 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 Th41:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace holds
  0.R_NormSpace_of_ContinuousFunctions(S,T)
    = 0.R_NormSpace_of_BoundedFunctions(the carrier of S,T)
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  thus 0.R_NormSpace_of_ContinuousFunctions(S,T)
= (the carrier of S)-->0.T by Th40
    .= 0.R_NormSpace_of_BoundedFunctions(the carrier of S,T) by RSSPACE4:15;
end;
