
theorem Th23:
  for X be non empty set, Y be ComplexNormSpace holds
  C_NormSpace_of_BoundedFunctions(X,Y) is
   reflexive discerning ComplexNormSpace-like
proof
  let X be non empty set;
  let Y be ComplexNormSpace;
  thus ||.0.C_NormSpace_of_BoundedFunctions(X,Y).|| = 0 by Th22;
  for x, y being Point of C_NormSpace_of_BoundedFunctions(X,Y) for c be
Complex holds ( ||.x.|| = 0 iff x = 0.C_NormSpace_of_BoundedFunctions(X,Y) ) &
  ||.c*x.|| = |.c.| * ||.x.|| & ||.x+y.|| <= ||.x.|| + ||.y.|| by Th22;
  hence thesis by CLVECT_1:def 13;
end;
