reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LM022:
  for I be LinearOperator of S, T,
  s1 being sequence of S
  st I is one-to-one onto isometric holds
  ( s1 is convergent iff I*s1 is convergent )
  proof
    let I be LinearOperator of S, T,
    s1 being sequence of S;
    assume AS: I is one-to-one onto isometric;
    consider J be LinearOperator of T, S such that
    P2: J = I" & J is one-to-one onto isometric by AS,LM020;
    thus s1 is convergent implies I*s1 is convergent by LM021,AS;
    assume P4: I*s1 is convergent;
    P6: rng s1 c= the carrier of S;
    J*(I*s1) = (J*I)*s1 by RELAT_1:36
    .= (id (the carrier of S))*s1 by AS,P2,FUNCT_2:29
    .= s1 by RELAT_1:53,P6;
    hence s1 is convergent by P2,P4,LM021;
  end;
