
theorem Th814:
  for X be RealBanachSpace, f be sequence of DualSp X
    st X is separable & ||.f.|| is bounded holds
    ex f0 be sequence of DualSp X st
       f0 is subsequence of f & f0 is weakly*-convergent
proof
  let X be RealBanachSpace, f be sequence of DualSp X;
  assume that
A1: X is separable and
A2: ||.f.|| is bounded;
  consider x0 be sequence of X such that
A3: rng x0 is dense by A1,NORMSP_3:21;
  set X0 = rng x0;
  consider f0 be sequence of DualSp X such that
AX: f0 is subsequence of f and
A31: for n be Nat holds f0#(x0.n) is convergent by A2,Lm814;
A21: for x be Point of X ex K be Real st
       0 <= K & for n be Nat holds |. (f#x).n .| <= K
  proof
    let x be Point of X;
    consider K0 be Real such that
B0:   0 < K0 & for n be Nat holds |. ||.f.||.n .| < K0 by A2,SEQ_2:3;
    reconsider K=K0*||.x.|| + 1 as Real;
    take K;
B11: K0*||.x.|| + 0 < K0*||.x.|| + 1 by XREAL_1:8;
    thus 0 <= K by B0;
    thus for n be Nat holds |. (f#x).n .| <= K
    proof
      let n be Nat;
B2:   |. ||.f.||.n .| <= K0 by B0;
      ||.f.||.n = ||.f.n.|| by NORMSP_0:def 4; then
B3:   ||.f.n.|| <= K0 by B2,ABSVALUE:def 1;
      reconsider h=f.n as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
B4:   |.h.x.| <= ||.f.n.|| * ||.x.|| by DUALSP01:26;
B51:  ||.f.n.|| * ||.x.|| <= K0 * ||.x.|| by B3,XREAL_1:64;
      |.(f#x).n .| = |.(f.n).x.|  by Def1; then
      |.(f#x).n .| <= K0 * ||.x.|| by B51,B4,XXREAL_0:2;
      hence thesis by B11,XXREAL_0:2;
    end;
  end;
  set T = rng f0;
  consider N be increasing sequence of NAT such that
AY: f0 = f * N by AX,VALUED_0:def 17;
  for x be Point of X
    ex K be Real st
      0 <= K
    & for g be Point of DualSp X st g in T holds |. g.x .| <= K
  proof
    let x be Point of X;
    consider K be Real such that
A4:   0 <= K & for n be Nat holds |. (f#x).n .| <= K by A21;
A5: for n be Nat holds |. (f0#x).n .| <= K
    proof
      let n be Nat;
B3:   f0.n = f.(N.n) by AY,ORDINAL1:def 12,FUNCT_2:15;
      reconsider n0=N.n as Nat;
      (f0#x).n = (f0.n).x by Def1
              .= (f#x).n0 by B3,Def1;
      hence thesis by A4;
    end;
    for g be Point of DualSp X st g in T holds |. g.x .| <= K
    proof
      let g be Point of DualSp X;
      assume g in T; then
      consider n be object such that
B1:     n in NAT & g = f0.n by FUNCT_2:11;
      reconsider n as Nat by B1;
      g.x = (f0#x).n by B1,Def1;
      hence thesis by A5;
    end;
    hence thesis by A4;
  end; then
  consider L be Real such that
A7: 0 <= L
  & for g be Point of DualSp X st g in T holds ||.g.|| <= L by Lm55;
  set M=L+1;
A8: L + 0 < M by XREAL_1:8;
A9: for g be Lipschitzian linear-Functional of X st g in T
      for x,y be Point of X holds |.g.x - g.y.| <= M * ||.x-y.||
  proof
    let g be Lipschitzian linear-Functional of X;
    reconsider g1=g as Point of DualSp X by DUALSP01:def 10;
    assume g in T;
    then ||.g1.|| <= L by A7; then
B1: ||.g1.|| < M by A8,XXREAL_0:2;
    let x,y be Point of X;
    |. g.x - g.y .| = |. g.x + (-1)*(g.y) .|; then
    |. g.x - g.y .| = |. g.x + (g.((-1)*y)) .| by HAHNBAN:def 3; then
    |. g.x - g.y .| = |. g.(x + (-1)*y) .| by HAHNBAN:def 2; then
B2: |. g.x - g.y .| = |. g.(x-y) .| by RLVECT_1:16;
B3: |. g.(x-y) .| <= ||.g1.|| * ||.x-y.|| by DUALSP01:26;
    ||.g1.|| * ||.x-y.|| <= M * ||.x-y.|| by B1,XREAL_1:64;
    hence thesis by B2,B3,XXREAL_0:2;
  end;
BX: for x be Point of X holds f0#x is convergent
