On the Generalizations of the Mazur–Ulam Isometric Theorem1

This paper contains several generalizations of the Mazur–Ulam isometric theorem in F∗-spaces which are not assumed to be locally bounded. Let X and Y be two real F∗-spaces, and let X be locally pseudoconvex or δ-midpoint bounded. Assume that a operator T maps X onto Y in a δ-locally 1/2i-isometric manner for all i ∈ 0 ∪ . Then T is affine. In addition, we give the sufficient conditions of a mapping between two topological vector spaces being affine