On non-linear mappings preserving the semi-inner product

We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ 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).