H(K) = li rn (Høø(U),ll IIu)= l m (H(U),rco)

Transcription

1 HOUSTON JOURNAL OF MATHEMATICS, Volume 10, No. 3, A REMARK ON THE REGULARITY OF SPACES OF GERMS Roberto Luiz Soraggi* We denote by H(K) the space of holomorphic germs on the non-void compact subset K of the Hausdorff locally convex space E, endowed with its natural topology defined as follows: H(K) = li rn (Høø(U),ll IIu)= l m (H(U),rco) UDK UDK where U runs over the collection of all open subsets of E which contain K. For basic information concerning the space H(K) we refer to [ 1 ] and [4]. In this article we continue the study of regularity of H(K), i.e., if each bounded subset of H(K) is contained and bounded in some/-frø(u) for some open U which contains K. Here /-frø(u) denotes the Banach space of all bounded holomorphic functions on U endowed with the supremum norm. The direction developed here is a natural consequence of the approach initiated in [8]. The study of the regularity of H(K) can be divided into two parts. The first concerns the existence of Cauchy's estimates. The second concerns coherence of the Taylor series expansions of elements of a bounded subset B of H(K), i.e., if f(x)(y) = o n_-.x) y) is the Taylor series expansions of f around x, does there exist a n=0 n! neighbourhood V of zero such that f(x)(y) = f(x')(y') for all x,x'e K; y,y'e V whenever x + y = x' + y' and f B? The first problem is related to regularity of H(0) and in [8] we gave some examples of locally convex spaces E where H(0) is regular. So, the study of the regularity of H(K) can be reduced to the following question: When does regularity of H(0) imply regularity of H(K), or in other words, when is the existence of Cauchy's estimates for the bounded subset B of H(K) sufficient for coherence of the Taylor expansions of elements of B? *This research, supported by UFRJ, Brazil, was performed when the author was a visiting professor at the University College, Dublin, Ireland, during the academic year

2 446 ROBERTO LUIZ SORAGGI In [8], we began to study this question and obtained a positive answer in the case of a metrizable compact subset K in a locally convex space satisfying a technical condition (property P). A locally convex space E satisfies property P if for each convex, balanced, open subset U of E and for any sequence oo (fn)n=0 of non-zero holomorphic functions U, there exist a subsequence (fnj )j:0 oo and a bounded sequence (yj)jøø 0 on U such that fnj(yj) =P- 0 for all j G N. Baire locally convex spaces, metrizable locally convex spaces, any product of metrizable locally convex spaces, the space C(X) (of all continuous complex valued functions on a completely regular Hausdorff topological space X endowed with the topology r o of uniform convergence the compact subsets of X) and any complete, Hausdorfflocally convex space with a Schauder basis which is very weakly convergent (in the sense of [5] ), are examples of locally convex spaces which satisfy property P (see [8] ). We proved in [8] that for complete locally convex spaces, we may consider a constant sequence (yj)7=0 in the definition of property P and using this result and Taylor series expansion we can easily show that it suffices in the definition of property P to consider a sequence of polynomials on E instead of a sequence of holomorphic functions (fn) '=0. It would be interesting to know if it is sufficient to consider a sequence of non-zero continuous linear functions on E in order to prove that E has property P. To study this question, we need a further characterization of property P. DEFINITION I. A sequence (Xn)n= 0 of elements of the locally convex space E is very strongly convergent if for any sequence of scalars (Xn)n= 0 the sequence (XnXn)n= 0 is a null sequence in E. It is easy to show that (Xn) = 0 is very strongly convergent in E if and only if for any continuous semi-norm p on E, there exists n o G N such that p(x n) = 0 for all n > no. For basic information on very strongly convergent sequences in a locally convex space we refer to [5] or [6]. One easily proves, using the definition of property P in terms of a constant sequence (yj)7=0, the following result: PROPOSITION 2. The complete locally convex space E has property P f and

