= Point evaluation functional
= Point evaluation
{synonym}
For a <function space> on a set $X$ and $x\in X$, the <point evaluation functional> is $\delta_x(f)=f(x)$. On $C(K)$ for a nonempty <compact space> $K$, it is a <bounded linear functional> for the <supremum norm>:
$$
|\delta_x(f)|\le\|f\|_\infty,\qquad\|\delta_x\|=1.
$$
The constant function one attains the bound. The same argument works for bounded <continuous functions> on any $X$.
Boundedness depends on the chosen <norm>. On $C([0,1])$ with the <L2 norm>, fix $x_0\in(0,1)$ and take $f_n(t)=\max(1-n|t-x_0|,0)$ for sufficiently large $n$. Then $f_n(x_0)=1$ but $\|f_n\|_2^2=2/(3n)$, so <point evaluation> is unbounded. On <L2 space> itself, it is not even well-defined on equivalence classes: changing a representative at the single point $x_0$ does not change its <Lebesgue almost everywhere> class.
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