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

Home , Equicontinuity, Uniform continuity

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∗ Lód´zUniversity of Technology, Institute of Mathematics Wólczańska 215, 90-924Lód´z, Poland

Abstract. In the paper, we generalize the Arzelà-Ascoli’stheorem in the setting of uniform spaces. At first, we recall the Arzelà-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 Arzelà-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 Arzelàand Giulio Ascoli provided the necessary 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 |f(x)|, x∈X where f ∈ C(X, R). Changing the norm topology to the topology of uniform convergence on compacta, one can obtain a version of Arzelà-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 uniform 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 mapping. 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 ε > 0, there exist T > 0 and δ > 0 such that if kx(T ) − y(T )k ≤ δ, then kx(t) − y(t)k ≤ ε for t ≥ T and if kx(−T ) − y(−T )k ≤ δ, then kx(t) − y(t)k ≤ ε 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 convergence (on compacta). ∗ Corresponding author: Mateusz Krukowski.

283 284 MATEUSZ KRUKOWSKI

R. Stańczydeveloped 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 Arzelà-Ascoli’s theorem in [5], p. 236 (Theorem 3.1 in this paper). We introduce a new concept 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) ∈ F × F :(f(x), g(x)) ∈ V where x ∈ X and V ∈ 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 † : V → 2F×F by the formula † ∀V ∈V V := (f, g) ∈ F × F : ∀x∈X (f(x), g(x)) ∈ V . (1)

† Then the family {V : V ∈ V} 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 † V [f] = g ∈ F : ∀x∈X g(x) ∈ V [f(x)] , (2) where V [y] = {z ∈ Y :(y, z) ∈ V } 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) ∈ F × F : ∀x∈D (f(x), g(x)) ∈ V (3) V ∈V,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 ∈ X if

∀V ∈V ∃Ux∈τX ∀f∈F f(Ux) ⊂ V [f(x)]. (4) The family F is said to be equicontinuous if it is equicontinuous at every x ∈ X. If one considers such a family, then the topology of pointwise convergence coincides with the compact-open topology, i.e. τpc|F = τco|F (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. τpc|F = τucc|F .

3. Arzelà-Ascoli’stheorem. Our objective in this section is to present the Arzelà- 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. (Arzelà-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. {f(x): f ∈ F} is relatively τY -compact at every x ∈ X; (AAucc2): F is equicontinuous. 286 MATEUSZ KRUKOWSKI

Our line of attack is motivated by the equicontinuity, which equates three topologies, namely the topology of pointwise convergence, the compact-open topology and the topology of uniform convergence on compacta. We look for a property that will do the same thing with the topology of uniform convergence. X For a topological space (X, τX ), uniform space (Y, V) and F ⊂ Y suppose that Wpc and Wucc are uniformities of pointwise convergence and uniform convergence on compacta on F, respectively. We say that F satisfies the finite extension property if † ∀V ∈V ∃W ∈Wpc W ⊂ V , (5) and that it satisfies the compact extension property if † ∀V ∈V ∃W ∈Wucc W ⊂ V . (6)

Intuitively, condition (5) means that the topology τpc coincides with τuc on F (the inclusion τpc ⊂ τuc always holds, so (5) guarantees the reverse inclusion). Condition (6) ‘merely’ implies that τucc coincides with τuc on F. Obviously if F satisfies the finite extension property then it satisfies the compact extension property. In order to strengthen our intuition and provide further motivation, we compare the above conditions with the one appearing in [7]. Condition (6) can be written explicitely as ∀V ∈V ∃D⊂⊂X ∀f,g∈F ∀x∈D g(x) ∈ U[f(x)] =⇒ ∀x∈X g(x) ∈ V [f(x)] . (7) U∈V If we suppose that Y is a metric space, then V [f(x)] becomes a ball B(f(x), ε) and U[f(x)] becomes a ball B(f(x), δ). Hence (7) turns into ∀ε>0 ∃D⊂⊂X ∀f,g∈F sup dY (g(x), f(x)) < δ =⇒ sup dY (g(x), f(x)) < ε δ>0 x∈D x∈X (8) which is the compact extension property studied in [7]. Reasoning analogously for (5), we obtain ∀ε>0 ∃D−finite ∀f,g∈F max dY (g(x), f(x)) < δ =⇒ sup dY (g(x), f(x)) < ε . δ>0 x∈D x∈X (9) Equipped with these tools, we are ready to prove the following theorem.

