If $\int f^*(x)g(x) dx$ is the scalar product in the vector space of $L^2$, what is $f(x)g(x)$? For two functions $v=f(x)$, $w=g(x)$ from the vector space $L^2$, the scalar product $v^Tw$ is defined to be 

$\int f^*(x)g(x) dx$.

However, I can also just multiply one function with another, $f(x)\cdot g(x)$. What is the corresponding algebraic operation?
It's not a vector space operation, because in a vector space $V$ the only $V \times V \to V$ operation is the addition.
It's not a vector space operation, because in a vector space $V$ the only $V \times V \to V$ operation is the addition.