  We consider the problem of isometric embedding of metric spaces to the Banach
spaces; and introduce and study the remarkable class of so-called linearly
rigid metric spaces: these are the spaces that admit a unique, up to isometry,
linearly dense isometric embedding into a Banach space. The first nontrivial
example of such a space was given by R. Holmes; he proved that the universal
Urysohn space has this property. We give a criterion of linear rigidity of a
metric space, which allows us to give a simple proof of the linear rigidity of
the Urysohn space and some other metric spaces. The various properties of
linearly rigid spaces and related spaces are considered.
