
theorem Th76:
  for X be RealBanachSpace, M be non empty Subset of X st
    X is Reflexive & M is linearly-closed closed holds
    NLin(M) is Reflexive
proof
   let X be RealBanachSpace, M be non empty Subset of X;
   assume that
A2: X is Reflexive and
A3: M is linearly-closed and
A4: M is closed;
   set M0 = NLin(M);
X1:the carrier of M0 = M by NORMSP_3:31,A3;
X2:the carrier of M0 c= the carrier of X by DUALSP01:def 16;
   for y be Point of DualSp DualSp M0
    ex x be Point of M0 st
      for g be Point of DualSp M0 holds y.g = g.x
   proof
    let y be Point of DualSp DualSp M0;
    reconsider y1=y as Lipschitzian linear-Functional of DualSp M0
      by DUALSP01:def 10;
    defpred P[Function,Function] means $2=$1|M;
P0: for x be Element of DualSp X ex y be Element of DualSp M0
      st P[x,y]
    proof
     let x be Element of DualSp X;
     reconsider x0=x as Lipschitzian linear-Functional of X
       by DUALSP01:def 10;
     reconsider y0=x0|M as Function of M0,REAL by X1,FUNCT_2:32;
B1:  y0 is additive
     proof
      let s,t be Point of M0;
      reconsider s1=s, t1=t as Point of X by X2;
C1:   s+t = s1+t1 by NORMSP_3:28;
      thus y0.(s+t) = x0.(s+t) by X1,FUNCT_1:49
                   .= x0.s1 + x0.t1 by C1,HAHNBAN:def 2
                   .= (x0|M).s + x0.t by X1,FUNCT_1:49
                   .= y0.s + y0.t by X1,FUNCT_1:49;
     end;
B2:  y0 is homogeneous
     proof
      let s be Point of M0, r be Real;
      reconsider s1=s as Point of X by X2;
C2:   r*s = r*s1 by NORMSP_3:28;
      thus y0.(r*s) = x0.(r*s) by X1,FUNCT_1:49
                   .= r*x0.s by C2,HAHNBAN:def 3
                   .= r*y0.s by X1,FUNCT_1:49;
     end;
     for s be Point of M0 holds |. y0.s .| <= ||. x .|| * ||. s .||
     proof
      let s be Point of M0;
      reconsider s1=s as Point of X by X2;
C3:   ||. s .|| = ||. s1 .|| by NORMSP_3:28;
      |. y0.s .| = |. x0.s .| by X1,FUNCT_1:49;
      hence thesis by C3,DUALSP01:26;
     end; then
     y0 is Lipschitzian; then
     reconsider y=y0 as Element of DualSp M0 by B1,B2,DUALSP01:def 10;
     take y;
     thus y = x|M;
    end;
    consider T be Function of DualSp X, DualSp M0 such that
P11:  for x being Element of DualSp X holds P[x,T.x] from FUNCT_2:sch 3(P0);
D1: T is additive
    proof
     let f,g be Point of DualSp X;
E1:  T.(f+g) is Function of M0,REAL
   & T.f + T.g is Function of M0,REAL by DUALSP01:def 10;
     for x be Point of M0 holds (T.(f+g)).x = (T.f + T.g).x
     proof
      let x be Point of M0;
      reconsider x1=x as Point of X by X2;
      T.f = f|M & T.g = g|M by P11; then
      reconsider fm=f|M, gm=g|M as Point of DualSp M0;
F2:   fm.x = f.x & gm.x = g.x by X1,FUNCT_1:49;
      thus (T.(f+g)).x = ((f+g)|M).x by P11
                      .= (f+g).x by X1,FUNCT_1:49
                      .= f.x1 + g.x1 by DUALSP01:29
                      .= (T.f).x + gm.x by P11,F2
                      .= (T.f).x + (T.g).x by P11
                      .= (T.f + T.g).x by DUALSP01:29;
     end;
     hence T.(f+g) = T.f + T.g by E1,FUNCT_2:def 8;
    end;
