I'm currently studying density matrices, and have been frequently coming across the construction
What is the formal meaning of this composition? I understand $\langle \psi|$ to be an element of the dual space (to that vector space for which $|\psi\rangle$ is a member) but I don't quite understand what it means to put them together.
I have been treating this object as a linear operator on the space of ket vectors, and assuming a certain associativity to its composition, such that
$$\bigg(|\psi\rangle \langle \psi|\bigg) |\phi\rangle = |\psi\rangle \bigg(\langle \psi|\phi\rangle \bigg) = \alpha |\psi\rangle \,,$$
but is there a way to make this more formal? Thank you.