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We study the holomorphic functions of bounded type defined on (DF)-spaces. We prove that they are of uniformly bounded type. The space of all these functions is a Frechet space with its natural topology. Some consequences and related results are obtained.

Given an entire mapping $f\in \mathcal{H}_b(X,X)$ of bounded type from a Banach space $X$ into $X$, we denote by $\overline{f}$ the Aron-Berner extension of $f$ to the bidual $X^{\ast\ast}$ of $X$. We show that $\overline{g\circ f} = \overline{g}\circ \overline{f}$ for all $f, g\in \mathcal{H}_b(X,X)$ if $X$ is symmetrically regular. We also give a counterexample on $l_1$ such that the equality does not hold. We prove that the closure of the numerical range of $f$ is the same as that of $\bar{f}$.

Several results and examples about locally bounded sets of hol(} morphic mappings defined on certain classes of locally convex spaces (Baire spaces, (DF)-spaces, C(X)-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theory is considered. Introduction. The main topic discussed in this article is the classification of locally convex spaces according to the holomorphic analogues of barrelled, bornological and quasibarrelled properties introduced for the linear theory. This question is related to the study of locally bounded sets of holomorphic mappings defined on open subsets of locally convex spaces. A satisfactory classification can only be obtained in the case of special classes of locally convex spaces, e.g., metrizable spaces, Baire spaces, (DF)-spaces and spaces of type C(X). Some positive results are presented and several examples will show the essential differences between the linear and the holomorphic theory. We shall use standard notations of locally convex spaces as in [21 and 22] and of infinite holomorphy as in [15 and 33]. The word "space" will mean separated locally convex topological vector space. If E is a space, cs(E) denotes the set of all continuous seminorms on E. A family X of mappings defined on an open subset U of a space E with values in a normed space F is called locally bounded if for every x E U there is a neighbourhood V of x contained in U and M > 0 with llf(Y)ll < M for every f E X and y E V. Now we describe briefly the organization of this paper. The first section discusses various extensions of the classical Hartogs' theorem on separate analyticity. According to results of Arias de Reyna [3] and Valdivia [39] a product of two metrizable Baire spaces need not be Baire. Consequently Theorem 1.2 is a proper extension of [6, Corollary 6.2]. In g2 we give a complete characterization of the spaces of type C(X) which are holomorphically barrelled and holomorphically quasibarrelled (cf. [5]). Dineen has recently proved in [16] that every weakly holomorphic mapping defined on any open subset of CN x C(N) is holomorphic. This result provides the first example of a holomorphically Mackey space (cf. [5]) which is not holomorphically quasibarrelled. In §3 the result of Dineen is extended and more examples and properties of holomorphically Mackey spaces are given. In the last section we present various results, on spaces related to (DF)-spaces, concerning the following topics: sequential completeness of H(U, F) endowed with the compact open topology, local boundedness of sequences of holomorphic mappings and the equivalence of local boundedness of sets and sequences on separable spaces. Received by the editors April 14, 1986 and, in revised form, July 15, 1987. 1980 Mathematics Subject Classification (1985 Relvision). Primary 46G20; Secondary 46E10. (D1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

[EN] In this paper we give general conditions on a countable family V of weights on an unbounded open set U in a complex Banach space X such that the weighted space HV (U) of holomorphic functions on U has a Frechet algebra structure. For such weights it is shown that the spectrum of HV(U) has a natural analytic manifold structure when X is a symmetrically regular Banach space, and in particular when X = C-n. (C) 2009 Elsevier Ltd. All rights reserved., We warmly thank Antonio Galbis for providing us with Example 2.6. We also thank the referee for useful comments which have improved the paper. The first author was partially supported by ANPCyT PICT 05 17-33042, UBACyT Grant X038 and ANPCyT PICT 06 00587. The three last authors were supported in part by MEC and FEDER Project MTM2008-03211. The third author was also supported by Prometeo 2008/101. Part of this work was performed while the first cited author was visiting the Departamento de Análisis Matemático (Universidad de Valencia) and Departamento de Matemática Aplicada (Universidad Politécnica de Valencia) during November 2008. He wishes to thank all the good friends in both Departments and also those outside them for such a pelasing time.

We show that on a complex Banach space X , the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon–Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k , it cannot be approximated by norm attaining polynomials with degree less than k . For X=d ∗ (ω,1) , a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm.

We obtain general lower and upper estimates for the first and the second Bohr radii of bounded complete Reinhardt domains in Cn.

The holomorphically ultrabornological spaces are introduced. Their relation with other holomorphically significant classes of locally convex spaces is established and separating examples are given. Some apparently new properties of holomorphically barrelled spaces are included and holomorphically ultrabornological spaces are utilized in a problem posed by Nachbin.

We prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L1(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.

A Banach space E is said to be (symmetrically) regular if every continuous (symmetric) linear mapping from E to E ′ is weakly compact. For a complex Banach space E and a complex Banach algebra F , let H b ( E , F ) denote the algebra of holomorphic mappings from E to F which are bounded on bounded sets. We endow H b ( E , F ) with the usual Frechet topology. M ( H b ( E , F ), F ) denotes the set of all non-null continuous homomorphisms from H b ( E , F ) to F . A subset of G EF on which the extension of Zalduendo is multiplicative is presented and it is shown that, in general, the sets G EF and M ( H b ( E , F ), F ) do not coincide. We prove that if E is symmetrically regular and every continuous linear mapping from E to F is weakly compact then there exists an analytic structure on M ( H b ( E , F ), F ).

It is proved that every quojection in the sense of Bellenot and Dubinsky [1] is the quotient of a countable product of copies of l 1 (I) for a suitable index set I.

A locally complete locally convex space E satisfies that every weakly meromorphic function defined on an open subset of \( \mathbb{C} \) with values in E is meromorphic if and only if E does not contain a countable product of copies of \( \mathbb{C} \). A characterization of locally complete spaces in the spirit of known characterizations of the (metric) convex compactness property is also given.

This paper contains characterizations of the bidual space of some closed subspaces of $\mathcal{H}_b(U)$, the space of holomorphic functions bounded type defined on an open subset $U$ of a Banach space $X$, where $U$ is either a bounded balanced open set or the whole space $X$.

We give asymptotically correct estimations for the cotype 2 constant C2(P(mXn)) ofthe spaceP(mXn) of allm-homogenous polynomials onXn, the span of the firstn sequencesek=(\gdkj)j in a Banach sequence spaceX. Applications to Minkowski, Orlicz and Lorentz sequence spaces are given.