    T is homogeneous
    proof
     let f be Point of DualSp X, r be Real;
E3:  T.(r*f) is Function of M0,REAL
   & r*(T.f) is Function of M0,REAL by DUALSP01:def 10;
     for x be Point of M0 holds (T.(r*f)).x = (r*(T.f)).x
     proof
      let x be Point of M0;
      reconsider x1=x as Point of X by X2;
      T.f = f|M by P11; then
      reconsider fm=f|M as Point of DualSp M0;
F4:   fm.x = f.x by X1,FUNCT_1:49;
      thus (T.(r*f)).x = ((r*f)|M).x by P11
                      .= (r*f).x by X1,FUNCT_1:49
                      .= r*(f.x1) by DUALSP01:30
                      .= r*((T.f).x) by P11,F4
                      .= (r*(T.f)).x by DUALSP01:30;
     end;
     hence T.(r*f) = r*(T.f) by E3,FUNCT_2:def 8;
    end; then
    reconsider T as LinearOperator of DualSp X, DualSp M0 by D1;
    for v be Point of DualSp X  holds ||. T.v .|| <= 1*||.v.||
    proof
     let v be Point of DualSp X;
     reconsider v1=v as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
B0:  T.v = v|M by P11; then
     reconsider vm=v|M as Point of DualSp M0;
     reconsider vm1=vm as Lipschitzian linear-Functional of M0
        by DUALSP01:def 10;
     now let r be Real;
      assume r in PreNorms(vm1); then
      consider t be VECTOR of M0 such that
B1:    r = |. vm1.t .| & ||. t .|| <= 1;
      reconsider t1=t as Point of X by X2;
B2:   |. vm.t .| = |. v.t1 .| by X1,FUNCT_1:49;
      ||. t1 .|| = ||. t .|| by NORMSP_3:28;
      hence r in PreNorms(v1) by B1,B2;
     end; then
     PreNorms(vm1) c= PreNorms(v1); then
     upper_bound PreNorms(vm1)
             <= upper_bound PreNorms(v1) by SEQ_4:48; then
     ||. vm .|| <= upper_bound PreNorms(v1) by DUALSP01:24;
     hence ||. T.v .|| <= 1*||. v .|| by B0,DUALSP01:24;
    end; then
    reconsider T as Lipschitzian LinearOperator of DualSp X, DualSp M0
       by LOPBAN_1:def 8;
P2: for f be Point of DualSp X, x be Point of X
      st x in M holds (T.f).x = f.x
    proof
     let f be Point of DualSp X, x be Point of X;
     assume x in M; then
     (f|M).x = f.x by FUNCT_1:49;
     hence thesis by P11;
    end;
    deffunc F(Element of DualSp X) = y.(T.$1);
    consider z be Function of the carrier of DualSp X, REAL such that
Q10:  for f be Element of the carrier of DualSp X holds z.f = F(f)
        from FUNCT_2:sch 4;
Q11:z is additive
    proof
     let s,t be Point of DualSp X;
     thus z.(s+t) = y.(T.(s+t)) by Q10
             .= y.(T.s + T.t) by VECTSP_1:def 20
             .= y1.(T.s) + y1.(T.t) by HAHNBAN:def 2
             .= z.s + y.(T.t) by Q10
             .= z.s + z.t by Q10;
    end;
Q12:z is homogeneous
    proof
     let s be Point of DualSp X, r be Real;
     thus z.(r*s) = y.(T.(r*s)) by Q10
             .= y.(r*(T.s)) by LOPBAN_1:def 5
             .= r*y1.(T.s) by HAHNBAN:def 3
             .= r*(z.s) by Q10;
    end;
    R_NormSpace_of_BoundedLinearOperators(DualSp X,DualSp M0)
      = NORMSTR (# BoundedLinearOperators(DualSp X,DualSp M0),
         Zero_(BoundedLinearOperators(DualSp X,DualSp M0),
             R_VectorSpace_of_LinearOperators(DualSp X,DualSp M0)),
