theorem Th31:
  for X be RealNormSpace holds (the carrier of X) --> 0 = 0.(DualSp X)
proof
  let X be RealNormSpace;
  (the carrier of X) --> 0
              = 0.R_VectorSpace_of_BoundedLinearFunctionals X by Th26
             .= 0.(DualSp X);
  hence thesis;
end;
