If $\left|\psi\right> \in L_2$, is $\left|\psi\right> ^2 \in L_2$, where $\left|\cdot\right>$ is Dirac notation.
Or for everyone to get what I am asking: $f(x) \in L_2$ if the following condition is met:
$$\int_{-\infty}^{\infty}|f|^2(x)dx<\infty$$
So my question consists in following: if $\psi(x) \in L_2$ will $\psi^2(x) \in L_2$ or in other words will the $\int_{-\infty}^{\infty}|\psi|^4(x)dx$ be less than $\infty$
The answer is not obvious to me, because if an integral converges it doesn't mean that it will do so with the squared integrand.