Consider the following operator: $L[f(x)]=\int_0^x f(s) ds$.
Is this operator linear?
I think in order to answer this question, I need to consider all possible functions, $f(x)$. The book says an operator is linear if you get a constant * the operator back: $L[f(x)]=cf(x)$.
I cannot integrate $f(s)ds$ without knowing what $f(s)$ is. So, suppose it is just $s$. Then the integral is $0.5s^2$ (evaluated from $0$ to $x$), or $0.5x^2$. This is not equal to $x$, so therefore the integral operator is not linear?