I recently read about how the Fourier series is essentially a projection of a function $f$ onto the basis $\{1, \sin(x),\cos(x)\}$. Since a vector $w$ in $\mathbb{R}^2$ can be represented as
$$w = \langle w,\textbf{i}\rangle\textbf{i} + \langle w,\textbf{j}\rangle\textbf{j}$$
is this analogous to the constant and Fourier coefficients $a_n$ and $b_n$ being the inner products $a_0=\langle f,1\rangle$, $b_n=\langle f,\sin nx\rangle$ and $a_n=\langle f,\cos nx\rangle$
If this is the case, why do we then take the infinite sum of each harmonic? Also, can this process be mimicked using any orthogonal basis?