On the Mazur-Ulam theorem (Q1715032)

The Mazur-Ulam theorem states that every isometry mapping from a real normed linear space onto another one is affine. In the article under review, the author proves the following extension of the Mazur-Ulam theorem: If $\varphi:X\to Y$ is an isometry from a real metrizable topological vector space $X$ with a translation invariant metric into a real normed linear space $Y$ such that the set $\{\varphi(x) - \varphi(0): x\in X\}$ is closed under addition, then $\varphi$ is an affine mapping.