ON BARRELLED TOPOLOGICAL MODULES

Some of the main theorems concerning ultrabarrelled topological vector spaces are extended to the context of topological modules.


INTRODUCTION AND PRELIMINARIES
The study of (real or complex) ultrabarrelled topological vector spaces, the topological vector spaces which replace barrelled locally convex spaces when local convexity is not presup- posed, was initiated by W. Robertson in [1].Since then, various authors have been considering the subject.The most important results concerning ultrabarrelled topological vector spaces may be found in the texts [2] and [3], the latter dealing with the case in which the fields of real or complex numbers are replaced by a non-trivially valued division ring.
In this article we introduce mid study the concept of barrelled topological module, the natural extension of the classical concept of ultrabarrelled topological vector space.The main results obtained here are extensions of the Bazlach-Steinhaus theorem mid of the Open Mapping md Closed Graph theorems to the context of topological m()dules.A version of Bourbaki's   criterion for the equicontinuity of sepaxately equicontinuous families of bilinear mappings m(l a version of Grothendieck's "Thdorme B" are also established.It shotfld also be mentioned that the methods used by W. Robertson in her flmdamental article just cited have strongly influenced the preparation of our article. We shall adopt the terminology of [4].Throughout, A denotes an arbitrary topological ring, unless otherwise specified.If A has m identity element, A* denotes the multiplicative group of its invertible elements.For every A-modules E m(l F, .T'(E; F) den()tes the A-module of *Partially supported by CNPq.
2. THE CLASS OF BARRELLED TOPOLOGICAL A-MODULES Definition 2.1.A tOllgi'd A-mdule (E, ) is sa,il t, }e }arrelled if every A-mdule topology on E which admits a fimdunentd system f neighln'hds f the origin consisting of -closed sets is weaker than r.
Remark 2.1.Assume that A is endowed with the discrete topology and let (E, r) be a sepated topological A-module.Then E is barrelled if m,d only if is discrete.Proposition 2.1.Let A be a sepu'ated topologicd ring m,d let E be a sepated bm'relled topological A-module.Then the completi,n E ,f E is a barrelled tpological A-module, where A designates the completion f A.
The following information will be needed in the sequel.
Proposition .. Let (E,)il be a family of topologicM A-modules, E m A-module md, for each I, let f, Ea(Ei; E).Then there exists a unique A-module topolo on E which is finM for gl,e fmnily (E,, f,),e.In pm'ficul, inductive limits exist in the categow of topologicM A-modules.
We now present certn stability properties of tle clmss of barrelled topologicM A-modules, some of which have been obtned in [1] (Proposition t3) m,d [6] (Corolly l, p.297) when A R OF .
Proo Leg * be A-module topolo on E which adnfits a fundmnentM system V of neighbor- hoods of 0 consisting of -closed ts.Fix m I m,d let Bi be the filter bse on Ei fomned by the sets f (V) (V V).By Theorem 12.$ of [4] there is a unique A-module topolo for wlfich Bi is a fimdmnentM system of neighborhoods of 0. Since earl, f(V) is ,-closed since (i,i) is belled, it follows flint C C .Hence f,'(E,,vi) (E, *) is continuous.By the bitrness of we obtn * C v, m,d so (E, ) is brelled.
Corollary .1.(a) An inductive limit of m, inductive system of brelled topologicM A-modtfles is a brelled topologicM A-module.
(b) A quofien by a submodule of a breHed t,,polgical A-module is a breHed gopologicM A-module.
(c) For a direct sum ,f a fmnily of tolhgical A-lnodules to be lm'relled it is necessary and sufficient that each of its m,mbers be barrelled.
The following pr,,p,,siti,,n c,,ntains The,,rem 2.37 ,,f [3] (hence Pr,,p,,siti,,n 12 ,,f [11) as a particular ca.se.Proposition 2.4.Let A le a tolmlogical ring with identity emd assume that there exists a countable subset of A* such that 0 C. If (, r) is a unitary tophgical A-module and M is a submodule of E which is non-meager in (E, r), then M is barrelled under the induced tol)oh)gy.
In particular, every Baire unitm'y tphgical A-module is barrelled.
Proof.Let r be the A-module tlmhgy on M induced by r, and let r* be an A-module topology on M which adnfits a flmdamental system of neighlmrhoods of 0 consisting of f-closed sets.Given an arbitrary V Y thereis a U Y with U-U C V. Since 0 C, thereis asequence (a,,),,e in A* such that M C a,,U.
Therehre some a.,U hs a non-empty interior, because each anU is r-closed and M is non- meager in (E, r).Hence U hs a non-empty interi,,r, since the mapping x (E, r) a,,, x (E,r) is a homeomorphism.If x int(U*), there is a neighborhood U' of 0 in (E,r) with x+U C U .Consequently, U C U U C Vr, andso V is aneighborhoodof0in(E,r).
Thus V is a neighborhood of 0 in (I, r), because V V r' M Vr.Therefore r* C r , and so (M, r') is belled.
Remark 2.2.(a) If A hms ma identity element, then the relatim 0 G A* implies that the topology of A is non-discrete (and is equivalent to this fact when A is a t,pologcal division ring).(b) Every topological division ring which possesses a null sequence of non-zero elements satis- fies the hypotheses of Proposition 2.4.Every topological ring with identity which contns invertible topologicMly nilpotent element satisfies the hylmtheses of Proposition 2.4.

