
theorem Th711:
  for X be RealBanachSpace, f be sequence of DualSp X
      st f is weakly*-convergent
    holds ||.f.|| is bounded & ||. w*-lim f .|| <= lim_inf ||.f.||
proof
  let X be RealBanachSpace, 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
      ex K be Real st
        0 <= K
      & for g be Point of DualSp X st g in rng f holds |. g.x .| <= K
  proof
    let x be Point of X;
    f#x is convergent by AS; then
    consider K be Real such that
A2:   for n be Nat holds |. f#x .|.n < K by SEQ_2:def 3;
A3: for g be Point of DualSp X st g in rng f holds |. g.x .| <= K
    proof
      let g be Point of DualSp X;
      assume g in rng f; then
      consider n be object such that
A4:     n in NAT & g=f.n by FUNCT_2:11;
      reconsider n as Nat by A4;
      (f.n).x = (f#x).n by Def1; then
      |. g.x .| = |. f#x .|.n by A4,SEQ_1:12;
      hence thesis by A2;
    end;
B6: |. f#x .|.0 < K by A2;
    0 <= |. (f#x).0 .| by COMPLEX1:46; then
    0 <= K by B6,SEQ_1:12;
    hence thesis by A3;
  end; then
  consider L be Real such that
A7: 0 <= L and
A8: for g be Point of DualSp X st g in rng f holds ||.g.|| <= L by Lm55;
Y1: for n be Nat holds |.||.f.||.n .| < (L+1)
  proof
    let n be Nat;
    n in NAT by ORDINAL1:def 12; then
    ||.f.n.|| <= L by A8,FUNCT_2:4; then
    ||.f.n.|| < L + 1 by XREAL_1:39; then
    |. ||.f.n.|| .| < L+1 by ABSVALUE:def 1;
    hence thesis by NORMSP_0:def 4;
  end; then
X1: ||.f.|| is bounded by A7,SEQ_2:3;
B1: for x be Point of X holds |.f0.x.| <= (lim_inf ||.f.||) * ||.x.||
  proof
    let x be Point of X;
B3: for n be Nat holds |.f#x.|.n <= (||.x.||(#)||.f.||).n
    proof
      let n be Nat;
      reconsider h=f.n as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
B4:   |. f#x .|.n = |. (f#x).n .| by SEQ_1:12;
      ||.f.n.|| = ||.f.||.n by NORMSP_0:def 4; then
      ||.f.n.|| * ||.x.|| = (||.x.||(#)||.f.||).n by SEQ_1:9; then
      |. h.x .| <= (||.x.||(#)||.f.||).n by DUALSP01:26;
      hence thesis by B4,Def1;
    end;
B6: lim_inf (||.x.||(#)||.f.||) = (lim_inf ||.f.||) * ||.x.||
       by X1,LOPBAN_5:1;
B7: f#x is convergent & lim(f#x) = f0.x by AS,Def2;
    ||.x.||(#)||.f.|| is bounded by A7,SEQ_2:3,Y1,SEQM_3:37; then
B9: lim_inf |. f#x .| <= lim_inf (||.x.||(#)||.f.||) by B3,RINFSUP1:91,B7;
    lim |. f#x .| = |. f0.x .| by B7,SEQ_4:14;
    hence thesis by B6,B7,RINFSUP1:89,B9;
  end;
  now
    let s be Real;
    assume B9: 0 < s;
    for k be Nat holds 0-s < ||.f.||.(0+k)
    proof
      let k be Nat;
      ||.f.k.||=||.f.||.k by NORMSP_0:def 4;
      hence thesis by B9;
    end;
    hence ex n be Nat st for k be Nat holds 0-s < ||.f.||.(n+k);
  end;
  then
B10: 0 <= lim_inf ||.f.|| by X1,RINFSUP1:82;
  reconsider g=f0 as Lipschitzian linear-Functional of X
    by DUALSP01:def 10;
  now let k be Real;
    assume k in PreNorms g; then
    consider x be Point of X such that
B11:  k = |.g.x.| & ||.x.|| <= 1;
B12: |.f0.x.| <= (lim_inf ||.f.||) * ||.x.|| by B1;
    (lim_inf ||.f.||) * ||.x.|| <= lim_inf ||.f.|| * 1
      by B10,B11,XREAL_1:64;
    hence k <= lim_inf ||.f.|| by B11,B12,XXREAL_0:2;
  end; then
  upper_bound PreNorms g <= lim_inf ||.f.|| by SEQ_4:45;
  hence thesis by A7,SEQ_2:3,Y1,DUALSP01:24;
end;
