hilbert space - Bra-ket notation and linear operators

Let $H$ be a hilbert space and let $\hat{A}$ be a linear operator on $H$.

My textbook states that $|\hat{A} \psi\rangle = \hat{A} |\psi\rangle$. My understanding of bra-kets is that $|\psi\rangle$ is a member of $H$ and that $\psi$ alone isn't defined to be anything, so $|\hat{A}\psi\rangle$ isn't defined.

Is $|\hat{A} \psi\rangle = \hat{A} |\psi\rangle$ just a notation or is there something deeper that I am missing?

Answer

This should be understood as a mere definition, i.e. a new label for the state you get when you apply the operator A to the ket psi.