
theorem
  for X be RealNormSpace
  st X is non trivial & X is Reflexive
  holds X is separable iff DualSp DualSp X is separable
  proof
    let X be RealNormSpace;
    assume
    A1: X is non trivial & X is Reflexive; then
    consider DuX be SubRealNormSpace of DualSp DualSp X,
             L be Lipschitzian LinearOperator of X, DuX such that
    A2: L is bijective & DuX = Im(BidualFunc X)
      & (for x be Point of X holds L.x = BiDual x)
      & for x be Point of X holds ||.x.|| = ||. L.x .|| by DUALSP02:10;
    A3: Im (BidualFunc X) = DualSp DualSp X by A1,DUALSP02:22;then
    reconsider L as Lipschitzian LinearOperator of X,DualSp DualSp X by A2;
    L is isomorphism by A2,A3;
    hence thesis by Th31;
  end;
