From my understanding, any inner product on $\mathbb{F}^n$ may be defined in the following form (citation):
$$\langle x,y \rangle = y^t M x$$ where $M$ is a positive definite, hermitian matrix.
We can then determine the adjoint of a linear operator $T$, with respect to the above inner product $T^*:=M^{-1}A^*M$
Now say we are working in the space of complex-valued functions $V$, where $T:V\to V$ is defined by $T(f) = hf$ $$\langle f,g\rangle = \int_{0}^{1} f(t)\overline{g(t)}dt$$
where $h \in V$.
How would we determine the adjoint of such an operator? Is there a similar method to the one mentioned above?