  proof
    let x be Point of X;
    for TK1 be Real st TK1 > 0
      ex m be Nat st
        for n be Nat st n >= m holds |.(f0#x).n - (f0#x).m.| < TK1
    proof
      let TK1 be Real;
      assume B1: TK1 > 0; then
C2:   0 < TK1/(3*M) by A7;
      set V={z where z is Point of X :||.x-z.|| < TK1/(3*M)};
      V c= the carrier of X
      proof
        let s be object;
        assume s in V; then
        ex z be Point of X st s = z & ||.x-z.|| < TK1/(3*M);
        hence thesis;
      end; then
      reconsider V as Subset of X;
      reconsider TKK=TK1 as Real;
      V is open Subset of TopSpaceNorm X by NORMSP_2:8; then
B31:  V is open by NORMSP_2:16;
      ||.x-x.|| < TKK/(3*M) by C2,NORMSP_1:6; then
      x in V; then
      consider s be object such that
B3:     s in X0 & s in V by XBOOLE_0:3,B31,A3,NORMSP_3:17;
      consider y be Point of X such that
B4:     s=y & ||.x-y.|| < TK1/(3*M) by B3;
      consider m be Element of NAT such that
B40:    s=x0.m by B3,FUNCT_2:113;
      consider m be Nat such that
B5:     for n be Nat st m <= n holds |.(f0#y).n -(f0#y).m.| < TK1/3
          by B1,A31,SEQ_4:41,B4,B40;
      take m;
      for n be Nat st n >= m holds |.(f0#x).n - (f0#x).m.| < TK1
      proof
        let n be Nat;
B6:     m in NAT by ORDINAL1:def 12;
B7:     n in NAT by ORDINAL1:def 12;
        reconsider g=f0.n as Lipschitzian linear-Functional of X
          by DUALSP01:def 10;
        reconsider h=f0.m as Lipschitzian linear-Functional of X
          by DUALSP01:def 10;
B8:     |. (f0#x).n - (f0#y).m .| + |. (f0#y).m - (f0#x).m.|
          <= |. (f0#x).n - (f0#y).n .| + |. (f0#y).n - (f0#y).m .|
               + |. (f0#y).m - (f0#x).m .| by XREAL_1:6,COMPLEX1:63;
        assume n >= m; then
        |. (f0#y).n - (f0#y).m .| < TK1/3 by B5; then
        |. (f0#x).n - (f0#y).n .| + |. (f0#y).n - (f0#y).m .|
          < |. (f0#x).n - (f0#y).n .| + TK1/3 by XREAL_1:8; then
B9:     |. (f0#x).n - (f0#y).n .| + |. (f0#y).n - (f0#y).m .|
          + |. (f0#y).m - (f0#x).m .| < |. (f0#x).n - (f0#y).n .| + TK1/3
          + |. (f0#y).m - (f0#x).m .| by XREAL_1:8;
        |. (f0#x).m - (f0#y).m .| = |. (f0.m).x-(f0#y).m .| by Def1; then
        |. (f0#x).m - (f0#y).m .| = |. h.x - h.y .| by Def1; then
B10:    |. (f0#x).m - (f0#y).m .| <= M * ||.x-y.|| by A9,FUNCT_2:4,B6;
        M * ||.x-y.|| < M * (TK1 / (3*M)) by A7,B4,XREAL_1:68; then
        M * ||.x-y.|| < TK1/3 by A7,XCMPLX_1:92; then
        |. (f0#x).m - (f0#y).m .| < TK1/3 by B10,XXREAL_0:2; then
        |. (f0#y).m - (f0#x).m .| < TK1/3 by COMPLEX1:60; then
B11:    TK1/3 + TK1/3 + |. (f0#y).m - (f0#x).m .| < TK1/3 + TK1/3 + TK1/3
          by XREAL_1:8;
        |. (f0#x).n - (f0#y).n .| = |. (f0.n).x - (f0#y).n .| by Def1; then
        |. (f0#x).n - (f0#y).n .| = |. g.x - g.y .| by Def1; then
B12:    |. (f0#x).n - (f0#y).n .| <= M * ||.x-y.|| by A9,FUNCT_2:4,B7;
        |. (f0#x).n - (f0#x).m .| <= |. (f0#x).n - (f0#y).m .|
          + |. (f0#y).m - (f0#x).m .| by COMPLEX1:63; then
        |. (f0#x).n - (f0#x).m .| <= |. (f0#x).n - (f0#y).n .|
          + |. (f0#y).n - (f0#y).m .| + |. (f0#y).m - (f0#x).m .|
          by B8,XXREAL_0:2; then
B13:    |. (f0#x).n - (f0#x).m .| < |. (f0#x).n - (f0#y).n .|
          + TK1/3 + |. (f0#y).m - (f0#x).m .| by B9,XXREAL_0:2;
        M * ||.x-y.|| < M * (TK1 / (3*M)) by A7,B4,XREAL_1:68; then
        M * ||.x-y.||< TK1/3 by A7,XCMPLX_1:92; then
