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On Schwartz Spaces and Mackey Convergence Publications Du Département De Mathématiques De Lyon, 1973, Tome 10, Fascicule 3 , P

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On Schwartz Spaces and Mackey Convergence Publications Du Département De Mathématiques De Lyon, 1973, Tome 10, Fascicule 3 , P PUBLICATIONS DU DÉPARTEMENT DE MATHÉMATIQUES DE LYON HANS JARCHOW On Schwartz Spaces and Mackey Convergence Publications du Département de Mathématiques de Lyon, 1973, tome 10, fascicule 3 , p. 79-97 <http://www.numdam.org/item?id=PDML_1973__10_3_79_0> © Université de Lyon, 1973, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ jj^ipe Anal. Fonct. Publications du (1973, Bordeaux) Departement de p. 233-251 Ma the ma ti qu es Lyon 1973 1.10-3 ON SCHWAK'TZ SPACES AND NiACKEY CONVERGENCE Hans Jarchow The purpose of this article is to give a survey on some of the re­ cent results in the theory of Schwartz spaces ana to show how they are connecteo with the theory of lackey convergence of sequences in locally convex spaces. These results are scattered over various po­ pe rs, see e.g. [l] , [l4] f [19-22] , [26] f [27-3cl f [33l f [34] . For some of them we will present here new and (or) simpl if iea proofs. There are at least two concepts which are highly appropriatec to provide us with a satisfactory language for our purpose, namely the concept of spaces with bornology, see Hogbe-Nlend D-31 , ana the con­ cept of limit spaces, see Fischer Cll3 . It is not hard to see thct in the case we are interested in the latter concept incluaes the first one in a natural way. Moreover, 3uchwalter's theory on corr.pactoloci- cal spaces [4} appears as a special case of the theory of limit spa­ ces. Since this theory is also important in our context, ana since we are concerned here mainly with questions of convergence rather than of bouncecness, we will work wholly within the category of limit spaces. 1. GENERALITIES ON LIMT SPACES We denote by F(h) the set of all filters on a given set M anc by P(F(M)) the power set of F(f>.). Following Fischer Cll3 we call a map ^: N; > F(F(K) ) a limit structure on M if it satisfies (Ll) anc (L2), for ail x€ «• : (LI) The ultrafilter generated by {x} belongs to ^(x) . (L2) For all r,(|e F(M) : f",|jc*(x) <=> Tf\§€ *X (x) . The ordered pair Q<\f ^3 is called a limit space. Since each topology on a set determines canonically a unique limit structure on this set, we will consider topological spaces as special limit spaces throughout this paper. Generalizing the corresponding 79 On Schwartz spaces and Mackey convergence notations from topology, we are led to the following definitions: Let[M,3Q be a limit space. If Je F(M) satisfies Te ft(x) for some xfif., then 3""is said to converge to x. If, for all x,ycM , ft(x)nft(y) is void unless x=y, then we say that[№,^] (or A ) is Hausdorff or separated. Let DM ,^1 be a second limit space. A map f :K »N is called (ft.j*)- -continuous, or simply continuous, if f(ft(x))c ^(f(x)) holds for all If ft and are limit structures on M such that the identity map of to is (ft -continuous, we say that ft is finer (or stronger) than ^i,: one that [L is coarser (or weaker) than ft, and we write ft^^tor £ ft . Together with their continuous mappings the limit spaces form a ca­ tegory L containing the category TOP of all topological spaces as a full subcategory. L is complete and co-complete. Products, co-productSi subobjects, quotients, and, more generally, projective and inductive limits in L are obtained in the usual way. Let now[Ni,ft3 be the inductive limit in L. of a family ((Vi^ tft^)^ j of limit spaces, with respect to the continuous mappings f^ > f i4 j. Possibly after some elementary manipulations, we may assume that M-CM. for i* j and that f.. is the corresponding inclusion mapping. We will reserve the notation tNi,ft]*= lim ind [M. ,ft .] i€I 1 1 exclusively for this special situation, and we will only consider in- cuctive limits in L of this type. The behavior of this "lim ind" is rather good. We start with the following proposition which is mainly due to Fischer [ll] (see also [36], £3] ) : (1.1} A filter