When formula_3 is a complex vector space, it is assumed that for all formula_5, formula_9 is real. As in the case when formula_3 is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace formula_11, and the partial order does not extend to all of formula_3, in which case the positive elements of formula_3 are the positive elements of formula_14, by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any formula_15 equal to formula_16 for some formula_17 to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such formula_18. This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.