reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th38:
  for X,Y be RealNormSpace holds R_NormSpace_of_BoundedLinearOperators(X,Y) is
   reflexive discerning RealNormSpace-like
proof
  let X,Y be RealNormSpace;
  thus ||.0.R_NormSpace_of_BoundedLinearOperators(X,Y).|| = 0 by Th37;
  for x, y being Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  for a be Real
    holds ( ||.x.|| = 0 iff x = 0.R_NormSpace_of_BoundedLinearOperators(X,Y
  ) ) & ||.a*x.|| = |.a.| * ||.x.|| & ||.x+y.|| <= ||.x.|| + ||.y.|| by Th37;
  hence thesis by NORMSP_1:def 1;
end;
