Putting convergence issues aside for the moment, for a periodic function $f: X \rightarrow \mathbb{R}$ its Fourier series is defined as $$f(x) = \sum_{k=-\infty}^\infty \langle f, x_k\rangle x_k(x) \tag{1}$$ where $\{x_k(x)\}$ is an orthonormal basis.
I am looking for an extension of the above to vector valued functions. For example, suppose $f: X \rightarrow \mathbb{R}^n$ and $f$ is periodic. Then there we can similarly decompose $f$ into a Fourier series: $$f = \sum_{k=-\infty}^\infty \textbf{a}_k \textbf{x}_k(x) \tag{2}$$ where $\{\textbf{x}_k(x)\}$ is an orthonormal basis.
In the real-valued case the coefficients $a_k$ are scalars and given by an inner product as in (1). Is there an explicit formula for computing the Fourier coefficients $\textbf{a}_k$ for vector valued functions such as (2)? The obvious choice is to do things component wise. That is, since $\textbf{a}_k$ and $\textbf{x}_k(x)$ both have $n$ components, then $$(\textbf{a}_k)_i = \langle f_i, (\textbf{x}_k)_i \rangle, \quad i = 1, \ldots n$$ but I could not find any such formula in any textbooks. Is this the right expression for $a_k$?