A locally convex space E is polynomially barrelled if and only if, for every positive integer m and for every Banach space F, the space of all continuous m-homogeneous polynomials from E into F is quasi-complete for the topology of pointwise convergence

Dinamerico P. Pombo Jr.

Miguel Caldas Cueva

English

Hrčak - Portal of scientific journals of Croatia

GLASNIK MATEMATICKIVol. 33(53) (1998). 93-96ON A CHARACTERIZATION OFPOLYNOMIALLY BARRELLED SPACESMiguel Caldas Cueva and Dinamerico P. Pombo Jr., Rio de Janeiro, BrasilAbstract. A locally convex space E is polynomially barrelled if and only if, for every positiveinteger m and for every Banach space F, the space of all continuous Ill-homogeneous polynomials fromE into F is quasi-complete for the topology of pointwise convergence.Pfister [4] has proved that a locally convex space E is barrelled if and only if, forevery Banach space F, the space of all continuous linear mappings from E into F isquasi-complete for the topology of pointwise convergence. The main purpose of thepresent note is to establish the corresponding result in the polynomial context. At theend of the note a discussion concerning the closed graph theorem for homogeneouspolynomials is also presented.In what follows we refer to [2] and [3] for the background on topological vectorspaces. All vector spaces under consideration are vector spaces over a field K whichis either R or C. If E and F are locally convex spaces and m is a positive integer,P(I1lE;F) denotes the vector space of all continuous m-homogeneous polynomialsfrom E into F, and Ts represents the locally convex topology of pointwise convergenceon P("'E; F). A locally convex space E is polynomially barrelled [5] if, for everypositive integer m, each Ts-bounded subset of P('"E) := P(11lE; K) is equicontinuous.THEOREM. Let E be a locally convex space. In order that E be a polynomiallybarrelled space it is necessary and sufficient that, for every positive integer m andfor every Banach space F, the space (P("'E; F), Ts) be quasi-complete.Proof. Since the necessity is a particular case of Proposition 3.26 of [5]' let usturn to the sufficiency.Let m be a positive integer and let Xc P("'E) be Ts-bounded. Let B(X) be thevector space of all K-valued bounded mappings on X, endowed with the supremumnorm: I[hll= sup{lh(f)I;f E X} for hE B(X). Then (B(X), 11'11) is a Banach space.Define P : E --+ B(X) by P(x)(f) = f(x) for x E E,j E X; note that P(x) E B(X)because X is Ts-bounded. It is easily seen that P is an m-homogeneous polynomialfrom E into B(X). We shall prove that P is continuous. In order to do so, we shallconstruct a set {PT,,; T E Q, f. > O} of continuous m-homogeneous polynomialsMathematics subject classification (1991): 46E40.Key words and phrases: Locally convex spaces, continuous Ill-homogeneous polynomials, topologyof pointwise convergence, equicontinuous sets, closed graph theorem.94 M. CALDAS CUEVA AND D. P. POMBO JR.from E into B(X) (where ,Q denotes the set of all non-empty finite subsets of E)satisfying the following properties:(1) For every T E ,Q and for every E > 0, IIPT,.(x) - P(x) II ~ E if x E T;(2) For every x E E, the set {PT..(X); T E ,Q, E > O} is bounded in B(X).Let T E ,Q and E > 0 be given. Consider on X the pseudo-metric dTif,g) =max{lf (x) - g(x)l;x E T} forf, g EX. Then the uniformity determined by dT is thesmallest uniformity on X which makes the mappingsf EX H f(x) E K (x E T)uniformly continuous. By Proposition 3.3, p.5 of [2], (X, dT) is precompact becauseX(x) = if (x);f E X} is a precompact subset of K for each x E T (recall that X is'l's-bounded). Moreover, if 'l'T denotes the topology induced by the pseudo-metricdT, then (X, 'l'T) is a normal space. Forf E X, put Bif, E) = {g E X; dTif, g) < E}.By the precompactness of (X, dT), there existf], ... J/1 E X such that X = Bifl, E) U... U Bif,,, E). By the Dieudonne-Bochner theorem, there exists a family (ei)l~i~/1of real-valued 'l'rcontinuous mappings on X satisfying the following properties:(3) e; ? 0 on X for i= 1, ... , n;/1(4) L e; = 1 on X;;=1(5) e; vanishes outside Bif;, E) for i= 1, ... , n./1Define PT" : E ~ B(X) by PT,' (x) if) = L e;if)f;(x) for x E E,f EX. It is;=1easily seen thatPT,. E P(I1IE;B(X)).Now, let us verify (1) and (2). Let T E ,Q and E > O. If x E T,1Z 11IPT,.