reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem Th22:
  for X be RealNormSpace holds BoundedLinearFunctionals X is linearly-closed
proof
  let X be RealNormSpace;
  set W = BoundedLinearFunctionals X;
A1: for v,u be VECTOR of X*' st v in W & u in W holds v + u in W
  proof
    let v,u be VECTOR of X*' such that
A2: v in W & u in W;
    reconsider f=v+u as linear-Functional of X by Def7;
    f is Lipschitzian
    proof
      reconsider v1=v, u1=u as Lipschitzian Functional of X by A2,Def9;
      consider K2 be Real such that
A4:   0 <= K2 and
A5:   for x be VECTOR of X holds |. v1.x .| <= K2*||. x .|| by Def8;
      consider K1 be Real such that
A6:   0 <= K1 and
A7:   for x be VECTOR of X holds |. u1.x .| <= K1*||. x .|| by Def8;
      take K3=K1+K2;
      now
        let x be VECTOR of X;
A8:     |. u1.x+v1.x .| <= |. u1.x .|+ |. v1.x .| by COMPLEX1:56;
A9:     |. v1.x .| <= K2*||. x .|| by A5;
        |. u1.x .| <= K1*||. x .|| by A7;
        then
A10:    |. u1.x .| + |. v1.x .| <= K1*||. x .|| +K2*||. x .||
          by A9,XREAL_1:7;
        |. f.x .| =|. u1.x+v1.x .| by Th20b;
        hence |. f.x .| <= K3*||. x .|| by A8,A10,XXREAL_0:2;
      end;
      hence thesis by A6,A4;
    end;
    hence thesis by Def9;
  end;
  for a be Real, v be VECTOR of X*' st v in W holds a * v in W
  proof
    let a be Real;
    let v be VECTOR of X*' such that
A11: v in W;
    reconsider f=a*v as linear-Functional of X by Def7;
    f is Lipschitzian
    proof
      reconsider v1=v as Lipschitzian Functional of X by A11,Def9;
      consider K be Real such that
A12:  0 <= K and
A13:  for x be VECTOR of X holds |. v1.x .| <= K*||. x .|| by Def8;
      take |.a.|*K;
A14:  now
        let x be VECTOR of X;
        0 <=|.a.| by COMPLEX1:46; then
A15:    |.a.|* |. v1.x .| <= |.a.|* (K*||. x .||) by A13,XREAL_1:64;
        |. a*v1.x .| = |.a.|* |. v1.x .| by COMPLEX1:65;
        hence |. f.x .| <= |.a.|* K*||. x .|| by A15,Th21b;
      end;
      0 <=|.a.| by COMPLEX1:46;
      hence thesis by A12,A14;
    end;
    hence thesis by Def9;
  end;
  hence thesis by A1;
end;
