Orthogonality and nonlinear functionals on Banach spaces

Abstract:

If B is a real Banach space and $x,y \in B$, then x is said to be orthogonal to $y\;(x \bot y)$ if $\left \| {x + \lambda y} \right \| \geqq \left \| x \right \|$ for all real numbers $\lambda$. A function $F:B \to E$, where E is a topological vector space, is said to be additive if it is continuous and $F(x + y) = F(x) + F(y)$ whenever $x \bot y$. The purpose of the present paper is to characterize additive functions.