1. ## Functional non-multiplicative

How can I show that the following functional is non-multiplicative:
$f(x)=\int_0^1{x(t)dt}$
(We are working in the space $X=C(0,1)$)

2. Let $Id$ be the identity map on $(0,1)$. Then

$f(Id\cdot Id)=\int_0^1t^2dt=\frac{1}{3}$.

However

$f(Id)\cdot f(Id)=\left ( \int_0^1 tdt\right )^2=\left( \frac{1}{2}\right)^2=\frac{1}{4}$,

so $f$ is not a multiplicative functional.