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 Th39:
  for X, Y be RealNormSpace holds
  R_NormSpace_of_BoundedLinearOperators(X,Y) is RealNormSpace
proof
  let X, Y be RealNormSpace;
  RLSStruct (# BoundedLinearOperators(X,Y), Zero_(BoundedLinearOperators(X
,Y), R_VectorSpace_of_LinearOperators(X,Y)), Add_(BoundedLinearOperators(X,Y),
    R_VectorSpace_of_LinearOperators(X,Y)), Mult_(BoundedLinearOperators(X,Y),
    R_VectorSpace_of_LinearOperators(X,Y)) #) is RealLinearSpace;
  hence thesis by Th38,RSSPACE3:2;
end;
