reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

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;