        |. (f0#x).n - (f0#y).n.| < TK1/3 by B12,XXREAL_0:2; then
        |. (f0#x).n-(f0#y).n .| + TK1/3 < TK1/3 + TK1/3 by XREAL_1:8; then
        |. (f0#x).n - (f0#y).n .| + TK1/3 + |. (f0#y).m - (f0#x).m .|
          < TK1/3 + TK1/3 + |. (f0#y).m - (f0#x).m .| by XREAL_1:8; then
        |. (f0#x).n - (f0#y).n .| + TK1/3 + |. (f0#y).m - (f0#x).m .|
          < TK1/3 + TK1/3 + TK1/3 by B11,XXREAL_0:2;
        hence thesis by B13,XXREAL_0:2;
      end;
      hence thesis;
    end;
    hence thesis by SEQ_4:41;
  end;
  defpred FP[Element of the carrier of X, object] means
    $2 = lim (f0#$1);
C01: for x being Element of the carrier of X
       ex y being Element of REAL st FP[x,y]
  proof
    let x be Element of the carrier of X;
    lim (f0#x) in REAL by XREAL_0:def 1;
    hence thesis;
  end;
  consider f1 be Function of the carrier of X,REAL such that
C0: for x be Element of the carrier of X holds FP[x,f1.x]
      from FUNCT_2:sch 3(C01);
C2: f1 is additive
  proof
    let x,y be Point of X;
D11: now let n be Nat;
      reconsider h=f0.n as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
      thus (f0#(x+y)).n = (f0.n).(x+y) by Def1
                       .= h.x + h.y by HAHNBAN:def 2
                       .= (f0#x).n + (f0.n).y by Def1
                       .= (f0#x).n + (f0#y).n by Def1;
    end;
D2: f0#x is convergent & f0#y is convergent by BX;
    thus f1.(x+y) = lim (f0#(x+y)) by C0
                 .= lim (f0#x + f0#y) by D11,SEQ_1:7
                 .= lim (f0#x) + lim (f0#y) by D2,SEQ_2:6
                 .= f1.x + lim (f0#y) by C0
                 .= f1.x + f1.y by C0;
  end;
C31: f1 is homogeneous
  proof
    let x be Point of X, r be Real;
D31: now let n be Nat;
      reconsider h=f0.n as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
      thus (f0#(r*x)).n = (f0.n).(r*x) by Def1
                       .= r*h.x by HAHNBAN:def 3
                       .= r*(f0#x).n by Def1;
    end;
    thus f1.(r*x) = lim (f0#(r*x)) by C0
                 .= lim (r(#)(f0#x)) by D31,SEQ_1:9
                 .= r*(lim (f0#x)) by BX,SEQ_2:8
                 .= r*f1.x by C0;
  end;

  consider M be Real such that
C4: 0 < M
  & for n be Nat holds |. ||.f.||.n .| < M by A2,SEQ_2:3;
  now let x be Point of X;
D5: f0#x is convergent by BX;
D7: |.f1.x.| = |.lim (f0#x).| by C0
            .= lim |.(f0#x).| by BX,SEQ_4:14;
D8: for n be Nat holds ||.f0.n.|| <= M
    proof
      let n be Nat;
      f0.n = f.(N.n) by AY,ORDINAL1:def 12,FUNCT_2:15; then
E3:   ||.f0.n.|| = ||.f.||.(N.n) by NORMSP_0:def 4;
      |. ||.f.||.(N.n) .| < M by C4;
      hence thesis by E3,ABSVALUE:def 1;
    end;
    reconsider s=M*||.x.|| as Real;
D91: now let n be Nat;
      reconsider h=f0.n as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
E3:   |.h.x.| <= ||.f0.n.||*||.x.|| by DUALSP01:26;
      ||.f0.n.||*||.x.|| <= M*||.x.|| by XREAL_1:64,D8; then
      |.h.x.| <= M * ||.x.|| by E3,XXREAL_0:2; then
E51:  |.(f0#x).n.| <= M*||.x.|| by Def1;
      (seq_const s).n = s by SEQ_1:57;
      hence |.(f0#x).|.n <= (seq_const s).n by E51,SEQ_1:12;
    end;
    lim (seq_const s) = (seq_const s).0 by SEQ_4:26
                     .= s;
    hence |.f1.x.| <= M*||.x.|| by D7,D91,D5,SEQ_2:18;
  end; then
  f1 is Lipschitzian by C4; then
  f1 is Point of DualSp X by DUALSP01:def 10,C31,C2; then
  f0 is weakly*-convergent by BX,C0;
  hence thesis by AX;
end;
