Would somebody mind clarifying the following for me please?
Suppose $\psi(f,g):=\int_a^b f(t)\overline{g(t)}dt$ where $f,g$ are complex functions of $t$, what does it mean to say that it is hermitian? How is the conjugate transpose defined in this case?
Thank you.
"Hermitian" is not referring here to a hermitian matrix but instead to a hermitian form.
Saying that the form $\psi$ is hermitian means that $$ \psi(g,f) = \overline{\psi(f,g)}, $$ for complex functions $f$ and $g$. It's easy to check that this holds here.
$\psi$ defines an inner product on the complex Hilbert space $L_2([a,b])$ and such an inner product must, by definition, be hermitian.