
theorem Th20:
  for X be ComplexNormSpace holds
  C_Normed_Algebra_of_BoundedLinearOperators(X) is
  reflexive discerning ComplexNormSpace-like Abelian
  add-associative right_zeroed right_complementable associative right_unital
  right-distributive vector-distributive scalar-distributive
  scalar-associative vector-associative vector-distributive
  scalar-distributive scalar-associative scalar-unital
proof
  let X be ComplexNormSpace;
  set C = C_Normed_Algebra_of_BoundedLinearOperators X;
  set BS=C_NormSpace_of_BoundedLinearOperators(X,X);
  thus C is reflexive
  proof
    thus ||.0.C.|| = ||.0.BS.|| .= 0;
  end;
  thus C is discerning
  proof
    let x be Point of C;
     reconsider y =x as Point of BS;
    assume ||.x.|| = 0;
     then ||.y.|| = 0;
     then y= 0.BS by NORMSP_0:def 5;
    hence x= 0.C;
  end;
  thus C is ComplexNormSpace-like
  proof
    let x,y be Point of C;
    let a be Complex;
    reconsider x1 =x, y1 =y as Point of BS;
A1: ||.x1.|| = ||.x.||;
    thus ||.a*x.|| = ||.a*x1.||
      .=|.a.|* ||.x.|| by A1,CLVECT_1:def 13;
    ||.x + y.|| = ||.x1 + y1.|| & ||.x1.|| + ||. y1.|| = ||.x.|| + ||.y.||;
    hence thesis by CLVECT_1:def 13;
  end;
  set RBLOP=C;
  (for x,y,z being Element of C for a,b be Complex holds
  x+y = y+x & (x+y)+z = x +(y+z) & x+(0.C) = x &
  x is right_complementable & (x*y)*z = x*(y*z) &
  x*(1.C) = x & (1.C)*x = x & x*(y+z) = x*y + x*z &
  (y+z)*x = y*x + z*x & a*(x*y) = (a*x) *y & (a*b)*(x*y)=(a*x)*(b*y) &
  a*(x+y) = a*x + a*y & (a+b)*x = a*x + b*x &
  (a*b)*x = a*(b*x) & 1r*x=x) &
  for a be Complex,
     v,w being VECTOR of RBLOP holds a * (v + w) = a * v + a * w
  by Th19;
  hence thesis;
end;