         Add_(BoundedLinearOperators(DualSp X,DualSp M0),
             R_VectorSpace_of_LinearOperators(DualSp X,DualSp M0)),
         Mult_(BoundedLinearOperators(DualSp X,DualSp M0),
             R_VectorSpace_of_LinearOperators(DualSp X,DualSp M0)),
         BoundedLinearOperatorsNorm(DualSp X,DualSp M0) #)
             by LOPBAN_1:def 14; then
    reconsider T1=T as Point of
      R_NormSpace_of_BoundedLinearOperators(DualSp X,DualSp M0)
        by LOPBAN_1:def 9;
    for s be Point of DualSp X holds
      |. z.s .| <= ||. y .|| * ||. T1 .|| * ||. s .||
    proof
     let s be Point of DualSp X;
B1:  |. z.s .| = |. y.(T.s) .| by Q10;
B2:  |. y1.(T.s) .| <= ||. y .||*||. T.s .|| by DUALSP01:26;
     ||. y .|| * ||. T.s .|| <= ||. y .|| * (||. T1 .|| * ||. s .||)
          by XREAL_1:64,LOPBAN_1:32;
     hence thesis by B1,B2,XXREAL_0:2;
    end; then
    z is Lipschitzian; then
    reconsider z as Point of DualSp DualSp X by Q11,Q12,DUALSP01:def 10;
    consider x be Point of X such that
Q2:   for f be Point of DualSp X holds z.f = f.x by A2,REFF1;
Q3: for f be Point of DualSp X holds y.(T.f) = f.x
    proof
     let f be Point of DualSp X;
     thus y.(T.f) = z.f by Q10
             .= f.x by Q2;
    end;
AX: x in the carrier of M0
    proof
     assume not x in the carrier of M0; then
     consider f be Point of DualSp X such that
B1:   (for x be Point of X st x in M holds (Bound2Lipschitz(f,X)).x = 0) and
B2:   (Bound2Lipschitz(f,X)).x = 1 by A3,A4,X1,Lm64;
     reconsider f1=f as Lipschitzian linear-Functional of X
        by DUALSP01:def 10;
B3:  f1 = (Bound2Lipschitz(f,X)) by DUALSP01:23;
B4:  for x be Point of X st x in M holds (T.f).x = 0
     proof
      let x be Point of X;
      assume C1: x in M; then
      f.x = 0 by B1,B3;
      hence thesis by C1,P2;
     end;
B8:  T.f is Function of M0,REAL by DUALSP01:def 10;
     for x be Point of M0 holds (T.f).x = (M --> 0).x
     proof
      let x be Point of M0;
      x in M by X1; then
      reconsider x1=x as Point of X;
      (T.f).x1 = 0 by X1,B4;
      hence thesis by X1,FUNCOP_1:7;
     end; then
B9:  T.f = M --> 0 by X1,B8,FUNCT_2:def 8
        .= 0.(DualSp M0) by X1,DUALSP01:25;
     f.x = y1.(0.(DualSp M0)) by B9,Q3
        .= 0 by HAHNBAN:20;
     hence contradiction by B2,B3;
    end;
Q4: for f be Point of DualSp X holds y.(T.f) = (T.f).x
    proof
     let f be Point of DualSp X;
     y.(T.f) = f.x by Q3;
     hence thesis by P2,X1,AX;
    end;
Q5: for f be Point of DualSp X holds y.(f|M) = (f|M).x
    proof
     let f be Point of DualSp X;
     T.f = f|M by P11;
     hence thesis by Q4;
    end;
    for g be Point of DualSp M0 holds y.g = g.x
    proof
     let g be Point of DualSp M0;
     reconsider g1=g as Lipschitzian linear-Functional of M0
        by DUALSP01:def 10;
     ex f1 be Lipschitzian linear-Functional of X,
        f be Point of DualSp X st
       f1 = f & f1|(the carrier of M0) = g1 & ||.f.||=||.g.|| by DUALSP01:36;
     hence thesis by X1,Q5;
    end;
    hence thesis by AX;
   end;
   hence thesis by REFF1;
end;
