I understand that, for example the inner product space $L^2(X)$ of complex - valued functions defined on $X$ has the inner product \begin{equation} (f,g) = \int f \, \overline{g\,}. \end{equation}
Now I am reading a text where the elements of my inner product space are vector - valued functions, and it is assumed the reader knows how to adjust the inner product so that it works in this case. I am not sure how to do this, here is a guess and it would be great if I could get feedback on whether this is the right way to generalize to vector - valued functions:
\begin{equation} (f,g) = \sum_{i = 1}^n \int f_i \, \overline{g_i} \end{equation}
(here, $n$ is the dimension of the range, and $f_i$ is the $i^{th}$ component of the function $f$). Many thanks !