# What is an evaluation functional?

What is an evaluation functional for another function $$f$$? The evaluation functional $$ev_{x}$$ satisfies $$ev_{x}(f) = f(x)$$.

Can you give me a concrete example?

A concrete example is the evaluation function on the ring of polynomials over a field, for example $$\Bbb{R}[X]$$. For every $$c\in\Bbb{R}$$ the evaluation functional $$\operatorname{ev}_c$$ is defined as $$\operatorname{ev}_c:\ \Bbb{R}[X]\ \longrightarrow\ \Bbb{R}:\ P\ \longmapsto\ P(c).$$ So for example, for $$c=1$$ we get $$\operatorname{ev}_1(X^2+3X+1)=5 \qquad\text{ and }\qquad \operatorname{ev}_1(X-6)=-5.$$
You can do the same with the ring of continuous functions $$C[0,1]$$ on the unit interval, defining for every $$c\in\Bbb{R}$$ the evaluation functional $$\operatorname{ev}_c$$ as $$\operatorname{ev}_c:\ C[0,1]\ \longrightarrow\ \Bbb{R}:\ f\ \longmapsto\ f(c).$$ A more abstract example is given by the vector space $$V^{\ast}$$ of linear functionals on a real vector space $$V$$. That is to say $$V^{\ast}$$ is the set of $$\Bbb{R}$$-linear functions $$V\ \longrightarrow\ \Bbb{R}$$. Here for every $$c\in\Bbb{R}$$ the evaluation functional $$\operatorname{ev}_c$$ is defined as $$\operatorname{ev}_c:\ V^{\ast}\ \longrightarrow\ \Bbb{R}:\ f\ \longmapsto\ f(c).$$ Even more abstractly, for a commutative unital ring $$R$$ and an $$R$$-module $$M$$, the evaluation functional $$\operatorname{ev}_m$$ for $$m\in M$$ on $$\operatorname{Hom}_R(M,R)$$ is given by $$\operatorname{ev}_m:\ \operatorname{Hom}_R(M,R)\ \longrightarrow\ R:\ f\ \longmapsto\ f(m).$$