Holomorphic functions with distinguished properties on infinite dimensional spaces

Abstract

In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

Holomorphic functions with large cluster sets

“…In the setting of complex holomorphic functions, several authors considered linear and algebraic structures in sets of non‐extendible functions (see [2, 8] and the references therein). Other distinguished properties of holomorphic functions were studied, for example, in [3, 9, 11] or [22].…”

Holomorphic functions with large cluster sets

“…In the setting of complex holomorphic functions, several authors considered linear and algebraic structures in sets of non-extendible functions (see [2,8] and the references therein). Other distinguished properties of holomorphic functions were studied, for example, in [3] or [19].…”