How can I show that the following functional is non-multiplicative:
$\displaystyle f(x)=\int_0^1{x(t)dt}$
(We are working in the space $\displaystyle X=C(0,1)$)
Let $\displaystyle Id$ be the identity map on $\displaystyle (0,1)$. Then
$\displaystyle f(Id\cdot Id)=\int_0^1t^2dt=\frac{1}{3}$.
However
$\displaystyle f(Id)\cdot f(Id)=\left ( \int_0^1 tdt\right )^2=\left( \frac{1}{2}\right)^2=\frac{1}{4}$,
so $\displaystyle f$ is not a multiplicative functional.