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 LM021:
  for I be LinearOperator of S, T,
  s1 being sequence of S
  st I is isometric &
  s1 is convergent holds
  I*s1 is convergent &
  lim (I*s1) = I.lim s1
  proof
    let I be LinearOperator of S, T, s1 be sequence of S;
    assume AS: I is isometric & s1 is convergent;
    P1: dom I = the carrier of S by FUNCT_2:def 1;
    P22: rng s1 c= dom I by P1;
    I is_continuous_in lim s1 by AS,LM010; then
    I/*s1 is convergent & I/.lim s1 = lim (I/*s1) by NFCONT_1:def 5,AS,P22;
    hence thesis by P22,FUNCT_2:def 11;
  end;
