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 pn whose consecutive difference pn+1 − pn is a bounded sequence.  we also introduce a new concept ”rigid banach space”. a rigid  banach space is a banach space x such that for  every continuous linear injection j : x → x, j(x) is either isomorphic to x or it does not contain any isometric copy of x. we prove that ℓ∞ is not a rigid banach space. we also discuss about  rigidity of banach algebras.