Theorem 3.2. (Arzelà-Ascoli’s theorem for uniform convergence) 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 τuc. A family F ⊂ C(X,Y ) is relatively τuc-compact if and only if (AA1): F is pointwise relatively compact and equicontinuous; (AA2): F satisfies the finite extension property.

Proof. The ‘if ’ part is easy. By (AA1) and Theorem 3.1 we obtain that F is relatively τucc-compact, consequently (AA2) implies that the topologies τpc and τucc coincide with τuc on F and we are done. We focus on ‘only if ’ part. We already noted that τuc is stronger than τucc, which implies that F is relatively τucc-compact. Hence, by Theorem 3.1, we obtain (AA1). ARZELA-ASCOLI’S` THEOREM IN UNIFORM SPACES 287

Suppose that (AA2) is not satisﬁed, which means that there exists V ∈ V such † 1 that W \V 6= ∅ for every W ∈ Wpc. In what follows, V 3 will mean a symmetric 1 1 1 1 set such that V 3 ◦ V 3 ◦ V 3 ⊂ V and likewise, by V 9 we denote a symmetric set 1 1 1 1 such that V 9 ◦ V 9 ◦ V 9 ⊂ V 3 , where A ◦ B = (a, b): ∃c (a, c) ∈ B, (c, b) ∈ A

1 1 as in [5], p. 176. The existence of sets V 3 and V 9 follows from the axioms of uniformity (see [4], p. 103). The negation of (AA2) means that

1 9 (f, g) ∈ F × F : ∀x∈D (f(x), g(x)) ∈ V , ∃x∗∈X (f(x∗), g(x∗)) 6∈ V 6= ∅ (10) for every ﬁnite set D ⊂ X. The family

1 g ∈ C(X,Y ): ∀x∈X (f(x), g(x)) ∈ V 9 (11) f∈F is a τuc-open cover of F, the closure of F in τuc. Indeed, if f ∈ F, then by the characterization of belonging to a closure, using (2), we get

1 † 1 (V 9 ) [f] ∩ F 6= ∅ ⇐⇒ ∃f∈F ∀x∈X (f(x), f(x)) ∈ V 9 .

Since we assume that F is τuc-compact (and we aim to reach a contradiction), we can choose a ﬁnite subcover from (11), which means that there is a sequence n (fk)k=1, fk ∈ F for k = 1, . . . , n, such that 1 9 ∀g∈F ∃k=1,...,n ∀x∈X (fk(x), g(x)) ∈ V . (12) For every k, l = 1, . . . , n, k 6= l, let us deﬁne a set

1 Dk,l := x ∈ X :(fk(x), fl(x)) 6∈ V 3 .

Let D be a set which consists of one element from each nonempty Dk,l. This set is ﬁnite and serves as a guard, watching whether each pair (fk, fl) ‘drifts apart’. More precisely, the following implication

1 1 ∀x∈D (fk(x), fl(x)) ∈ V 3 =⇒ ∀x∈X (fk(x), fl(x)) ∈ V 3 (13) holds for every k, l = 1, . . . , n. Indeed, if for some k, l = 1, . . . , n there exists x ∈ X 1 such that (fk(x), fl(x)) 6∈ V 3 , then Dk,l 6= ∅. Hence D ∩ Dk,l 6= ∅ and there exists 1 xD ∈ D such that (fk(xD), fl(xD)) 6∈ V 3 . This proves the implication (13). By (10), we pick (f∗, g∗) such that

1 9 ∀x∈D (f∗(x), g∗(x)) ∈ V and ∃x∗∈X (f∗(x∗), g∗(x∗)) 6∈ V. (14)

Let kf , kg ∈ N be constants chosen as in (12) for f∗ and g∗, respectively. We have 1 9 ∀x∈D (fkf (x), f∗(x)), (f∗(x), g∗(x)), (g∗(x), fkg (x)) ∈ V 1 3 =⇒ ∀x∈D (fkf (x), fkg (x)) ∈ V .

