reserve X,G for RealNormSpace-Sequence,
          Y for RealNormSpace;
reserve f for MultilinearOperator of X,Y;

theorem Th43:
  for X be RealNormSpace-Sequence for Y be RealBanachSpace holds
  R_NormSpace_of_BoundedMultilinearOperators(X,Y) is RealBanachSpace
  proof
    let X be RealNormSpace-Sequence;
    let Y be RealBanachSpace;
    for seq be sequence of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
    st seq is Cauchy_sequence_by_Norm
    holds seq is convergent by Th42;
    hence thesis by LOPBAN_1:def 15;
  end;
