
theorem Lm710A:
  for X be RealNormSpace, f be sequence of DualSp X
    st f is weakly-convergent holds f is weakly*-convergent
proof
  let X be RealNormSpace, f be sequence of DualSp X;
  assume AS: f is weakly-convergent;
  reconsider f0=w-lim f as Point of DualSp X;
  for x be Point of X holds f#x is convergent & lim (f#x) = f0.x
  proof
    let x be Point of X;
    reconsider G=BiDual x as
      Lipschitzian linear-Functional of DualSp X by DUALSP01:def 10;
C3: G*f is convergent & lim (G*f) = G.f0 by DefWeaklim,AS;
B4: for r be Real st 0 < r
      ex m be Nat st
        for n be Nat st m <= n holds |.(f#x).n - f0.x.| < 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 |.(G*f).n - G.f0.| < r by C3,SEQ_2:def 7;
      take m;
      thus for n be Nat st m <= n holds |.(f#x).n - f0.x.| < r
      proof
        let n be Nat;
        assume D3: m <= n;
B1:     G.f0 = f0.x by DUALSP02:def 1;
        (G*f).n = G.(f.n) by FUNCT_2:15,ORDINAL1:def 12
                   .= (f.n).x by DUALSP02:def 1; then
        (G*f).n = (f#x).n by Def1;
        hence thesis by B1,C1,D3;
      end;
    end; then
    f#x is convergent;
    hence thesis by B4,SEQ_2:def 7;
  end;
  hence f is weakly*-convergent;
end;
