Isometry and phase-isometry of non-Archimedean normed spaces

In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry f : S_r (X) → S_r (X), where X is a finite-dimensional non-Archimedean normed space and S_r(X) is a sphere with radius r ∈ ∥X∥, is surjective if and only if \(\mathbb{K}\) is spherically complete and k is finite. Moreover, we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with |2| = 1, any phase-isometry f: X → Y is phase equivalent to an isometric operator.