3 A REMARK ON THE REGULARITY OF SPACES OF GERMS 447 only if every very strongly convergent sequence in [H(E),r o] is a trivial sequence (i.e., eventually zero). It follows from Proposition 2 for E a reflexive D. F: space that the non-existence of a non-trivial very strongly convergent sequence in E/ is equivalent to implying that E has property P. We do not know if this holds for arbitrary locally convex spaces. Recently, we proved in [9] that in a fully nuclear space E with a Schauder basis whose strong dual E/ is a K-space (in the sense of [6]), property P is related to [H(E ),r o] being infrabarreled. On the other hand, for E = C IN] since E/ = C N has no continuous norm (this is easily seen to be equivalent to the existence of a non-trivial very strongly convergent sequence in E}) we see that E = C [N] does not have property P. In this case we know (see [4] or [7] ) that H(0), 0 C E, is not regular. It is known [8] that if E does not satisfy property P then E/ has no continuous norm and hence (by Corollary of [8] )H(0), 0 C E, is not regular if E is also a reflexive D.F. space. In the case of a fully nuclear space E with a Schauder basis it is also known, [2], that regularity of H(0), 0GE, is equivalent to [H(E ),r o] being infrabarreled and so, in the case of a fully nuclear space E with a Schauder basis whose strong dual is a K-space, the technical property P is related to the regularity of H(0), 0 E. In other words, when E is a reflexive D.F. space or a fully nuclear space whose strong dual is a K-space, regularity of H(0), 0 E, implies that E satisfies property P. All these examples strengthen our conjecture that property P is just a working tool and can be always dropped when H(0) is regular. Here, we obtain from internal properties of H(0) - non existence of non trivial very strongly convergent sequences in H(0) and existence of Cauchy's estimates for the bounded subsets of H(0) - coherence of the Taylor series expansions of elements of a bounded subset of H(K), K a metrizable compact subset of E. This result strengthens our conjecture that regularity of H(0) always implies regularity of H(K), for metrizable compact subsets of any Hausdorff locally convex space E. REMARK 3. For any bounded subset B of H(K) the subset B = -- fcb, xck, mcn) " (d _ f (x), is also a bounded subset of H(0). This follows from the fact that for every continuous

4 448 ROBERTO LUIZ SORAGGI semi-norm p on H(O), --, mf(x), p(f)= Z n=0 sup Pt m J x K ß is also a continuousemi-norm on H(K). Continuity of follows since (f) is finite for each f6 H(K) and H(K) is a barreled space. [For a proof of this, see [3], Proposition 10]. THEOREM 4. Let E be a Hausdorff locally convex space E and let K be a metrizable compact subset of E. If H(O) is regular and if there is no non-trivial very strongly convergent sequence in H(O) then H(K) is also regular. PROOF. Let B be a bounded subset of H(K). Since H(0) is regular and = (d Ix), fcb, xck, m N) is a bounded subset of H(0), there exist M> 0 and a balanced, convex,- open neighbourhood of zero in E such that Hence, we have for each f C B " im w dmf(x)llv < M. ß 2 m oo dmffx)f. f(x)(y) = f(x +y) = Zm=0 m. ty, y G V. Now, suppose we do not have coherence of the Taylor series expansions of elements of B, then, for each balanced, convex open neighbourhood Vo of zero in E, V ac(1/4)v, there exist fagb;xa, x agk;ya, yagv asuchthat x a+ya and fo (Xo )(yo ) :P- fa(x )(yc[). Ordering the sets(vo ) by set inclusion, Hence Xo t. t - Xo Yo - Yo! we have ya-> 0 and Yo xa Ya -> 0 as a-> oo Since K is metrizable, K - K is also metrizable and so, we can choose sequences (Xn)n= 0 and (Xn)n= 0 such that the sequence (x/ - Xn)n= 0 is a null sequence corresponding! in E and x n- x n (1/2)V for all n. For each! fn B; x n + Yn = Xn + Yn such that n N we consider the We define, for each n N, fn(xn)(yn) :P- fn(x )(yr ).

