
theorem Lm813A:
  for X be RealNormSpace, x be sequence of X st X is Reflexive holds
    x is weakly-convergent iff (BidualFunc X)*x is weakly*-convergent
proof
  let X be RealNormSpace, x be sequence of X;
  assume AS: X is Reflexive;
  set f=(BidualFunc X)*x;
  hereby assume AS: x is weakly-convergent;
    reconsider x0=w-lim x as Point of X;
    for g be Point of DualSp X
      holds f#g is convergent & lim (f#g) = (BiDual x0).g
    proof
      let g be Point of DualSp X;
      reconsider f0=BiDual x0 as Point of DualSp DualSp X;
A3:   for n be Nat holds (f#g).n = (g*x).n
      proof
        let n be Nat;
        reconsider f1=BiDual(x.n) as Point of DualSp DualSp X;
        f.n = (BidualFunc X).(x.n) by FUNCT_2:15,ORDINAL1:def 12; then
B2:     f.n = BiDual(x.n) by DUALSP02:def 2;
        (f#g).n = (f.n).g by Def1; then
        (f#g).n = g.(x.n) by B2,DUALSP02:def 1;
        hence (f#g).n = (g*x).n by FUNCT_2:15,ORDINAL1:def 12;
      end;
      reconsider g1=g as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
A41:  g1*x is convergent & lim (g1*x) = g1.x0 by DefWeaklim,AS;
A5:   for r be Real st 0 < r
        ex m be Nat st
          for n be Nat st m <= n holds |.(f#g).n - f0.g.| < r
      proof
        let r be Real;
        assume 0 < r; then
        consider m be Nat such that
B1:       for n be Nat st m <= n holds |.(g1*x).n - g1.x0.| < r
            by A41,SEQ_2:def 7;
        take m;
        thus for n be Nat st m <= n holds |.(f#g).n - f0.g.| < r
        proof
          let n be Nat;
          assume C3: m <= n;
          f0.g = g.x0 by DUALSP02:def 1; then
          |.(f#g).n - f0.g.| = |.(g1*x).n - g1.x0.| by A3;
          hence thesis by B1,C3;
        end;
      end; then
      f#g is convergent;
      hence thesis by A5,SEQ_2:def 7;
    end;
    hence f is weakly*-convergent;
  end;
    assume f is weakly*-convergent; then
    consider f0 be Point of DualSp DualSp X such that
A0:   for h be Point of DualSp X
        holds f#h is convergent & lim (f#h) = f0.h;
    consider x0 be Point of X such that
A1:   for g be Point of DualSp X holds f0.g = g.x0 by AS,DUALSP02:21;
    for g be Lipschitzian linear-Functional of X
      holds g*x is convergent & lim (g*x) = g.x0
    proof
      let g be Lipschitzian linear-Functional of X;
      reconsider g1=g as Point of DualSp X by DUALSP01:def 10;
A3:   for n be Nat holds (g*x).n = (f#g1).n
      proof
        let n be Nat;
        reconsider f1=BiDual(x.n) as Point of DualSp DualSp X;
B2:     f.n = (BidualFunc X).(x.n) by FUNCT_2:15,ORDINAL1:def 12
           .= BiDual(x.n) by DUALSP02:def 2;
        thus (g*x).n = g1.(x.n) by FUNCT_2:15,ORDINAL1:def 12
                    .= f1.g1 by DUALSP02:def 1
                    .= (f#g1).n by B2,Def1;
      end;
B4:   f#g1 is convergent & lim (f#g1) = f0.g1 by A0;
A5:   for r be Real st 0 < r
        ex m be Nat st
          for n be Nat st m <= n holds |.(g*x).n - g1.x0.| < r
      proof
        let r be Real;
        assume 0 < r; then
        consider m be Nat such that
B1:       for n be Nat st m <= n holds |.(f#g1).n - f0.g1.| < r
            by SEQ_2:def 7,B4;
        take m;
        hereby let n be Nat;
          assume C3: m <= n;
          f0.g1 = g1.x0 by A1; then
          |.(g*x).n - g1.x0.| = |.(f#g1).n - f0.g1.| by A3;
          hence |.(g*x).n - g1.x0.| < r by B1,C3;
        end;
      end; then
      g*x is convergent;
      hence thesis by A5,SEQ_2:def 7;
    end;
    hence x is weakly-convergent;
end;
