
theorem
  for X be RealNormSpace, f be sequence of DualSp X
    st X is Reflexive holds
      f is weakly-convergent iff f is weakly*-convergent
proof
  let X be RealNormSpace, f be sequence of DualSp X;
  assume AS: X is Reflexive;
  thus f is weakly-convergent implies f is weakly*-convergent by Lm710A;
  thus f is weakly*-convergent implies f is weakly-convergent
  proof
    assume AS1: f is weakly*-convergent;
    reconsider f0=w*-lim f as Point of DualSp X;
    for F be Lipschitzian linear-Functional of DualSp X
      holds F*f is convergent & lim (F*f) = F.f0
    proof
      let F be Lipschitzian linear-Functional of DualSp X;
      reconsider G=F as Point of DualSp DualSp X by DUALSP01:def 10;
      consider x be Point of X such that
B1:     for f be Point of DualSp X holds G.f = f.x by AS,DUALSP02:21;
C4:   f#x is convergent & lim (f#x) = f0.x by AS1,Def2;
B5:   for r be Real st 0 < r
        ex m be Nat st
          for n be Nat st m <= n holds |.(F*f).n - F.f0.| < r
      proof
        let r be Real;
        assume 0 < r; then
        consider m be Nat such that
C1:       for n be Nat st m <= n holds |.(f#x).n - f0.x.| < r
            by C4,SEQ_2:def 7;
        take m;
        hereby let n be Nat;
          assume D3:m <= n;
          (F*f).n = G.(f.n) by FUNCT_2:15,ORDINAL1:def 12; then
          (F*f).n = (f.n).x by B1; then
          (F*f).n = (f#x).n by Def1; then
          |.(F*f).n - F.f0.| = |.(f#x).n - f0.x.| by B1;
          hence |.(F*f).n - F.f0.| < r by C1,D3;
        end;
      end; then
      F*f is convergent;
      hence thesis by B5,SEQ_2:def 7;
    end;
    hence f is weakly-convergent;
  end;
end;