5 A REMARK ON THE REGULARITY OF SPACES OF GERMS 449 d fn(xn ) d n oo fn(xn ), Qn- where the integer m n will be defined later. mn! n n. [Z] =0 j! l(xn- x n) CLAIM 1. For each n, there exists m n C N such that Qn 0. Suppose that for every m C N m, (*) d fn(xn) d m dlfn(x n) m! = -. [ ½0 j! ](Xn- Xn)' Since we have absolute and uniform convergence of the series oo fn(xn ) Zi:0 j (Y) in V we have absolute and uniform convergence of the series m oo djfn(x n), d m djfn(x n) [Z j=0 j! ](Xn- x n) = Z 0 V.[ ](x n - x n) in a neighbourhood of zero W contained in (1/4)V. So, if (*) holds for every m N, then for y in W we have m, --' oo d fn(xn) djfn(x n) f(x )(y) = Zm= 0'. (y) = -0[Z øø dm - j=0- T [ j! ](x - Xn)] (y) _oo m djfn(x n) = Z7=012..;m=0. [ j! l(x - Xn)l (y) ^. oo djfn(xn), --,, = Zi= 0 (Xn- x n + y) = fn(xn)(x n - x n + y). that Since { y + x n - x n, y W} c V, the uniqueness of analyticontinuation implies fn(xn)(y) = fn(xn)(xn- x n + y) for every (1/4)V. But Yn Vn C (1/4)V and so fn(x )(yr ) = fn(xn)(xn- x n + y )= fn(xn)(yn) which is a contradiction and this completes the proof of Claim 1. CLAIM 2. The sequence (Qn)n=0 is a non-trivial very strongly convergent sequence in H(0). Let p be a continuousemi-norm in H(0). We show that p(qn )= 0 for n

6 450 ROBERTO LUIZ SORAGGI sufficiently large. Suppose otherwise, then we can find a strictly increasing sequence of positive integers (nœ)œoo= 0 such that p(qnœ) 4:0 for all œ. We also denote by (Qn)n=0 the corresponding subsequence. Let p(qn ) = $n > 0. such that We remark that Qn is the limit in H(0) of "mn,,,. d fn(xn ) _k mnrdjfn(xn), Rn k = -- mn! - 2.;j=0 n n. [ j! ](x n ' x n) as k oo. Hence, we can choose a strictly increasing sequence of positive integers (kn)n> 0,-m,,. (1) p[ d -- nfn(xl ) - Z n0 n [ djfn(xn) j! ](x - Xn)] > $n mn!. - -, (2) 2kn$½ > n. We define the following semi-norm on H(K):,,m,, ". -,- I2oo 2kn r d nffx ) k n dmn djf(x n) P(f)= n=0 pt' - I;j=0 nn.[ - -. l(x - Xn)l. The semi-norm p is the supremum of a sequence of continuous semi-norms on H(K). Since H(K) is barreled it suffices, in order to prove p is continuous, to show p(f) is finite for each f C H(K). To show this, let f C H(K). Choose a balanced, convex and open neighbourhood U of zero such that f GHøø(K+ 6U). By Cauchy's inequalities we get for all m N and x K. "5.< Since x n - x n 0, we can choose n o N such that x n - x n C U for all n > no. By Cauchy's inequalities we have for all n > no: mn d J f(x n) d J f(x n), dj f(x n) I[- n. [ ji ](x - Xn)11U < II j Ilxn_Xn+ U < II j7112u ß djf(x n) < 2JII--. IIu < l llfllr+6u. The uniform and absolute convergence of the Taylor series of the derivatives in a neighbourhood W of zero implies that for y G W we also have 3 J

