A sequence of compact metric spaces and an isometric embedding into the Gromov-Hausdorff space

Abstract

For a convergent series with positive terms, we prove that the ${\ell }^{\infty }$ product space of bounded subspaces of the Gromov-Hausdorff space can be isometrically embedded into the Gromov-Hausdorff space, where each subspace consists of compact metric spaces with the diameter less than or equal to the term.