In his famous book "Th$\acute{\mbox{e}}$orie dees po$\acute{\mbox{e}}$rations lin$\acute{\mbox{e}}$aires" Stephan Banach
posed a question of characterization of isometries between metric linear spaces ($F$-spaces he called). In 1932, the same year of the publication of the Banach's book, Mazur and Ulam answered the question in the case of, now we call, normed linear spaces.

**the Mazur-Ulam theorem**
Let $X_j$ be real normed linear spaces for $j=1,2$. Suppose that $U:X_1\to X_2$ is a surjective isometry. Then $U$ is an affine map : a linear map followed by a translation.

After this theorem many authors have studied isometries on Banach spaces and metric linear spaceses. Even in this century alternative simple proofs of the theorem of Mazur and Ulam is given.
In this talk I exhibit related topics.