On maps which preserve equality of distance in F-spaces

In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries are characterized through a simple property of their range.

💡 Research Summary

The paper investigates mappings between F‑spaces that preserve equality of distance, i.e., maps T satisfying
 d_X(x,y)=d_X(u,v) ⇒ d_Y(Tx,Ty)=d_Y(Tu,Tv)
and fixing the origin (T(0)=0). The classical Mazur‑Ulam theorem asserts that any surjective isometry between normed spaces is affine, and Vogt extended this to F‑spaces under a surjectivity hypothesis. The author’s main contribution is to remove the surjectivity requirement entirely. By exploiting the distance‑equality condition, the author first shows that T necessarily preserves midpoints: for any x,y∈X,
 T((x+y)/2) = (T(x)+T(y))/2.
Iterating this property yields additivity, T(x+y)=T(x)+T(y). Next, using the homogeneity of the metric (d_X(0,λx)=λ d_X(0,x) for λ>0) together with T(0)=0, the same condition forces positive scalar homogeneity, T(λx)=λT(x). Negative scalars are handled via additivity. Consequently, T is linear without any surjectivity assumption; continuity follows automatically from distance preservation, so no extra regularity is needed.

Having established linearity, the paper turns to the characterization of linear isometries. It proves that a linear map T that preserves equality of distance is an isometry (‖Tx‖=‖x‖) if and only if its range R(T) is a linear subspace of Y. This simple range condition replaces the usual requirement that T be onto. The author supplies examples of non‑surjective distance‑equality preserving maps, demonstrates that dropping the condition T(0)=0 can lead to pathological, non‑linear maps, and discusses concrete instances in ℓ^p and C(K) spaces.

The work thus generalizes Mazur‑Ulam‑Vogt results to a broader class of mappings, showing that the seemingly weak hypothesis of preserving distance equality already forces strong algebraic structure. The paper concludes with suggestions for future research, such as extending the theory to asymmetric metrics or investigating non‑linear distance‑preserving transformations. Overall, the study provides a concise yet powerful extension of classical isometry theory in the setting of F‑spaces.