Topological and Banach Space interpretation for real sequences whose consecutive terms have a bounded difference

In this paper we give a topology-dynamical interpretation for the space  of all integer sequences P_n whose consecutive difference P_{n+۱}-P_n is a bounded sequence.  We also introduce a new concept textit{"Rigid Banach space"}. A rigid  Banach space is a Banach space X  such that for  every continuous linear injection j:Xto X,;overline{J(X)} is either isomorphic to X or it does not contain any isometric copy of X. We prove that ell_{infty} is not a rigid Banach space. We also  discuss about  rigidity of Banach algebras.

Keywords:

Rigid Banach space , sequence space , Primes