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.