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ARZEL`A-ASCOLI's THEOREM in UNIFORM SPACES Mateusz Krukowski 1. Introduction. Around 1883, Cesare Arzel`

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ARZEL`A-ASCOLI's THEOREM in UNIFORM SPACES Mateusz Krukowski 1. Introduction. Around 1883, Cesare Arzel` DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018020 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 1, January 2018 pp. 283{294 ARZELA-ASCOLI'S` THEOREM IN UNIFORM SPACES Mateusz Krukowski∗ L´od´zUniversity of Technology, Institute of Mathematics W´olcza´nska 215, 90-924L´od´z, Poland Abstract. In the paper, we generalize the Arzel`a-Ascoli'stheorem in the setting of uniform spaces. At first, we recall the Arzel`a-Ascolitheorem for functions with locally compact domains and images in uniform spaces, coming from monographs of Kelley and Willard. The main part of the paper intro- duces the notion of the extension property which, similarly as equicontinuity, equates different topologies on C(X; Y ). This property enables us to prove the Arzel`a-Ascoli'stheorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki's theorem, we establish the relative compactness of 0 n subfamily of C(R; D (R )). 1. Introduction. Around 1883, Cesare Arzel`aand Giulio Ascoli provided the nec- essary and sufficient conditions under which every sequence of a given family of real-valued continuous functions, defined on a closed and bounded interval, has a uniformly convergent subsequence. Since then, numerous generalizations of this result have been obtained. For instance, in [10], p. 278 the compact families of C(X; R), with X a compact space, are exactly those which are equibounded and equicontinuous. The space C(X; R) is given with the standard norm kfk := sup jf(x)j; x2X where f 2 C(X; R). Changing the norm topology to the topology of uniform convergence on com- pacta, one can obtain a version of Arzel`a-Ascoli'stheorem on C(X; Y ), where X is a locally compact Hausdorff space and Y is a metric space ([10], p. 290). The author, together with Bogdan Przeradzki endeavoured to retain the topology of uni- form convergence (cf. [7]). The idea of using the extension property goes back to [11], where B. Przeradzki studied the existence of bounded solutions to the equation x0 = A(t)x + r(x; t), where A is a continuous function taking values in the space of bounded linear operators in a Hilbert space and r is a nonlinear continuous map- ping. The paper gives a characterization of relatively compact sets K ⊂ Cb(R;E), where E is a Banach space. In addition to pointwise relative compactness and equicontinuity, the following condition was introduced: (P): For any " > 0, there exist T > 0 and δ > 0 such that if kx(T ) − y(T )k ≤ δ, then kx(t) − y(t)k ≤ " for t ≥ T and if kx(−T ) − y(−T )k ≤ δ, then kx(t) − y(t)k ≤ " for t ≤ −T , where x and y are arbitrary functions in K. 2010 Mathematics Subject Classification. Primary: 46E10; Secondary: 54E15. Key words and phrases. Arzel`a-Ascoli'stheorem, uniform spaces, uniformity of uniform con- vergence (on compacta). ∗ Corresponding author: Mateusz Krukowski. 283 284 MATEUSZ KRUKOWSKI R. Sta´nczydeveloped this approach, while investigating the existence of solutions to Hammerstein equations in the space of bounded and continuous functions (cf. [14]). The author, together with B. Przeradzki rewrote (P) for σ-locally compact Haus- dorff space X and metric space Y (cf. [7]). In this paper, we improve the result obtained in [7] by considering the uniform spaces (more precisely, Y will be a Hausdorff uniform space), which are thoroughly described e.g. in [4] Chapter 7 and 8. The concept of uniform spaces generalizes the concept of metric spaces and provides a convenient setting for studying the uniform continuity and the uniform convergence, as well. A brief recap on the uniformity of uniform convergence and the uniformity of uniform convergence on compacta is presented in section2. We highlight the importance of equicontinuity, which equates the topology of pointwise convergence, the compact-open topology and the topology of uniform convergence on compacta. This property of `bringing the topologies together' is a motivation for the extension property, whose objective is to equate the topology of uniform convergence with the aforementioned ones. The main part of the paper is in Section3, where we build on the Arzel`a-Ascoli's theorem in [5], p. 236 (Theorem 3.1 in this paper). We introduce a new con- cept of extension property and investigate how it relates to the Cb(X; Y )-extension property studied in [7]. The culminating point is Theorem 3.2, which characterizes compact subsets of C(X; Y ) with the topology of uniform convergence (rather than uniform convergence on compacta). The Hausdorff space X is assumed to be locally compact, while Y is assumed to be a Hausdorff uniform space. Finally, in Section4, we present two possible applications of Theorem 3.2 in the theory of distributions. Let us recall that the space D0(Rn) is known to be nonmetrizable ([2], p. 81), yet it can be considered as a uniform structure. 