Edward Poon, Embry-Riddle Aeronautical University

Abstract: Given a norm $\| \cdot \|$ on a real Banach space $X$, there is a smallest ‘reasonable’ complexification norm $\| \cdot \|_C$ on the complexified space $X_C$, defined by $$\| x+ iy \|_C = \sup \{\|x \cos \theta + y \sin \theta \| : \theta \in [0, 2\pi]\}$$ for $x,y \in X$. Provided $X$ has a certain finiteness condition (possessed by all finite-dimensional spaces) we characterize the isometries for $\| \cdot \|_C$ in terms of the isometries for $\| \cdot \|$.