Metric Geometry in Normed Spaces

Abstract

Banach’s book, p. 160, gives a theorem of Mzaur and Ulam that an isometry of one normed space onto another which carries 0 to 0 is linear. This is true only for real-linear spaces, and is proved by characterizing the midpoint of a segment in a normed space in terms of the distance function. Using the same proof a slightly stronger result can be attained.