Mazur's sphere theorem shows that the unit sphere of the L_p-space can be Hölder homeomorphic to the unit sphere of the Hilbert space with explicit Hölder estimates. In this talk, we will prove that Mazur's sphere theorem can be inherited by spheres of its subspaces. As its applications, we provide a quantitative refinement of the work of Ribe and Aharoni-Lindenstrauss.