
theorem Th23:
  for X be non empty set for Y be RealNormSpace holds
  R_NormSpace_of_BoundedFunctions(X,Y) is RealNormSpace
proof
  let X be non empty set;
  let Y be RealNormSpace;
  RLSStruct (# BoundedFunctions(X,Y), Zero_(BoundedFunctions(X,Y),
RealVectSpace(X,Y)), Add_(BoundedFunctions(X,Y), RealVectSpace(X,Y)), Mult_(
    BoundedFunctions(X,Y), RealVectSpace(X,Y)) #) is RealLinearSpace;
  hence thesis by Th22,RSSPACE3:2;
end;
