One of the answers in this link gives an existence proof for the adjoint operator
Existence is straightforward to establish if you have an orthonormal basis, say $e_k$:
Then, with $v= \sum_k v_k e_k$, we have $\langle f(v), w \rangle = \sum_k \overline{v_k} \langle f(e_k), w \rangle = \langle \sum_k v_k e_k, \sum_k \langle f(e_k), w \rangle e_k \rangle$, that is, $\langle f(v), w \rangle = \langle v, w' \rangle $, where $w'=f^*(w) = \sum_k \langle f(e_k), w \rangle e_k $.
I don't understand how this works if the vector spaces are different, because you could otherwise not equate the right sides of the inner product, $f^*(w) = \sum_k \langle f(e_k), w \rangle e_k$.
Is there something I'm missing that lets you do this. How would you prove this if the inner products were different for the vector spaces?