Geometric Aspects of the Theory of Bounds for Entire Functions in Normed Spaces

Abstract

This chapter describes the geometric aspects of the theory of bounds for entire functions in normed spaces. A phenomenon in infinite-dimensional complex analysis, which has no counterpart in finite dimensions, is that an entire function may be unbounded on a bounded set. This usually leads to the definition of a bounding set, that is., a set which is mapped onto a bound set by every entire function, and this concept has been studied by many authors. A fundamental result is that in certain spaces only the relatively compact sets are bounding, bounding sets have no interior. If on the other hand, those sets are considered where an individual entire function is bounded it is clear that these sets may have interior points; they are the subsets of the open sets.