
theorem Th712A:
  for X be RealBanachSpace, X0 be Subset of X, vseq be sequence of DualSp X
    st ||.vseq.|| is bounded & X0 is dense
  & (for x be Point of X st x in X0 holds vseq#x is convergent)
  holds vseq is weakly*-convergent
proof
  let X be RealBanachSpace, X0 be Subset of X, vseq be sequence of DualSp X;
  assume that
A1: ||.vseq.|| is bounded and
A2: X0 is dense and
A3: for x be Point of X st x in X0 holds vseq#x is convergent;
  reconsider vseq1=vseq as
    sequence of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
      by DUALSP02:14;
  reconsider X01=X0 as Subset of LinearTopSpaceNorm(X) by NORMSP_2:def 4;
B1: for x be Point of X st x in X01 holds vseq1#x is convergent
  proof
    let x be Point of X;
    assume x in X01; then
B11: vseq#x is convergent by A3;
    vseq1#x = vseq#x by RNSBH1;
    hence vseq1#x is convergent by RNS8,B11;
  end;
B2: for x be Point of X
      ex K be Real st 0 <= K
        & for n be Nat holds ||.(vseq1#x).n.|| <= K
  proof
    let x be Point of X;
    consider K0 be Real such that
B20:  0 < K0
    & for n be Nat holds |. ||.vseq.||.n .| < K0 by A1,SEQ_2:3;
    reconsider K=K0*||.x.|| + 1 as Real;
    take K;
C0: K0*||.x.|| + 0 < K0*||.x.|| + 1 by XREAL_1:8;
    thus 0 <= K by B20;
    thus for n be Nat holds ||.(vseq1#x).n.|| <= K
    proof
      let n be Nat;
      |. ||.vseq.||.n .| < K0 by B20; then
      |. ||.vseq.n.|| .| < K0 by NORMSP_0:def 4; then
A5:   ||.vseq.n.|| < K0 by ABSVALUE:def 1;
      reconsider h=vseq.n as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
B1:   |.h.x.| <= ||.vseq.n.|| * ||.x.|| by DUALSP01:26;
C3:   ||.vseq.n.|| * ||.x.|| <= K0 * ||.x.|| by A5,XREAL_1:64;
      |.(vseq#x).n .| = |.(vseq.n).x.| by Def1; then
      |.(vseq#x).n .| <= K0 * ||.x.|| by C3,B1,XXREAL_0:2; then
B4:   |.(vseq#x).n .| <= K by C0,XXREAL_0:2;
      vseq#x = vseq1#x by RNSBH1;
      hence ||.(vseq1#x).n .|| <= K by B4,EUCLID:def 2;
    end;
  end;
  X01 is dense by A2,NORMSP_3:15; then
  consider tseq be Point of
    R_NormSpace_of_BoundedLinearOperators(X,RNS_Real) such that
B4: ( for x be Point of X holds vseq1#x is convergent
   & tseq.x=lim (vseq1#x)
   & ||.tseq.x.|| <= lim_inf ||.vseq1.|| * ||.x.|| )
   & ||.tseq.|| <= lim_inf ||.vseq1.|| by B1,B2,LOPBAN_5:8;
  reconsider g0=tseq as Point of DualSp X by DUALSP02:14;
  for x be Point of X holds vseq#x is convergent & lim (vseq#x) = g0.x
  proof
    let x be Point of X;
B7: vseq1#x = vseq#x by RNSBH1;
    vseq1#x is convergent & tseq.x=lim (vseq1#x) by B4;
    hence vseq#x is convergent by B7,RNS8;
    then lim (vseq#x) = lim (vseq1#x) by RNSBH1,RNS9;
    hence g0.x=lim (vseq#x) by B4;
  end;
  hence vseq is weakly*-convergent;
end;
