theorem Th43:
  for X be RealNormSpace for Y be RealBanachSpace holds
  R_NormSpace_of_BoundedLinearOperators(X,Y) is RealBanachSpace
proof
  let X be RealNormSpace;
  let Y be RealBanachSpace;
  for seq be sequence of R_NormSpace_of_BoundedLinearOperators(X,Y) st seq
  is Cauchy_sequence_by_Norm holds seq is convergent by Th42;
  hence thesis by Def15;
end;
