Bounded sets in (LF)-spaces

Abstract

Abstract. The behaviour of bounded sets is important in the theory of countable inductive limits of Fréchet spaces, the (LF)-spaces, and its applications. An (LF)-space is called regular if every bounded set is contained and bounded in one of the steps. In the present paper necessary conditions and sufficient conditions are given for the regularity of an (LF)-space. The conditions are expressed in terms of the behaviour of the neighbourhoods of the steps. It is proved that the conditions are equivalent for (LF)-spaces of sequences or of continuous functions.The purpose of the present article is to give necessary (and also sufficient) conditions to ensure that an (LF)-space satisfies that every bounded set is contained and bounded in some of the steps. The countable inductive limits of Fréchet spaces, called (LF)-spaces, were thoroughly studied by Dieudonné and Schwartz [7] and by Grothendieck [8]. They were motivated by their relevance in the theory of distributions of Schwartz and its applications to partial differential equations. More recently several authors have analyzed the structure of (LF)-spaces and of (LF)-spaces of sequences and of continuous or holomorphic functions. These spaces are important in connection with applications to spaces of ultradistributions and convolution equations. We refer to [1], [9] and [17] for excellent presentations of the theory of (LF)-spaces.The space (E, t) = ind"(7s", tn) is an (LF)-space if (En, t")"em is an increasing sequence of Fréchet spaces with continuous inclusions (E", t") c (En+X, tn+x), E = \Jn€NEn and (E, t) is endowed with the finest locally convex topology such that the injections from (E", t") into E are continuous. Here an (LF)-space is always assumed to be Hausdorff. If every step (En, t") is a Banach space, then the inductive limit E is called an (LB)-space. In what follows, for each n £ N, we let (Un tk)k&n denote a basis of absolutely convex O-neighbourhoods in E" . We will assume without loss of generality (a) U",k DU",k+xVn,k£ N;(b) U",k c Un+X^n, k £ N. We will use n, m, p, v, M, N for natural numbers in the first index (steps) and k, I, K, L for natural numbers in the second index (neighbourhoods in a