1 3 By (13) we know that (fkf (x), fkg (x)) ∈ V for every x ∈ X. Finally,

1 3 ∀x∈X (f∗(x), fkf (x)), (fkf (x), fkg (x)), (fkg (x), g∗(x)) ∈ V

=⇒ ∀x∈X (f∗(x), g∗(x)) ∈ V 288 MATEUSZ KRUKOWSKI which is a contradiction with (14). Hence, we conlcude that the ﬁnite extension property must hold.

The next corollary characterizes the relation between the ﬁnite and the compact extension property.

Corollary 1. Let (X, τX ) be a locally compact Hausdorﬀ space, (Y, V) be a Haus- dorﬀ uniform space and let C(X,Y ) be endowed with the topology of uniform convergence τuc. For a family F ⊂ C(X,Y ) satisfying (AA1), the following conditions are equivalent:

(C1): F is relatively τuc-compact; (C2): F satisfies the finite extension property; (C3): F satisfies the compact extension property.

Proof. Theorem 3.2 claims the equivalence of (C1) and (C2). Moreover, we already observed that the ﬁnite extension property implies the compact extension property ((C2) implies (C3)). Finally, if (C3) is satisﬁed, then the topologies τucc and τuc conicide on F and realtive τuc-compactness follows from Theorem 3.1.

Before presenting the applications of Theorem 3.2, we compare it with the result from the paper [7]. Let X be a σ-locally compact Hausdorﬀ space and let (Y, dY ) be a metric space. By Cb(X,Y ) we denote the metric space of bounded and continuous functions with the distance

dCb(X,Y )(f, g) = sup dY (f(x), g(x)). x∈X We recall that theorem without the proof, because it is an evident corollary of Theorem 3.2. Theorem 3.3. (Arzelà-Ascoli’s theorem for σ-locally compact Hausdorff space) Let (X, τX ) be σ-locally compact Hausdorff space and (Y, dY ) be a metric space. A family b F ⊂ C (X,Y ) is relatively dCb(X,Y )-compact if and only if (KP1): F is pointwise relatively compact; (KP2): F is equicontinuous; (KP3): F satisfies the compact extension property.

The first apparent difference is that Theorem 3.2 does not require X to be a countable union of compact sets. This is due to the fact, that the proof of Theorem 3.3 focuses on sequences (and choosing convergent subsequences) and the authors needed countability in their argument. However, the proof of Theorem 3.2 has rather a topological character, focusing on open covers. Hence, no countability is assumed. The most striking difference is that Theorem 3.2 replaces a metric space (Y, dY ) by a Hausdorff uniform structure (Y, V). This is because uniform spaces serve perfectly as generalizations of metric spaces. Finally, in [7], assuming that Y is a Banach space, we are dealing with a Banach space Cb(X,Y ). Hence the functions needed to be bounded. However, Theorem 3.2 showed that we can consider C(X,Y ) which need not be a metric space. Conse- quently, in theorem 3.3 we could replace Cb(X,Y ) by C(X,Y ), accepting the loss of metricity. ARZELA-ASCOLI’S` THEOREM IN UNIFORM SPACES 289

4. Applications in distribution theory. In the final section of the paper, we present two possible applications of Theorem 3.2. They were motivated by Schwartz’s distribution theory, described e.g. [13] Chapter 2 or [17] Chapter 1. We briefly recall the essentials that will be used in the sequel. n After [17], p. 28 by DK (R ) we denote the set of all real-valued functions ∞ n n f ∈ C0 (R ) with supp(ϕ) ⊂ K ⊂⊂ R . This is a locally convex space with a family of seminorms |s| ∂ kϕkK,m = sup s1 sn ϕ(x) . |s|≤m ∂x1 . . . ∂xn x∈K n Moreover, if K1 ⊂ K2 then the topology of DK1 (R ) coincides with the relative n n topology of DK1 (R ) as a subset of DK2 (R ). Hence, we may construct the inductive n limit, namely the space of test functions D(R ). A set U0 is an open neighbourhood n n n of 0 in D(R ) if it is absorbing, balanced, convex and U0∩DK (R ) is open in DK (R ) for every K ⊂⊂ Rn. The space of distributions, denoted by D0(Rn), comes with the weak∗ topology ([13], p. 94 or [3], p. 51). As such, it can be defined by the pseudometrics

