
theorem Th6:
  for X be non empty set for Y be RealNormSpace holds
  BoundedFunctions(X,Y) is linearly-closed
proof
  let X be non empty set;
  let Y be RealNormSpace;
  set W = BoundedFunctions(X,Y);
A1: RealVectSpace(X,Y) = RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,Y
    )),FuncAdd(X,Y),FuncExtMult(X,Y)#) by LOPBAN_1:def 4;
A2: for v,u be VECTOR of RealVectSpace(X,Y) st v in W & u in W holds v + u in W
  proof
    let v,u be VECTOR of RealVectSpace(X,Y) such that
A3: v in W and
A4: u in W;
    reconsider f=v+u as Function of X,the carrier of Y by A1,FUNCT_2:66;
    f is bounded
    proof
      reconsider v1=v as bounded Function of X, the carrier of Y by A3,Def5;
      consider K2 be Real such that
A5:   0 <= K2 and
A6:   for x be Element of X holds ||. v1.x .|| <= K2 by Def4;
      reconsider u1=u as bounded Function of X, the carrier of Y by A4,Def5;
      consider K1 be Real such that
A7:   0 <= K1 and
A8:   for x be Element of X holds ||. u1.x .|| <= K1 by Def4;
      take K3=K1+K2;
      now
        let x be Element of X;
        ||. u1.x .|| <= K1 & ||. v1.x .|| <= K2 by A8,A6;
        then
A9:     ||. u1.x .|| + ||. v1.x .|| <= K1 +K2 by XREAL_1:7;
        ||. f.x .|| =||. v1.x+u1.x .|| & ||. u1.x+v1.x .|| <= ||. u1.x
        .||+ ||. v1.x .|| by LOPBAN_1:11,NORMSP_1:def 1;
        hence ||. f.x .|| <= K3 by A9,XXREAL_0:2;
      end;
      hence thesis by A7,A5;
    end;
    hence thesis by Def5;
  end;
  for a be Real
   for v be VECTOR of RealVectSpace(X,Y) st v in W holds a *  v in W
  proof
    let a be Real;
    let v be VECTOR of RealVectSpace(X,Y) such that
A10: v in W;
    reconsider f=a*v as Function of X,the carrier of Y by A1,FUNCT_2:66;
    f is bounded
    proof
      reconsider v1=v as bounded Function of X, the carrier of Y by A10,Def5;
      reconsider a as Real;
      consider K be Real such that
A11:  0 <= K and
A12:  for x be Element of X holds ||. v1.x .|| <= K by Def4;
       reconsider aK =  |.a.|*K as Real;
      take aK;
A13:  now
        let x be Element of X;
A14:    0 <=|.a.| by COMPLEX1:46;
        ||. f.x .|| =||. a*v1.x .|| & ||. a*v1.x .|| = |.a.|* ||. v1.x
        .|| by LOPBAN_1:12,NORMSP_1:def 1;
        hence ||. f.x .|| <= |.a.|* K by A12,A14,XREAL_1:64;
      end;
      0 <=|.a.| by COMPLEX1:46;
      hence thesis by A11,A13;
    end;
    hence thesis by Def5;
  end;
  hence thesis by A2,RLSUB_1:def 1;
end;
