Let K be a nonempty convex compact subset of a separable real Hilbert space \(({{\mathcal {H}}},\langle \cdot ,\cdot \rangle )\) with an orthonormal basis \((e_n)_{n\ge 1}\) and let \(\pi _{n}:K\longrightarrow \mathbb {R}\) be the canonical projection defined by \(\pi _{n}(x)=\langle x,e_n\rangle \) for all \(x\in K\) . We prove that any positive linear functional \(\phi :{{\mathcal {C}}}(K)\longrightarrow \mathbb {R}\) satisfying \(\phi (\mathbf{{1}})=1\) and \(\begin{aligned} \phi ({\pi _n}^2)=\phi (\pi _n)^2 {\text { for all integer }} n\ge 1 \end{aligned}\) is an evaluation functional. As a consequence, we show that every positive linear operator \(T:{{\mathcal {C}}}(K)\longrightarrow {{\mathcal {F}}}(X)\) satisfying \(\begin{aligned} T(\mathbf{{1}})(x){>}0 {\text { for all }} x{\in } X {\text { and }} T(\mathbf{{1}})T({\pi _n}^2){=}T(\pi _{n})^2 {\text { for every integer }} n{\ge }1 \end{aligned}\) is a weighted composition operator.