0 n ∀T,S∈D (R ) pϕ(T,S) = |T (ϕ) − S(ϕ)|, where ϕ ∈ D(Rn). It is easy to see that the family −1 0 n 0 n pϕ ([0, ε)) = (T,S) ∈ D (R ) × D (R ): |T (ϕ) − S(ϕ)| < ε , for ϕ ∈ D(Rn) and ε > 0, is a subbase for the uniformity on D0(Rn). Let F be a family of functions from R×Rn to R such that the following conditions are satisﬁed: n (EQ): For every t∗ ∈ R, ε > 0 and K ⊂⊂ R , there exists δ > 0 such that

∀|t−t∗|<δ |f(t, x) − f(t∗, x)| < ε. x∈K f∈F 1 n (PRC): For every t∗ ∈ R, the set f(t∗, ·): f ∈ F is Lloc(R )-relatively compact. (EP): For every ε > 0 and K ⊂⊂ Rn, there exists R > 0 such that for every f, g ∈ F we have Z sup |f(t, x) − g(t, x)| dx < ε. |t|>R K We consider a family of distributions defined by Z n ∀ϕ∈D(R ) Tf,t(ϕ) := f(t, x) ϕ(x) dx, n R where f ∈ F and moreover, we denote Ff : t 7→ Tf,t (a distribution-valued function). With Theorem 3.2 at our disposal, we are able to prove the following: Theorem 4.1. Under the assumptions (EQ), (PRC) and (EP), the family 0 n FF := (Ff )f∈F is relatively τuc-compact in C(R, D (R )). Proof. The first two conditions are stronger than the Carathéodory conditions (cf. [1], p. 153). Condition (EQ) will turn out to be very useful when proving the equicontinuity of FF . (PRC) will be used to establish the pointwise relative 290 MATEUSZ KRUKOWSKI compactness in the space D0(Rn). Finally, with the use of (EP), we will justify (AA2). First of all, we check that FF is equicontinuous, so in particular Ff ∈ C(R, 0 n D (R )) for every f ∈ F. Condition (4) at a fixed point t∗ ∈ R reads as Z

∀ n ∃ ∀ f(t, x) − f(t , x) ϕ(x) dx < ε. (15) ϕ∈D(R ) δ>0 |t−t∗|<δ ∗ n ε>0 f∈F R n For a ﬁxed ϕ∗ ∈ D(R ) and ε > 0, we use (EQ) to choose δ > 0 such that ε ∀ |t−t∗|<δ |f(t, x) − f(t∗, x)| < , λ(supp ϕ ) kϕ k b n x∈supp ϕ∗ ∗ ∗ C (R ) f∈F where λ is the Lebesgue measure on Rn. Hence, Z

∀|t−t∗|<δ f(t, x) − f(t∗, x) ϕ∗(x) dx n f∈F R Z ≤ |f(t, x) − f(t∗, x)| |ϕ∗(x)| dx supp ϕ∗ Z |ϕ (x)| < ε ∗ dx ≤ ε λ(supp ϕ ) kϕ k b n supp ϕ∗ ∗ ∗ C (R ) and we infer that FF is equicontinuous. Next, we prove the relative compactness of FF at t∗, i.e. we show that Z

Tf,t∗ : f ∈ F = ϕ 7→ f(t∗, x) ϕ(x) dx : f ∈ F (16) n R is relatively compact in D0(Rn). By theorem 39.9 in [16], p. 262 it is sufficient and n m 0 n necessary that for a fixed ε > 0 and ϕ∗ ∈ D(R ), we find (Tk)k=1 ⊂ D (R ) such that the family p−1([0, ε))[T ] for k = 1, . . . , m, covers the set (16). Without loss of ϕ∗ k generality, we may assume that ϕ 6= 0. ∗ 1 n Since f(t∗, ·): f ∈ F is relatively compact in Lloc(R ), which is a complete m 1 n metric space (cf. [8], p. 5), we can find (fk)k=1 ⊂ Lloc(R ) such that ε 1 ∀f∈F ∃k=1,...,m kfk − f(t∗, ·)kL (supp ϕ∗) < . λ(supp ϕ ) kϕ k b n ∗ ∗ C (R ) This implies that Z

