In this paper we prove several results on the existence of analytic functions on an infinite dimensional real Banach space which are bounded on some given collection of open sets and unbounded on others. In addition, we also obtain results on the density of some subsets of the space of all analytic functions for natural locally convex topologies on this space.

Abstract. Let U be an open subset of a separable Banach space. Let T be the collection of all holomorphic mappings / from the open unit disc DcC into U such that /(D) is dense in U. We prove the lineability and density of T in appropriate spaces for different choices of U.

Abstract. Let X be an infinite-dimensional complex Banach space. Very recently, several results on the existence of entire functions on X bounded on a given ball B\ C X and unbounded on another given ball B2 C X have been obtained. In this paper we consider the problem of finding entire functions which are uniformly bounded on a collection of balls and unbounded on the balls of some other collection. Introduction.Throughout the article, X will denote an infinitedimensional complex Banach space and TL{X) will be the space of all entire (holomorphic) functions on X. If x G X and r > 0, then B(x, r) will denote the open ball in X with center x and radius r. If / e H(X) and S C X, let \\f\\s = sup xeS \f(x)\. When one considers a continuous linear form or a continuous polynomial on X, it is well-known that both are bounded on every bounded subset of X. But, as a consequence of the Josefson-Nissenzweig theorem (see [4, p. 219]) that yields the existence of a sequence (

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