (Tietze 1915). Let X be a normal topological space. If F is a closed subset of X and f : F → R is continuous, then there exists a continuous extension ˜ ˜ f : X → R of f. Furthermore, the extension can be chosen such that infF f ≤ f ≤ supF f on X. Note that the assumptions in the second theorem are more general than the assumptions in the result of McShane mentioned above, as the domain space is allowed to be a member of the larger class of normal topological spaces (instead of a metric space) and just continuity is required for the function f (instead of Lipschitz continuity). Also the additional requirement that f is defined on a closed subset of the space is not really a restriction, as Lipschitz continuous functions into complete metric spaces allow a unique Lipschitz continuous extension to the closure. However, the conclusion gets weaker, too, as the extension provided by the theorem is only continuous (not Lipschitz continuous). It is the purpose of Chapter 3 to establish those results. The first major problem dealt with in this master’s thesis is the question whether it is possible to combine the theorems of McShane and Tietze. Problem. Let (X, τ) be a normal topological space and d a metric on X. Let F be a subset of X which is closed with respect to τ, and let g be a real-valued function defined on F that is both τ|F -continuous and L-Lipschitz, for some L > 0. Then by the result of McShane there exists an extension f1 of g to the whole space that is L-Lipschitz, and by Tietze’s Extension Theorem there exists an extension f2 of g to the whole space that is τ-continuous and such that infF g ≤ f2 ≤ supF g on X. Is it possible to have f1 = f2? Again not really surprisingly, it turns out that in general this is not possible. However, E. Matoušková [11] could show that it is possible if we restrict our- selves to bounded functions and require that the topology τ and the metric d are “compatible” in a certain sense. Theorem (Matoušková 1997). Let (X, τ) be a normal topological space and d a metric on X such that the set B(A, α) = {x ∈ X; d-dist(A, x) ≤ α} is τ-closed for every τ-closed A ⊆ X and every α > 0. Suppose that F ⊆ X is τ-closed and g : F → R is a bounded τ|F -continuous function that is L-Lipschitz in d for some L > 0. Then there exists a τ-continuous function f : X → R such that f|F = g, infF g ≤ f ≤ supF g on X, and that is L-Lipschitz in d. The idea here is to adapt the proof of Urysohn’s Lemma. As corollaries to this theorem one gets that such “double extensions” in particular exist in compact Hausdorff spaces with lower semi-continuous metrics and in duals 2 of Banach spaces with the weak* topology, which is also a canonical example of spaces that naturally carry a topology (the weak* topology) and a metric (the norm-distance).
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