 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem
  for S be non empty compact TopSpace,
      T be NormedLinearTopSpace st T is complete holds
   R_NormSpace_of_ContinuousFunctions(S,T) is complete
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  set Z = R_NormSpace_of_ContinuousFunctions(S,T);
  assume T is complete; then
  for seq be sequence of Z st seq is Cauchy_sequence_by_Norm holds
    seq is convergent by Th51;
  hence thesis by LOPBAN_1:def 15;
end;
