On Non-linear Mappings Preserving the Semi-inner Product

Abstract

We say that a smooth normed space X has a property (SL), if every mapping f:X rightarrow X preserving the semi-inner product on X is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space ell _p, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space ell _p have the property (SL) for every 1< p < infty . Finally, using a variant of the Gowers–Maurey space, we construct an infinite-dimensional uniformly smooth Banach space X such that every smooth Banach space isomorphic to X has the property (SL).