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In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space {\displaystyle } is a linear functional f {\displaystyle f} on V {\displaystyle V} so that for all positive elements v ∈ V , {\displaystyle v\in V,} that is v ≥ 0 , {\displaystyle v\geq 0,} it holds that
In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.
When V {\displaystyle V} is a complex vector space, it is assumed that for all v ≥ 0 , {\displaystyle v\geq 0,} f {\displaystyle f} is real. As in the case when V {\displaystyle V} is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace W ⊆ V , {\displaystyle W\subseteq V,} and the partial order does not extend to all of V , {\displaystyle V,} in which case the positive elements of V {\displaystyle V} are the positive elements of W , {\displaystyle W,} by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any x ∈ V {\displaystyle x\in V} equal to s ∗ s {\displaystyle s^{\ast }s} for some s ∈ V {\displaystyle s\in V} to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such x . {\displaystyle x.} This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.