
theorem
  for X be RealBanachSpace st X is non trivial holds
    X is Reflexive iff DualSp X is Reflexive
proof
   let X be RealBanachSpace;
   assume AS: X is non trivial;
   hence X is Reflexive implies DualSp X is Reflexive by Lm77R;
    assume AS2: DualSp X is Reflexive;
    DualSp X is non trivial by AS,Lm65A1; then
C2: DualSp DualSp X is Reflexive by AS2,Lm77R;
    consider L be Lipschitzian LinearOperator of X, Im(BidualFunc X)
     such that
C3: L is isomorphism by AS,Th74A;
    set f = BidualFunc X;
    set V = DualSp DualSp X;
D0: rng f is linearly-closed by NORMSP_3:35;
D1: rng f <> {}
    proof
      assume rng f = {}; then
      dom f = {} by RELAT_1:42;
      hence thesis by FUNCT_2:def 1;
    end; then
C4: the carrier of Im(f) = rng f by NORMSP_3:31,D0;
    Im(f) is complete by C3,NORMSP_3:44; then
    rng f is closed by C4,NORMSP_3:48; then
    Im(f) is Reflexive by C2,D0,Th76,D1;
    hence X is Reflexive by C3,NISOM12;
end;
