
theorem Lm55:
  for X be RealBanachSpace, T be Subset of DualSp X
    st ( for x be Point of X
         ex K be Real st
           0 <= K
         & for f be Point of DualSp X st f in T holds |. f.x .| <= K ) holds
  ex L be Real st
    0 <= L
  & for f be Point of DualSp X st f in T holds ||.f.|| <= L
proof
  let X be RealBanachSpace, T be Subset of DualSp X;
  assume
AS: for x be Point of X
      ex K be Real st 0 <= K
      & for f be Point of DualSp X st f in T holds |. f.x .| <= K;
  reconsider T1=T as Subset of
    R_NormSpace_of_BoundedLinearOperators(X,RNS_Real) by DUALSP02:14;
  for x be Point of X
    ex K be Real st
      0 <= K
    & for f be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
        st f in T1 holds ||. f.x .|| <= K
  proof
    let x be Point of X;
    consider K be Real such that
B1:   0 <= K
    & for f be Point of DualSp X st f in T holds |. f.x .| <= K by AS;
    take K;
    for f be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
          st f in T1 holds ||. f.x .|| <= K
    proof
      let f be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
      assume C2: f in T1; then
      reconsider f1=f as Point of DualSp X;
      |. f1.x .| = ||. f.x .|| by EUCLID:def 2;
      hence thesis by C2,B1;
    end;
    hence thesis by B1;
  end; then
  consider L be Real such that
A1: 0 <= L
  & for f be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
     st f in T1 holds ||.f.|| <= L by LOPBAN_5:5;
  take L;
  for f be Point of DualSp X st f in T holds ||.f.|| <= L
  proof
    let f be Point of DualSp X;
    assume C5: f in T; then
    f in T1; then
    reconsider f1=f as Point of
      R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
    ||.f1.|| = ||.f.|| by DUALSP02:18;
    hence thesis by C5,A1;
  end;
  hence thesis by A1;
end;
