The Fourier transform $\hat{f}$ of some function $f$ is often presented as a change of basis to a basis of complex exponentials. This however begs the question: since $\hat{f}$ is expressed with respect to a basis, is $f$ too expressed with respect to some basis?

In a linear algebra setting, the initial and final bases are clearly defined, for example from the standard basis to some linear combination of it. Here, however, it seems that the initial basis is just left out of the discussion.