2. Pointwise convergence, uniform convergence and uniform convergence on compacta. We begin with the theorem, which describes the uniformity of pointwise convergence. This result summarizes the discussions in [4], p. 279 and [5], p. 182. Theorem 2.1. (Uniformity of pointwise convergence) Let X be a set and let (Y; V) be a uniform space and F ⊂ Y X . The uniformity of pointwise convergence on F, which is denoted by Wpc, is the product uniformity on F. The subbase set of this uniformity is of the form (f; g) 2 F × F :(f(x); g(x)) 2 V where x 2 X and V 2 V. We recall the concept of the uniformity of uniform convergence. A comprehensive study of this notion can be found in [5], p. 226 or [16], p. 280. For convenience of the reader, we extract the essence of these discussions in the following: Theorem 2.2. (Uniformity of uniform convergence) Let X be a set, (Y; V) be a uniform space, F ⊂ Y X and let us define a mapping y : V! 2F×F by the formula y 8V 2V V := (f; g) 2 F × F : 8x2X (f(x); g(x)) 2 V : (1) y Then the family fV : V 2 Vg is a base for the uniformity Wuc on F, which we call the uniformity of uniform convergence. ARZELA-ASCOLI'S` THEOREM IN UNIFORM SPACES 285 The topology induced by the uniformity of uniform convergence is called the topology of uniform convergence, which we denote by τuc. The base sets for this topology are of the form y V [f] = g 2 F : 8x2X g(x) 2 V [f(x)] ; (2) where V [y] = fz 2 Y :(y; z) 2 V g as in [5], p. 176. A concept, which is closely related to the uniformity of uniform convergence is the uniformity of uniform convergence on compacta. It appears for example in [5], p. 229 or [16], p. 283. Theorem 2.3. (Uniformity of uniform convergence on compacta) Let (X; τX ) be a topological space, (Y; V) be a uniform space and let F ⊂ Y X . The family (f; g) 2 F × F : 8x2D (f(x); g(x)) 2 V (3) V 2V;D⊂⊂X is a subbase for the uniformity Wucc on F, which we call the uniformity of uniform convergence on compacta. Remark 1. The notation D ⊂⊂ X stands for `D is a compact subset of X'. The topology induced by the uniformity of uniform convergence on compacta is called the topology of uniform convergence on compacta, which we denote by τucc. It is an easy observation that τucc is weaker than τuc. In fact, if we restrict our attention to C(X; Y ) rather than to Y X , which we are going to do in the sequel, τucc becomes a familiar compact-open topology (see the proof in [5], p. 230 or [16], p. 284). Another important concept is that of equicontinuity (see [5], p. 232 or [16], p. 286). For a topological space (X; τX ) and a uniform space (Y; V), we say that the family F ⊂ Y X is equicontinuous at x 2 X if 8V 2V 9Ux2τX 8f2F f(Ux) ⊂ V [f(x)]: (4) The family F is said to be equicontinuous if it is equicontinuous at every x 2 X. If one considers such a family, then the topology of pointwise convergence coincides with the compact-open topology, i.e. τpcjF = τcojF (see the proof in [5], p. 232 or [16], p. 286). Together with what we said above and the fact that an equicontinuous family consists of continuous functions, we infer that the topology of pointwise convergence coincides with the topology of uniform convergence on compacta on F, i.e. τpcjF = τuccjF . 3. Arzel`a-Ascoli'stheorem. Our objective in this section is to present the Arzel`a- Ascoli's theorem for the topology of uniform convergence in the setting of uniform spaces. Our starting point is the theorem provided in [5], p. 236. Theorem 3.1. (Arzel`a-Ascoli's theorem for uniform convergence on compacta) Let (X; τX ) be a locally compact Hausdorff space, (Y; V) be a Hausdorff uniform space and let C(X; Y ) be endowed with the topology of uniform convergence on compacta τucc. A family F ⊂ C(X; Y ) is relatively τucc-compact if and only if the following two conditions are satisfied: (AAucc1): F is pointwise relatively compact, i.e. ff(x): f 2 Fg is relatively τY -compact at every x 2 X; (AAucc2): F is equicontinuous. 286 MATEUSZ KRUKOWSKI Our line of attack is motivated by the equicontinuity, which equates three topolo- gies, namely the topology of pointwise convergence, the compact-open topology and the topology of uniform convergence on compacta.
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