THE BANACH-STEINHAUS THEOREM AND SOME CONSEQUENCES
The Banach-Steinhaus theorem holds in our setting (previous results in this direction may be found in [1], Theorem 5; [7], Theorem 3; [3], Theorem 2.58; [4], Theorem 25.6): Theorem 3.1.Let E be a barrelled topological A-module, F an arbitrary topological A-module and X C (E;F).If,(x) {f(x);f X} is bounded in r for each x e E, then, is equicontimous.
Proof.Let r be the given topology of E and let l/be a fundamental system of closed neighbor- hoods of 0 in F. For each V Y, let u.N f-'(v)" Obviously,/3 (Uv) 'ev is a filt,-r 1,Le (,n E. By The(,rem 12.3 (f [4] th,.r-is a unique A-moduh topology '* on E fr which/3 is fundamntM system f neighborhods f 0 (th condition (TMN 2) of Theorem 12.3 h,,l,ls 1,e,'ause the sets ,U(x) are 1,,,un,led in F).Since (E, r) is barrelled mcl since each Uv is r-closed, it fillws that r* C r. Thus ertch Uv is a neigh|orhood of 0 in (E, r), and so 2" is equicontinuous.
Corollary 3.1.Under the a.ssumptions of The(rem 3.1, a.ssum-additionally that F is separated. Let (f'),e be a net in :(E; F) such that (f,(z)),e is bounded in F for each z 6 E and such that (f'),e is pointwise convergent t,, a mappings f: E F. Then f e :(E; F) and (fi),e converges to f uniformly on every l)recompact su]set of E. In particular, if (f"),,eN is a sequence in :(E; F) pointwise converg,-nt to a mapping f: E F, then f e :(E; F) mad (f"),,eN converges to f uniformly on every precompact subset f E.
Proof.By Theorem 3.1, (f),et is an eqficontimmus net.Thus it sull:ices to apply Proposition 6 mad Theorem 1 of [8], chap.X, 2, and the fact that :,(E; F) is simply closed in .T'(E; F), to conclude the proof.
Remark 3.1.Let E and F |e topological A-modules and let/3 be a fmnily of bounded subsets of E. Then the topolo of B-c,nvergence on (E; F) (denoted by r) is m lditive oup topology which is sepated if B is a cvering of E md F is sel)ated.Moreover, if A is conmmtative, then (E; F)is an A-module md r is m A-module topology on (E; F) ([9], Proposition (a)).
When B is the family of M1 finite (resp.bounded) subsets of E, we write r, (resp.re r).
Corollary 3.2.Let A be a commutative topologicM ring with identity such that 0 A*.Let E be a bmelled topologicM A-modtfle, md let F be a separated locally compt tmity topologicM A-module.If , C (E; F), the fifllong statements e equient: (i) X is n-bounded in (E; f); (ii) .Y is r,-bounded in (E; r); (iii) X is r,-relatively compact in (E; F); (iv) X is equicontinuous.
In order to prove Corollary 3.2 we shall need a lemma which is aJ extesion of the Alaoglu- Bourbaki theorem: Lemma 3.1.Let A be a topological ring with identity such that 0 -;.Let E be a topological A-module and let F be a separated locally compact tufitary topological A-module.If X C .(E;F) is equicontinuous, then , is r-relatively compact in/:(E; F).
Non-locally convex versions of a classical theore,n of Bourbaki ([10], TVS III. 29)have been obtained in [7] (Theorem 5) and [11] (Corollary 9).Ottr next go',d is to prove that it remains valid in our context.Theorem 3.2.Lrt E and F be m,trizabh tlohgical A-modules, E being assumed lrrelled, a,nd let G be m arbitr'y topflgical A-module.If,l' C ,(E, F; G) is such that ,l is equicontinuous for each x E, then , is equicmtinuous.
In order to prove the a]w, theorem we shall need two lemma.. Lemma 3.2.Let E, F and G be tplogical A-modules, and let .V le a separately equicmtinums family of A-bilinear mappings frm E F int G.If ,V is vquicmtimus at the origin, then .l' is equicontinuous.
Proof.It suffices to recall the identity f(x, y)f(Xo, Yo) f(x Xo, y Yo) + f(x -.to, Yo) + f(.ro, y Yo), which holds for every A-l,ilinear mapping f: E x F G and every p,,ints (x, y), (Xo, yo) in E x F. Remark 3.2.When A is a non-trivially valued field, the equicontinuity of a family of A-bilinem" mappings at the origin is sufficient to ensure its equicontinuity ([10], Proposition 6, TVS 1.9).Lemma 3.3.Let E, F and G be topological A-modules, E being ,,sumed barrelled.Let X C (E, F; G) be such that , is equicontinous for each x E. Then "v is equicontinuous for each y F.
Proofi Fix a y F. If x e E, ,l(x) ,(y) is bounded in G since , is equicontinuous.By Theorem 3.1 , is equicontinuous.
Proof of Theorem 3.2.In view of Lemmms 3.2 mid 3.3, it is enough to establish the equiconti- nuity of X at (0,0).If ,Y is not equicontinuous at (0,0), there e a neighborhood W of 0 in G, a ndl sequence (x,,)neN in E, a null sequence (yn)ne N in F mad a sequence (L,)neN in X sud that f(x,,,y,,) W for all n e N (remember that E md F e metrizable).By Theorem 25.5 of [4] mad Theorem 3.1, the fmnily {h,;f A.',n N} is equicontinous.Therefore, there is an integer no such that f(x,,, y,,) W for all f , mad all n > no, a contradiction.Thus ,:t" is equicontimous.
From Theorems 3.1 mad 3.2 we derive: Corollary 3.3.Let E mad F be barrelled metrizable topological A-modules, aaxd let G be arbitrary topological A-module.If X C ,,(E,F; G) is such that ,(x,y) is bounded in G for all (x, y) E x F, then 2( is equicontinuous.