Consider $\mathbf{f}:\mathcal{S}\rightarrow\mathbb{R}^{n}$ and $\left(\phi_k\right)_{k=1}^{\infty}$, where $\phi:\mathcal{S}\rightarrow\mathbb{C}$, and $\mathcal{S}\subset\mathbb{R}^{n}$.
My intent is to write $\mathbf{f}$ as a projection onto the basis formed by $\phi$:
$$ \mathbf{f}=\sum_{k=1}^{\infty}\phi_k\mathbf{w}_k, $$
with $\mathbf{w}_k$ being:
$$ \mathbf{w}_k=\begin{bmatrix}\langle\phi_k,f_1\rangle\\ \langle\phi_k,f_2\rangle\\ \vdots\\ \langle\phi_k,f_n\rangle\end{bmatrix} $$
Can I compactify this notation somehow, maybe embedding it into the summation? I was thinking about something as $\langle\phi_k,\mathbf{f}\rangle$. As $\phi$ is a scalar function, and $\mathbf{f}$ is a vector function, is it obvious to relate the two notations? Do you have any other suggestions?
Moreover, from the definitions of $\mathbf{f}$ and $\phi$, is it correct to say that $\mathbf{w}\in\mathbb{C}^n$?