(x)if) - P(x)if)1 = II:e;if)f;(x) - I:e;if)f(x) I;=1 ;=1It /1~ I:e;if)lf;(x) -f(x)1 ~ E(I:e;if)) = E;=1 ;=1for allf E X (by (3), (4) and (5)). Therefore (1) holds. If x E E, (3) and (4) yieldIl IIIIPT"(x)II = sup{lI: eiV)f;(x)l;f E X} ~ I: If;(x) I;=1 ;=1for all T E ,Q and for all E > O. Therefore (2) holds.For T, Tr E ,Q and E, EI > 0, put (T, E) ~ (TI, Ed if and only if T C TI andEI ~ E. In this way, ,Q x ]0, +00 [ becomes a directed set and the set {PT,,; T E ,Q, E >O} may be regarded as a net in P(11IE;B(X)). By (2), the set {PT,.; T E ,Q, E > O} is'l's-bounded in P(11IE;B(X)). Hence, by hypothesis, its closure in (P(11IE;B(X)), 'l's)is 'l's-complete. On the other hand, (1) ensures that (PT,.) is a 'l's-Cauchy net.Consequently, there exists a PI E P(11IE;B(X)) such that (PT,,) converges to PI for'l's. By (1), P = PI and the continuity of P is established. It then follows that Xis uniformly bounded on a neighborhood of zero in E, and so X is equicontinuousby Theorem 1 of [6]. We have proved that every 'l's-bounded subset of P(I1IE) isequicontinuous, and hence E is polynomially barrelled. This completes the proof ofthe Theorem.ON A CHARACTERIZATION OF POLYNOMIALLY BARRELLED SPACES 95It is known [3] that a separated locally convex space E is barrelled if and onlyif, for every Banach space F, each linear mapping from E into F with a closed graphis continuous. In the same spirit, it is possible to establish a sufficient condition for alocally convex space to be polynomially barrelled (Proposition 1). We do not knowif this condition is also necessary, although we have been able to obtain a partialresult in this direction (Proposition 2).PROPOSITION 1. Let E be a locally convex space. In order that E be a poly-nomially barrelled space it is sufficient that, for every positive integer m and forevery Banach space F, each m-homogeneous polynomial from E into F with a closedgraph be continuous.Proof Let m be a positive integer and let X C P(IIIE) be 't's-bounded. LetB(X) be the Banach space considered in the proof of the Theorem, and defineP : E -+ B(X) by P(x) if) = f (x) for x E E,f E X. Then P is an m-homogeneouspolynomial from E into B(X) whose graph is obviously closed. By hypothesis,P is continuous, and the argument used in the proof of the Theorem ensures theequicontinuity of X. Therefore E is polynomially barrelled, as was to be shown.PROPOSITION 2. Let m be an integer, m ? 2, E1, ••• , EIII separated locally111convex (DF)-spaces, F a separated locally convex space, and u : IT Ek -+ Fak=Imultilinear mapping with a closed graph. If(i) E 1, , EIIIare barrelled spaces and F is an infra- (s) -space, or(ii) EI, ,Em are ultrabomological spaces and F is a webbed space, then uis continuous.Proof Let k E {I, ... , m} and Xj E Ej (1 ~ j ~ m,j i:- k) be fixed. Arguingas in the proof of the Theorem obtained in [1], we see that the linear mappingt E Ek M U(XI, ... ,Xk-I, t,Xk+h'" ,XIII) E Fhas a closed graph. By (4) a), pA5 of [3] (case(i)) or (2) , p.57 of [3] (case (ii)), thismapping is continuous. By Corollary 1, p.226 of [2] and Exercise 1, p.228 of [2], uis continuous, as was to be shown.Remark. (i) Under the hypotheses of Proposition 2, u is an m-homogeneous111 111polynomial from IT Ek into F by Proposition 1 of [6]. Moreover, IT Ek is polyno-k=l k=Imially barrelled. In fact, using the argument of the proof of Proposition 4.1 of [5],we see that every barrelled (DF)-space is polynomially barrelled.(ii) In the non-linear case, the Theorem proved in [1] is a special case of bothparts of Proposition 2; in the linear case, it is the classical closed graph theorem inthe context of Banach spaces.REFERENCES[1] C. S. Fernandez, The closed graph theorem for multilinear mappings, Internal. J. Math. & Math.Sci. 19 (1996). 407-408.96 M. CALDAS CUEVA AND D. P. POMBO JR.[2J A. Grothendieck, Espaces Vectoriels Topologiques, 3:" ed., Pub!. Soc. Mat., Sao Paulo, 1964.[3] G. Kothe, Topological Vector Spaces II, Grundlehren der mathematischen Wissenschaften Band237, Springer-Verlag, 1979.[4] H. Pfister, A new characterization of barrelled spaces, Proc. Amer. Math. Soc. 65 (1977), 103-104.[5] D. P. Pombo Jr., On polynomial classification of locally convex spaces, Studia Math. 78 (1984),39-57.[6] , Polynomials in topological vector spaces over valued fields, Rend. Circ. Mat. Palermo 37(1988),416-430.(Received August 29,1996) Instituto de MatematicaUniversidade Federal FluminenseRua Siio Paulo, s/n~, 24020-005RJ-Brasil

On a characterization of polynomially barrelled spaces

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia