Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space ${\displaystyle (V,\leq )}$ is a linear functional ${\displaystyle f}$ on ${\displaystyle V}$ so that for all positive elements ${\displaystyle v\in V,}$ that is ${\displaystyle v\geq 0,}$ 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 ${\displaystyle V}$ is a complex vector space, it is assumed that for all ${\displaystyle v\geq 0,}$ ${\displaystyle f(v)}$ is real. As in the case when ${\displaystyle V}$ is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace ${\displaystyle W\subseteq V,}$ and the partial order does not extend to all of ${\displaystyle V,}$ in which case the positive elements of ${\displaystyle V}$ are the positive elements of ${\displaystyle W,}$ by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any ${\displaystyle x\in V}$ equal to ${\displaystyle s^{\ast }s}$ for some ${\displaystyle s\in V}$ to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such ${\displaystyle x.}$ This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.