T in. (K'I ,ft3 = 1 iro ind |V;. ,ft 3 converges to x€ M if and only id 1 1 if we have xCN^ , MiT and J^cft^x) for some id, where is the trace of 7 ojn M^. It follows that [Ni,ft3is Hausdorff iff this is true for all O^^jl • Another important property of L, namely to be a cartesian closed category, is expressed in the following theorem, the proof of which 80 On Schwartz spaces and Mackey convergence can be found in [?3 and [z\ : (1.2) Let |V, A1 one psi,^] be limit spaces. Let l(M,N) denote the set of all (Hdiscontinuous maps from M into N. Among all limit struc­ tures A oj2 ^(h,N) such that the evaluation map «:W(Ni.N).A]x[Mf^] >DsifjJ: (ffx) I *f(x) is continuous, there is a coarsest one which we denote by A . c A is called the limit structure of continuous convergence. It is Hausdorff iff fx has this property. A filter $ in «£(M,N) belongs to /\ (f) for some fcftC^tN) if and only if, for every xeM and every 3"€^(x), we have u($xT)€ ^.(f(x)). C fk The limit structure inauced from A on HC 'S(MiN) will be denoted by A° again. Sometimes we will write H instead of 1h»A°3 % and £(M.N) c c instead of ,N) , A 3 . Also, if (Ki,A]is any limit space, and if no confusion is to be feared, we sometimes will omit the % and write № instead of [N,^3 • For limit spaces M and N, (1.2) yields the following universal property of the speces H for HC^(Ni.N) : c (1.3) If_ Z is any limit space, then a map f:Z *H is continuous if and only if f : Z x Ni • N : (z,x) l—• f (z) (x) is continuous. It is clear that (1.3) actually characterizes the limit structure /\ on H. We take the occasion to establish the connection with the compec- tological spaces of Buchwalter [4]: v A compactological space is an ordered pair 0 'f3C)i where F*. is a set and 3cf the cornpactology of (f itfc) • is an upwards directed cover­ ing of M, each of its members K being supplied with a fixed Hausdorff compact topology % such that (Cl) ana (C2) are satisfied: (CI) If K,L€3cand KCL , then [K , T, ,3 is a subspace of fL.trJ . (C2) If KC3C and HCK , then the tf -closure H of Hf supplied with the topology induced from , belongs to Qfo . Y/e will call the cornpactology 3t of (h,0C) maximal if there exists a limit structure ^4. on l<\ such that all compact subsets of 0v«ijO are 81 On Schwartz spaces and Mackey convergence topological spaces under the induced limit structure, and 3t consists exactly of these compact sets. A subset S in [M,^*] is said to be compact if every ultrafilter in M containing S converges to some x€S . f f Let (M,3fc) and (M ,3C ) be compactologicql spaces. A map f:M » M» is called compactologicql if it maps continuously each member of 3t into some member of 3C' • *n this case, f(K) is an element of under the final topology of K —* f(K), for every KfiOfc. If (M,3t) rs a compactological space, we can introduce a limit structure 0^ on M by means of CM,XJ « lim indD</0. KeX K 1 f 1 If (M fX ) is another compactological space and f;M » M is a com— pactological map, then f is clearly ,5^f)-continuous. Conversely f if f is (X^f^yt)-continuous and if %' is maximal, then it is easy to see that f is also compactological. In this case, JR1 consists exactly f So we of the compact subsets of tM ,V#3« have: (1.4) The category of all maximal compactological spaces (compacto­ logical maps) can be considered as a full subcategory of L. 2. LIMIT STRUCTURES ON DUAL SPACES A limit vector space is a limit space (JEAl consisting of a (real x$r complex) vector space E and a limit structure A on E such that addi­ tion Ex E >E and scalar multiplication KxE *E are continuous for the respective product limit structures. Here K denotes the field of real or complex numbers, with its euclidean topology. If [E,A] is a limit vector space, then, for every xcE, ^(x) is obtai­ ned from %(o) by A(x) - x + 5i(o) . Therefore a linear map T:E *F , ££,X! and [F limit vector spaces, is 9JL )-continuous iff it satisfies T(a(o))Cp(o).. Again, the limit vector spaces form, together with their continuous linear maps, a category LVR having arbitrary limits and co-limits. For details we refer to [ll] , [36] , [23] , [l6] , and others.
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