
theorem Th75:
for X be RealNormSpace, L be Subset of X
  st X is non trivial
   & (for f be Point of DualSp X holds
        ex Kf be Real st
          0 <= Kf
        & for x be Point of X st x in L holds |. f.x .| <= Kf)
holds
  ex M be Real st 0 <= M
  & for x be Point of X st x in L holds ||.x.|| <= M
proof
   let X be RealNormSpace, L be Subset of X;
   assume
AS: X is non trivial
  & for f be Point of DualSp X holds
     ex Kf be Real st 0 <= Kf &
       for x be Point of X st x in L holds |. f.x .| <= Kf;
   set T = (BidualFunc X).:L;
XX:T is Subset of
    R_NormSpace_of_BoundedLinearOperators(DualSp X,RNS_Real) by Th75A;
   for f be Point of DualSp X
    ex Kf be Real st
      0 <= Kf &
      for g be Point of
        R_NormSpace_of_BoundedLinearOperators(DualSp X,RNS_Real)
        st g in T holds ||. g.f .|| <= Kf
   proof
    let f be Point of DualSp X;
    consider Kf be Real such that
A0:  0 <= Kf &
     for x be Point of X st x in L holds |. f.x .| <= Kf by AS;
    for g be Point of R_NormSpace_of_BoundedLinearOperators(DualSp X,RNS_Real)
      st g in T holds ||. g.f .|| <= Kf
    proof
     let g be Point of
       R_NormSpace_of_BoundedLinearOperators(DualSp X,RNS_Real);
     assume g in T; then
     consider x be object such that
A1:    x in dom (BidualFunc X) & x in L & g = (BidualFunc X).x
       by FUNCT_1:def 6;
     reconsider x as Point of X by A1;
A2:  |. f.x .| <= Kf by A1,A0;
     g = BiDual x by A1,Def2; then
     f.x = g.f by Def1;
     hence ||. g.f .|| <= Kf by A2,EUCLID:def 2;
    end;
    hence thesis by A0;
   end; then
   consider M be Real such that
B0: 0 <= M
  & for g be Point of R_NormSpace_of_BoundedLinearOperators(DualSp X,RNS_Real)
      st g in T holds ||.g.|| <= M by XX,LOPBAN_5:5;
B1:for x be Point of X st x in L holds ||.x.|| <= M
   proof
    let x be Point of X;
    assume B2: x in L;
    x in the carrier of X; then
B3: x in dom(BidualFunc X) by FUNCT_2:def 1;
    reconsider g=(BidualFunc X).x as Point of
        R_NormSpace_of_BoundedLinearOperators(DualSp X,RNS_Real) by Th75A;
B4: g in T by B2,B3,FUNCT_1:def 6;
    ||.x.|| = ||. (BidualFunc X).x .|| by AS,LMNORM
           .= ||. g .|| by LMN11;
    hence thesis by B0,B4;
   end;
   take M;
   thus thesis by B1,B0;
end;