∀f∈F ∃k=1,...,m Tk(ϕ∗) − f(t∗, x) ϕ∗(x) dx < ε, n R where Z n ∀ϕ∈D(R ) Tk(ϕ) := fk(x) ϕ(x) dx. n R We conclude that each distribution in set (16) is in p−1([0, ε))[T ] for some k = ϕ∗ k 1, . . . , m. Lastly, we need to verify (AA2). The uniformity of uniform convergence on 0 n FF ⊂ C(R, D (R )) has the base sets of the form −1 † −1 (pϕ ([0, ε))) = (Ff ,Fg) ∈ FF × FF : ∀t∈R (Ff (t),Fg(t)) ∈ pϕ ([0, ε)) ARZELA-ASCOLI’S` THEOREM IN UNIFORM SPACES 291 = (Ff ,Fg) ∈ FF × FF : ∀t∈R |Ff (t)(ϕ) − Fg(t)(ϕ)| < ε Z

= (Ff ,Fg) ∈ FF × FF : ∀t∈R f(t, x) − g(t, x) ϕ(x) dx < ε , n R where ε > 0, ϕ ∈ D(Rn). Analogously, the uniformity of uniform convergence on compacta has base sets of the form −1 Wϕ,R,δ = (Ff ,Fg) ∈ FF × FF : ∀|t|≤R (Ff (t),Fg(t)) ∈ pϕ ([0, δ)) Z

sup f(t, x) − g(t, x) ϕ∗(x) dx n t∈R R Z

In order to provide an example of a family F satisfying (EQ), (PRC) and (EP), we consider F := f(x) e−|c(t)−x| : f ∈ G ,

∞ n n where G ⊂ L (R ) is relatively compact and c ∈ C(R, R ) satisfies lim|t|→∞ |c(t)| = ∞. (PRC) is obviously satisfied as the relative compactness in L∞(Rn) implies 1 n the relative compactness in Lloc(R ). For the remaining two conditions, fix t∗ ∈ n ∞ n R, ε > 0,K ⊂⊂ R and put M := supf∈G kfkL (R ). The last constant is finite by the relative compactness of G. By continuity, choose δ > 0 such that |c(t)−c(t∗)| ∀|t−t∗|<δ M e − 1 < ε. (18)

Using the well-known estimate −u −v −v −(u−v) −v |u−v| ∀u,v≥0 e − e ≤ e e − 1 ≤ e e − 1 ,

292 MATEUSZ KRUKOWSKI we can verify (EQ) as follows:

−|c(t)−x| −|c(t∗)−x| ∀|t−t∗|<δ f(x) e − f(x) e x∈K f∈G ≤ M e−|c(t∗)−x| e |c(t)−x|−|c(t∗)−x| − 1

(18) ≤ M e|c(t)−c(t∗)| − 1 < ε.

It remains to prove (EP). Without loss of generality, we assume that λ(K) > 0, because for λ(K) = 0, the condition (EP) is trivial. Since lim|t|→∞ |c(t)| = ∞, we can choose R > 0 such that ε ∀|t|≥R |c(t) − x| > − ln . (19) x∈K 2M λ(K) For every f, g ∈ G we have Z −|c(t)−x| −|c(t)−x| sup f(x) e − g(x) e dx |t|

n ∀ϕ∈D(R ) ∃R>0 ∀ |t|>R |ϕ ◦ f(t) − ϕ ◦ g(t)| < ε, (20) ε>0 f,g∈F 0 n then the family (δf(·))f∈F ⊂ C(R, D (R )) is relatively τuc-compact. Proof. At ﬁrst, we check (AA1), i.e. equicontinuity and relative pointwise compactness at a ﬁxed point t∗ ∈ R. The condition (4) reads as ∀ n ∃ ∀ |ϕ ◦ f(t) − ϕ ◦ f(t )| < ε. (21) ϕ∈D(R ) r>0 |t−t∗|0 f∈F n Fix ε > 0 and ϕ∗ ∈ D(R ). By the uniform continuity of ϕ∗, we obtain

n ∃ρ>0 ∀u,v∈R |u − v| < ρ =⇒ |ϕ(u) − ϕ(v)| < ε. (22) By the equicontinuity of F, we ﬁnd that

