
theorem IMDDX:
for X be RealNormSpace st X is non trivial holds
  ex DuX be SubRealNormSpace of DualSp DualSp X,
     L be Lipschitzian LinearOperator of X, DuX
 st 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 .||
proof
   let X be RealNormSpace;
   assume A0: X is non trivial;
   set F = BidualFunc X;
    set V1 = rng F;
D0: V1 is linearly-closed by NORMSP_3:35;
    V1 <> {}
    proof
      assume V1 = {}; then
      dom F = {} by RELAT_1:42;
      hence thesis by FUNCT_2:def 1;
    end; then
    the carrier of Lin(V1) = V1 by NORMSP_3:31,D0; then
C4: the carrier of Im(F) = rng F; then
   reconsider L = BidualFunc X as Function of X, Im(F) by FUNCT_2:6;
A3:F is additive homogeneous;
B0:L is additive
   proof
    let x,y be Point of X;
    L.(x+y) = F.x + F.y by A3;
    hence L.(x+y) = L.x + L.y by NORMSP_3:28;
   end;
   L is homogeneous
   proof
    let x be Point of X, r be Real;
    L.(r*x) = r*(F.x) by LOPBAN_1:def 5;
    hence L.(r*x) = r*(L.x) by NORMSP_3:28;
   end; then
   reconsider L as LinearOperator of X,Im(F) by B0;
P5:for x be Point of X holds ||.x.|| = ||. L.x .||
   proof
    let x be Point of X;
    ||.x.|| = ||. (BidualFunc X).x .|| by A0,LMNORM;
    hence thesis by NORMSP_3:28;
   end; then
   for x be Point of X holds ||. L.x .|| <= 1* ||.x.||; then
   reconsider L as Lipschitzian LinearOperator of X, Im(BidualFunc X)
      by LOPBAN_1:def 8;
   take Im(BidualFunc X), L;
   L is one-to-one onto by C4;
   hence thesis by Def2,P5;
end;
