On Non-linear Mappings Preserving the Semi-inner Product

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