Schmets, J.

Czech digital mathematics library

On spaces of vector-valued continuous functions

Institute of Mathematics AS CR, v. v. i.

Toposym 4-BJean SchmetsOn spaces of vector-valued continuous functionsIn: Josef Novák (ed.): General topology and its relations to modern analysis and algebra IV,Proceedings of the fourth Prague topological symposium, 1976, Part B: Contributed Papers. Societyof Czechoslovak Mathematicians and Physicist, Praha, 1977. pp. [400]--406.Persistent URL: http://dml.cz/dmlcz/700637Terms of use:© Society of Czechoslovak Mathematicians and Physicist, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czON SPACES OP VECTOR-VALUED CONTIIÏUOUS PШTCTIONS J. SCHMETS LIEGE Let X be a completely regular and Hausdorff spree, end 3 be a locally convex topological vector space. Then we denote by tf(X) the linear- spree of the continuous functions on X and by Co(x) the space tf(X) endowed with the topo­logy of pointwise or simple convergence. As far as E is concerned, we denote by P its .oyster of semi-norms and by P1 a systen of seni-norrs on 3 v.-hich is finer thnn P. If R is a locally convex property which is strblo for Hr.us-dorff inductive limits and which is satisfied by any linear space when it is equipped with its finest locally convex topology, then Y, Komura [2] has introduced t1 r R-space associated to 3 as the linear space E endowed with the coarsest system of semi-norms Pf which (is finer than P and which) makes 3 satisfy property R. As examples of such properties R let us mention the following ones : the space E is ultrabornological, bornologi ca.1, barreled or eva­luable • Let us now recall the following results from [l]« Theorem 1. a) The space C (x) is always evaluable. b) The barreled space associated to C (x) is C (uX). In particu-s c lar, C (x) is barreled if and only if X is a u-space which compact subsets are finite. c) The bornological space associated to C (x) is C (uX). In parti­cular, C (X) is bornological if and only if X is rcalcorpact. 401 d) The ultrabornological space associated to C o(X) is C ^ ( u X ) . In particular, C (X) is ul tr ah ornol o<;i c al if and only if X is real-compact and such that all its compact subsets : re finite. In this theorem, C (x) stands of course for tf(X) endowed with the com pact-open topology coning from X, and u X for the realcompac-tification of X. Moreover, calling bounding in X the s u b s e t s of X where every f £ tf(x) is bounded, uX is the smallest oubspace of uX containing X and where every bounding subset is relatively c oir.pact o Definiti ons. Let us denote by tf(X;E) the linear space of the continuous functions from X into E and by C p, o(X;E) the space tf(X;E) endowed with the semi-norms p!(p'^P', ACX finite) defined by P|(q>) = sup p'[cp(x)], Vtp £ tf(X) . x£A Then C_t (X;E) is of course a locally convex topological vector r 9 S space, which we write C (X;E) if P< « P. Ve are interested in characterizing the s o c i a t e d t o C D , o ( X ; E ) . I n [ 4 ] > we h a v e a l r e a d y g o t t h e f o l l o w i n g r e s u l t s < r , S Theorem 2, a) The space CD, (X;E) is a Mackey space if and only if (E,P') r , s is a Tv'ackey space, b) The evaluable space associated to C p t (X;E) is the space r , S 0 (X;5) where P* is the system of semi-norms of the evaluable r , s e e ' space associated to (E,P*)» In particular, Cpl ^(X;E) is evalua-ble if and only if (E,P!) is evaluable. c) The space Cp. (X;3) is barreled if and only if Co(X) and r , s s (E,Pf) are barreled. Moreover, if C (X) is barreled, then the s barreled space associated to C^, (X;E) is C , (X;E) where P! r',S rIjS X is the system of semi-norms of the barreled space associated to (E,P<). 