For two given square-integrable wave functions $\phi(x)$ and $\psi(x)$, Schwarz Inequality states that $$|\int_a^bdx\phi^*\psi|\le\sqrt{\int_a^bdx\phi^*\phi\int_a^bdx\psi^*\psi}.$$ This guarantees that the inner product between the functions exists and is finite. What about the expectation value? Does the relation guarantee the same thing when I am working instead with the integral form of expectation value for some operator $A$?
I think it does, since we are just taking the inner product between a wave function and that wave function worked on by $A$. But is my thought correct?