The power of Gâteaux differentiability

Abstract

The search for useful non standard minimization conditions on C 1 functionals defined on Banach spaces lead us to a very simple argument which shows that if a C 1 function f: E → F between Banach spaces is actually Gâteaux differentiable on finite points along finite vectors, then it is uniformly continuous on bounded sets if and only if it is lipschitzian on bounded sets. The following is a development of these ideas starting from locally convex spaces. © 2007 Springer-Verlag Wien.