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

theorem Th42:
  for X be ComplexNormSpace, Y be ComplexBanachSpace holds
  C_NormSpace_of_BoundedLinearOperators(X,Y) is ComplexBanachSpace
proof
  let X be ComplexNormSpace;
  let Y be ComplexBanachSpace;
  for seq be sequence of C_NormSpace_of_BoundedLinearOperators(X,Y) st seq
  is Cauchy_sequence_by_Norm holds seq is convergent by Th41;
  hence thesis by Def13;
end;
