Abstract: The result of Tian can be interpreted by saying that the space of positive metrics H on the line bundle L can be seen as a direct limit of spaces of Hermitian metrics Hk on H0(X; Lk). It turns out that this correspondence becomes an isometry when Hk is endowed with the natural metric coming from the locally symmetric space structure and H is endowed with the so-called Mabuchi metric. The main goal of this section is to recall the definition and basic properties of Mabuchi metric on H and to establish some facets of the above metric correspondence. The lecture is based on the works of Phong-Sturm and Berndtsson about the relation between the geodesics in the space of K ̈ahler metrics and the space of Hermitian metrics on the section ring, and on the results of Chen-Sun, Berndtsson and Darvas-Lu-Rubinstein concerning the identification of the natural distances on H and Hk.