7 A REMARK ON THE REGULARITY OF SPACES OF GERMS 451 mf(x ) 2 n=o m! (Y) =f(x +y)=ffx n+x -.x n+y), djf(x n), = f(xn)(x n - x n + y) = Z- j_-0 ' (x n - x n + y).- j M -oo rdmr d f(xn).., = L =02.,m=0t.t j! lt-xn- Xn)](y) = Z n=0e70 [ m djf(xn) ^. ml (y) _oo r mr oo djf(xn) = Zm=0t.l ;j= 0 j! ](xr[- Xn)](Y). Hence mf(x )_oo m. Jf(xn)., m! - j=0. t j! l(x n- Xn). have Now, we can find C(U) > 0 such that p(g) < C(U)llgll U for all g G H(U). We now _k n mnf(x ) k n mn Jf(x n) Z ø=no 2 p[. - Zj=0 - n. [ j! ](x - Xn)] oo k n mn t dj f(x n) = Zn=no 2 p[zj> kn+l- n.[ j! ](x - Xn)] no2kn "mn.,., < C(U) ; o= ;j >kn+ 1 ii n. [djf(xn) ](Xn_ Xn)llu oo k n <C(U)IlflIK+6U Zn=no2 Zj> kn+l(1/3)j < C(U)IlflIK+6U Zn> no(2/3) kn <- C(U)IIflIK+6U. And so, p(f) is finite and p is a continuous semi-norm on H(K). On the other hand, for all n, we have ^m n,.-. d fn(xn) vknd.-m n djfn(x n) ' (fn ) > 2knp[ mn - j=0 n n.[ j! ](x - Xn)] > 2 kn - > n. Hence ' (fn) -->oo as n --> oo. This is a contradiction since {fn' n G N} C B C H(K) and so (Qn)n=0 must be a non-trivial very strongly convergent sequence in H(0) and

8 452 ROBERTO LUIZ SORAGGI this completes the proof of Theorem 4. REMARK 5. Observe that when E-C [N] the sequence (fn)n=l defined by fn(z) = Zngn(Z 1) where gn(3,)= n/(n3,-l)for 3,E U n = ( 3, E C; 13,1 < l/n) is a non-trivial very strongly convergent sequence in H(0), 06 E = C [N]. Thisequence gives in H(0) a bounded subset which is not uniformly bounded in any open neighbourhood of zero. (See[4], Chapter III or [7], counterexample 5.1.) The following question naturally suggests itself. QUESTION 6. Is it true that the existence of a non-trivial very strongly convergent sequence in H(0) implies that H(0) is not regular? The author gratefully thanks Professor S. Dineen and Dr. P. J. Boland for many encouraging and helpful discussions during the preparation of this article. REFERENCES 1. K. D. Bierstedt, R. Meise, Aspects of inductive limits in spaces of germs of holomorphic functions on locally convex spaces and applications to the study of [H(U},r(o], Advances in Holomorphy, J. A. Barroso (ed.), Notas de Matem itica, North-Holland, 34(1979), P. J. Boland, S. Dineen, Duality theory of spaces of germs and holomorphic functions on nuclear spaces, Advances in Holomorphy, J. A. Barroso (ed.), Notas de Matemfitica, North-Holland, 34(1979), S. Dineen, Holomorphic germs on compact subsets of locally convex spaces, Funct. Anal., Holomorphy and Approx. Theory, Proceedings, Rio de Janeiro, 1978, S. Machado (ed.), Springer-Verlag, Lecture Notes in Math., 843(1981), , Complex analysis in locally convex spaces, North-Holland Math. Studies, 57(1981). 5., Surjective limits of locally convex spaces and their application to inf vaite dimensional holomorphy, Bull. Soc. Math. France, 103(1975), , Holomorphic functions and surjective limits, Proc. on Infinite Dimensional Holomorphy, University of Kentucky, Springer-Verlag, Lecture Notes in Math., 364(1974), R. L. Soraggi, Partes limitadas nos espa os de germes de aplica 3es holomorfas, Anais da Academia Brasileira de Cie hcias, 49(1977), , Holomorphic germs on certain locally convex spaces, to appear. 9., On entire functions of nuclear type, Proc. of the RoyalIrish Acad.,83A(1983), University College Belfield Dublin 4, I teland and Universidade Federal do Rio de Janeiro Rio de Janeiro, Brazil Received March 24, 1983