We show that holomorphic mappings of bounded type defined on Frechet spaces extend to the bidual. The relationship between holomorphic mappings of bounded type and of uniformly bounded type is discussed and some algebraic and topological properties of the space of all entire mappings of (uniformly) bounded type are proved, for example a holomorphic version of Schauder's theorem.
For a FrCchet space E , X b ( E ) denotes the space of all entire functions which are bounded on bounded sets. It is endowed with the topology q, of uniform convergence on bounded sets. Clearly, Xb(E) is a FrCchet space if E is a normed space. The elements of X b ( E ) are called entire functions of bounded type. An entire function on E is called of uniformly bounded type if it is bounded on the multiples of some O-neighbourhood in E. They were introduced by COLOMBEAU and MATOS in [9]. We denote by Xub(E) the space of all these functions and it is endowed with a natural topology Tub (see Section 2). In this paper we show, in the setting of distinguished FrCchet spaces, that functions in Xb(E) extend to functions in i%?b(E'), E' the strong bidual of E. So, we generalize a result of ARON and BERNER [l] for Banach spaces (see also [ll]). For those spaces the Davie-Gamelin [ 113 norm preserving extension of bounded holomorphic functions is also obtained.
In studying the topological structure of (Xb(E), q,) for a Frkchet space E we prove that every prequojection E (see Section 2) satisfies the equality %?b(E) = X u b ( E ) . Moreover this relationship in the vector valued case characterizes these spaces (see Proposition 2.4) as in the linear case (see [ti]). The regularity of the (LF)-structure of (Xub(E), rub) is investigated, and for the case of FrBchet-Schwartz spaces E, its equivalence with the property of being Monte1 is shown. Furthermore we also prove that the continuous spectrum of Xub(E) coincides (via the Gelfand transform) with E, if, in addition, E has the approximation property.
This article is organized in two sections. In the first one we study the mentioned 1) Supported in part by DGICYT pr. no. P.S. 88/0144. The third author is also supported in part by DGICYT pr. no. P.S. 88/0050.