∃r>0 ∀|t−t∗|

δf(t∗) : f ∈ F (24) is relatively compact in D0(Rn). Putting n U = ϕ ∈ D( ): kϕk b n < 1 0 R C (R ) ARZELA-ASCOLI’S` THEOREM IN UNIFORM SPACES 293 we immediately see that sup |ϕ ◦ f(t∗)| ≤ 1. ϕ∈U0 f∈F Consequently, (24) is relatively compact in D0(Rn) by Banach-Alaoglu-Bourbaki’s theorem ([6], p. 248 or [9], p. 264). The family −1 † (pϕ ([0, ε))) = (δf(·), δg(·)): ∀t∈R |ϕ ◦ f(t) − ϕ ◦ g(t)| < ε , where ε > 0, ϕ ∈ D(Rn), is a base for the uniformity of uniform convergence on (δf(·))f∈F and the uniformity of uniform convergence on compacta has the base sets of the form Wϕ,R,r = (δf(·), δg(·)): ∀|t|≤R |ϕ ◦ f(t) − ϕ ◦ g(t)| < r ,

n n where ϕ ∈ D(R ), R > 0, r > 0. If we ﬁx ε > 0 and ϕ∗ ∈ D(R ), then by (20) there exists R > 0 such that ε ∀ |t|>R |ϕ∗ ◦ f(t) − ϕ∗ ◦ g(t)| < . f,g∈F 2 Finally, since

∀ t∈R |ϕ∗ ◦ f(t) − ϕ∗ ◦ g(t)| f,g∈F

≤ sup |ϕ∗ ◦ f(t) − ϕ∗ ◦ g(t)| + sup |ϕ∗ ◦ f(t) − ϕ∗ ◦ g(t)| |t|≤R |t|>R ε ≤ sup |ϕ∗ ◦ f(t) − ϕ∗ ◦ g(t)| + , |t|≤R 2 −1 † it becomes evident that W ε ⊂ (p ([0, ε))) , meaning (AA2) is satisﬁed. We ϕ∗,R, 2 ϕ∗ conclude, via Theorem 3.2, that (δf(·))f∈F is relatively τuc-compact. Condition (20) may seem ugly due to the appearance of the test functions. How- ever, it turns out to be rather ﬂexible. In fact, any family F ⊂ C(R, Rn) such that

∃f∗∈F lim |f∗(t)| = ∞ |t|→∞ and ∀f∈F |f(t)| ≥ |f∗(t)|, t∈R satisﬁes (20). Indeed, ﬁx ε > 0, ϕ∗ ∈ D(R) and choose R > 0 such that

∀|t|>R supp(ϕ∗) ⊂ B(0, f∗(t)).

It is easy to see that for |t| > R we have ϕ∗ ◦f(t) = 0 for every f ∈ F. Consequently, condition (20) is satisﬁed. Another condition which implies (20) is the following: ∀ r>0 ∃R>0 ∀ |t|>R |f(t)| < M ∨ |g(t)| < M =⇒ |f(t) − g(t)| < r. (25) M>0 f,g∈F n To see this, ﬁx ε > 0 and ϕ∗ ∈ D(R ). Pick M > 0 large enough so that supp(ϕ) ⊂ B(0,M). Again, by the uniform continuity of ϕ∗ we have

n ∃r>0 ∀u,v∈R |u − v| < r =⇒ |ϕ∗(u) − ϕ∗(v)| < ε. (26) 294 MATEUSZ KRUKOWSKI

For the above choice of M, r > 0, by (25) we choose a suitable R > 0. Observe that for every |t| > R, f, g ∈ F, we have the implication |f(t)| ≥ M ∧ |g(t)| ≥ M =⇒ |ϕ∗ ◦ f(t) − ϕ∗ ◦ g(t)| = 0, so (20) is satisﬁed. However, if it turns out that either |f(t)| < M or |g(t)| < M, then by (25) we have |f(t) − g(t)| < r, which together with (26) implies (20).

Acknowledgments. I would like to acknowledge the invaluable suggestions of my supervisor Bogdan Przeradzki. Our discussions of Arzel`a-Ascoli’stheorem were truly inspiring and thought-provoking. I would also like to thank Robert Stegli´nski,who helped to verify the proof of Theorem 3.2 and improved the notation. He also suggested writting explicitely Corollary1.

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