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Extension of entire functions on nuclear locally convex spaces

We prove that for a nuclear locally convex complex vector space every entire function is the pull-back of some entire function on an appropriate Banach space if and only if the entire functions on E have the following universal extension property: whenever E is a topological linear subspace of a locally convex space F with a fundamental system of seminorms induced by semi-inner products, then every entire function f on E can be extended to an entire function on F. For a complex locally convex space E let H(E) denote the vector space of all entire functions on E, i.e. of all continuous complex-valued functions on E which are Gateaux-analytic. An entire function on E is called of uniformly bounded type if it is bounded on all multiples of a suitable zero neighbourhood in E. By Hub(E) we denote the linear space of all entire functions on E which are of uniformly bounded type. The main result of the present article is the characterization of the nuclear locally convex spaces E satisfying the identity H(E) = Hub(E) by the following universal extension property of the entire functions on E: Whenever E is a topological linear subspace of a locally convex space F which has a fundamental system of continuous seminorms induced by semi-inner products, then each f E H(E) has an extension to an entire function on F. This characterization is interesting as a certain contrast to a result of [6] by which, in every infinite-dimensional (FM)-space F f CN, there exists a closed linear subspace X such that not every entire function on X has a holomorphic extension to F. The proof of the main result is based in one direction on an elementary lemma showing the universal extension property for every entire function of uniformly bounded type. For the other direction we use a result of Colombeau and Mujica [4], by which H(F) = Hub(F) for each (DFM)-space F, together with permanence properties of the relation H(E) = Hub(E) and the remark that every nuclear locally convex space E is isomorphic to a subspace of F', where F is a (DFS)-space with certain properties. The lemma and the result of Colombeau and Mujica [4] mentioned above can also be used to obtain the following improvement of the extension theorem of Boland [2]: Let E be a (DFS)-space which has a fundamental system of continuous seminorms induced by semi-inner products and let X be a closed linear subspace of E. Then every f E H(X) can be extended to an entire function g on E. This shows Received by the editors December 12, 1983. 1980 Mathematics Subject Classification. Primary 46G20, 46A12.