Preface.
In the present article we investigate linear topological invariantsnamely A f ( a ) -, A,(a)and A,(a)-nuclearity -of spaces of holomorphic functions on open subsets of locally convex (1.c.) vector spaces and analytic functionals on compact subsets of metrizable nuclear spaces. These invariants have recently been shown to be important for the structure theory of I.c. spaces and can also be used to prove that certain 1.c. spaces of holomorphic functions are not isomorphic. The treatment of this subject was suggested by the results of our article [6], in which we have shown that for a nuclear power series space &4-(a) the space (H(A,(a)i), To) of entire functions on A,(a)i endowed with the compact open topology zo is isomorphic to A, (,S(a)). The isomorphism is obtained by a simultaneous arrangement of the TAYLOR coefficients of the holomorphic functions into a sequence. Moreover we have shown in [6] that (H(SZ), T o ) is isomorphic to A,(p(a)), if Al(a) is nuclear and if SZ is the open unit ball of I" inbersected with Al(a)i. The nuclearity results of the present article are the following: Let E be a 1.c. space with the property that the closed absolutely convex hull of any compact set is compact. If (A", zo) is A(a)-nuclear (where A equals A,, A , or AJ then (I?@), to) is A(,S(a))-nuclear for any open subset SZ of E. If P is a A(a)-nuclear metrizable I.c. space, then H(K),' -the space of analytic functionals on K -is A(B(a))-nuclear for any compact subset K of F . These two theorems extend the results on nuclearity resp. strong nuclearity of BOLAND [7] and WAELRROECK [ 2 5 ] , COLOMBEAU and MEISE [lo] and BIERSTEDT and MEISE [2]. The proofs of both theorems rely on an idea of WAELBROECK [25], a factorization argument and on a precise knowledge of the growth properties of the exponent sequences of type p(a). They are new even in the case of ordinary nuclearity.
The article is divided in four sections. In the first one we recall some definitions and results and fix the notation. Then in the second section we introduce the exponent sequence /?(a) and prove two propositions on the growth properties of P ( E ) . Section three contains the main lemmas which are needed for the proof of the two theorems. In section four we finally state and prove our theorems and show that they are optimal. Furthermore we indicate how they can be used t o $1 Xath. Sattu. Bd 1O(i Borgens/Meise/Vogt, &a)-nuclearity prove e.g. that for k * n and arbitrary non-empty open subsets U of H(Ck)i and V of H(Cm)i the spaces ( H ( U ) , zo) and ( H ( V ) , zo) are not isomorphic.
We want to emphasize that the results of this article will be used in connection with the structure theory of nuclear FRECHET spaces which has been established by DUBINSKY, VOGT and WAGNER (see e.g. [12], 1211, [ 2 2 ] , [24] and [26]) to obtain a number of results on the structure of ( H ( U ) , zo), where U isan open polydisc in the strong dual of a A(a)-nuclear FRECHET space with a hasis. This will be the subject of a forthcoming article [15]. Parts of the results of this article have been announced in [5].