402 The tornolo^ical and ultrabornol epical cases seen hnrcler to study. In [4] > v/e ^ive sore results about the borr.clogicol c.se. It is the purpose of this note to show that an analogous rethod allows to get results about the ultrabornological case. A more extended version is under preparation [5]-The hint goes on as follows. If B is an absolutely convex subset of E, then E^ denotes the linear hull of B when it is equipped v/ith the gauge of B; moreover 3 is celled a Ba--g>ch •j'.isk if E~ is a Banach space. Then a sequence is Frcke.Y (res-., fĵ sjt) convergent in E Ĵ o e if there is a bounded click (resp. a b curded Banach disk) B of E such that the sequence converges to c in 3^. Finally E is bornological (resp. ultrabornological) if end only if every absolutely convex subset of E which absorbs the Lackey (resp. fast) convergent sequences is a neighborhood of 0 i:\ I!. The following result indicates then where to look for. Theorem 5- If the space C-. (X;S) is ultraborn cl c ic-1 r , s (resp. bornological)i then the spaces Ct(x) and (E,P') are ultra-s bornological (resp. bornological)* The bornological result is theorem V „ 4 » 1 of [4]» the proof of the ultrabornological ccce goes on anclogously by use of the preceding remark. The aim is then to get converse results to t h e o r e . . i 3 o This can be done v/ith the help of the following generalization of a result of Nachbin [3]* Theorem 4-. If T is an absolutely convex subset of tf(X;E), there is a smallest compact subset K(T) of pX such that cp 6 tf(X;E) belongs to T if <p is equal to 0 on a neighborhood cf K(T) in pX (cp represents the unique continuous extension of cp from pX into pS)« If moreover there are p €• P and r > 0 such that T ) {cp 6 *(X;S) : p[cp(x)] -* r, V:: * J!} , then K(T) is the smallest corpsct subset of pX such that cp €• tf(X;E) belongs to T if cp vanishes on K ( T ) , and o;:o hrs then 403 T Э {ç. * tf(X;E) . ^(x) € Ъ (<r), Vx Є K(T)}. Proposition 5» If T is an absolutely convex subset of tf(X;E) which absorbs the fast convergent sequences of Cp. (X;E), then r , S K(T) is a subset of uX0 Proof o Suppose there is an element x £ (3X\uX which belongs to K(T). Then we know there are open subsets 0 of pX which are increasing, constitute a cover of uX and do not contain x. By theorem 4> there is then a sequence cp € tf(X;E) such that cp j£ nD and q> (G ) = {0} for every n. One can prove then that -» oo B = ( S c n« : E I c I ^ 1 } 1 . n ^n . « n» } n=1 n-=1 is an absolutely convex compact subset, hence a bounded. Banach disk of C , (X;E)0 Of course the sequence cp tends to 0 in tf(X;E)-n and cannot be absorbed by T, which is contradictory. Pronosition 6. If T is an absolutely convex subset of tf(X;E) which absorbs the Mackey (resp. fast) convergent sequences of C p, (X;E) [resp, of C (X;E)] and such that K(T) is contained in X, and if X satisfies the first axiom of countability [resp. and if (E,p) is metrizable], then an element cp belongs to T if 9 vanishes on K(T). Proof.. Let us prove the "Llackey" version. By proposition V„4«7 of [4]> we know that K(T) is finite. So there is a sequence f € tf(X), with values in [0,1], equal to 1 on neighborhoods of K(T) and to 0 outside decreasing neighborhoods in X of K ( T ) , which intersection, is K(T). Let now cp £ tf(X;E) vanish on K(T)<> The abso-2 lutely convex hull of the sequence n f qp is then a bounded disk of C p f (X;E) : therefore the sequence n f cp is absorbed by To Moreover for every n, [(1~£ )<p] vanishes on a neighborhood in pX of K(T), S O (1-f )cp belongs to eT for every e > 0. Hence the con-elusion since cp equals -7 [(l*~fn)q> + ^n^l *>or every n* 404 Let us now consider the "fast" version. Let <p vanish on K(T) and {p s ntfU} be a countable system of semi-norms on E, equiva-lent to P. Then there is a sequence f € tf(X), with values in [0,1]f equal to 1 on a neighborhood of K(T) and to 0 outside Gn -» {x e X 1 pn[<p(x)] < n~4}. Then 00 CO B - { 2 c n2 f f « E |o | * 1} n»1 n«1 is a bounded Banach disk of C-. (X;E). But the sequence nf <p con-Jt f s n verges to 0 in tf(X;E)-gf so it is absorbed by T. Hence the conclu-sion since £ ("I'-*)*?]" vanishes on a neighborhood in pX of K ( T ) . Theorem 7. a) If C0(X) is bornological and (EfP) metrizable, then C (X;E) s s is bornological* b) If C (X) is ultrabornological and (E,P) a Prechet space, then Ca(X;E) is ultrabornological. Proof, a) is theorem Y.4«11 of [4]» However one can proceed similarly to b) f which simplifies the proof. b) Let T be an absolutely convex subset of tf(X;E) which absorbs the fast convergent sequences of G (X;E). Combining theorem 4 and part d) of theorem 1f we get that K(T) is a finite subset of X. Then there are p £ P and r > 0 such that T O {q> * *(X;E) 1 p[<p(x)] *. r, Vx 6 K(T)} ; by contradiction : if this is not the case, there is a sequence sup x*X Yn £ y(X|E)\D such that Pn[<p(x)] -*- n"4; then B - { - on n2 „ . z |cn| - 1} n«1 n-1 is a bounded Banach disk of Cfl(X;E) and the sequence n<p tends to 0 in tf(X;E)Bf which is contradictory to ajn $ D Q rpne conclusion then follows from the last part of theorem 4. 405 .Theorem 8* If C (X) is hornological and if X satisfies the first axiom of oountabilityf then the bornological space associa-ted to C-., (X;B) is the space CBt _(X;B)f where PJ is the system if $ S h9 of semi-norms of the bornological space associated to (liP1)* In particularf if C (X) and (EfPf) are bornological, then Cp , (Xfl) is bornological® Proof* This is theorem ¥#4*12 of [4] but one can proceed as in theorem 9 for the proof of the particular case, which simpli-fies considerably the developments used there* Theorem 9 0 If CQ(X) is ul trabor.no! crical and (E,P) a Frechet space, and if X satisfies the first axiom of oountability- then G (X;E) is ultrabornological. s Proofs By part a) of theorem 2, C (X;E) is already a Mackey space. To concludes it is then sufficient to show that every li-near functional Z on tf(X;E) which is "bounded on the fast conver-gent sequences of C (XfE) is continuous* But then o T - {? 6 *(X;E) 1 |*(<p)| £ 1} is absolutely convex in tf(X;E) and absorbs the fast convergent sequences of C (X;E). By theorems 4 and 5 and by part d) of theo-s * * rem 1, K(T) is a finite subset of X. Proposition 6 permits then to define a functional @ on the ultrabornological space P - II (EfP) by X^K(T) «U) - ffU)t vf e P t if -p 6 tf(X;E) is such that <p(x) = f(x) for every x £ K(T). Then @ is a linear continuous functional on P since it is bounded on the fast convergent sequences of P, Therefore there are conti-nuous linear functionals Z on (E,P) for x £ K(T) such that * U ) - E / Zr [<p(x)]f Vq> € tr(XfE) . x^K(T) x Hence the conclusion. 406 REFERENCES [1] H. BUCHWALTER - J. SCHMETS, Sur le theoreme de Nachbin-Shirota. J. Math. Pures et Appl., $2. (1973), 337-352. [2] Y. KOMURA, On linear topological spaces, Kumamoto J. of so., ̂ 1 (196*2), 1 JfrTyf7~~^ [3] Ii« NACHBIN, Topological vector spaces of continuous func-tions. Proc. Nat. Acad. USA, 40 (1954), 471-474. [4] J« SCHMETS, Espaces de fonctions continues. Lecture Notes in Mathematics 519 (1976), Springer, Berlin-Heidelberg-New York. [5] J. SCHMETS, Bornologrical and ultrabornological C(X;E) spaces « (in preparati on). [6] T« SHIROTA, On locally convex vector spaces of continuous functions. Proc. Japan Acad.- 50 (1954). 294-296. ïïniversité de Liège Institut de Mathématique Avenue des Tilleuls, 15 B-4000 LIEGE / BELGIÏÏM 

English

http://dml.cz/bitstream/handle/10338.dmlcz/700637/Toposym_04-1976-2_85.pdf

On spaces of vector-valued continuous functions

Abstract

Similar works

Full text

Available Versions

Czech digital mathematics library

Institute of Mathematics AS CR, v. v. i.