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?
$\endgroup$1 Answer
$\begingroup$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$$\operatorname{ev}_m:\ \operatorname{Hom}_R(M,R)\ \longrightarrow\ R:\ f\ \longmapsto\ f(m).$$
$\endgroup$
