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.
On
Building on what mcd has written, the usual way to expand in the orthonormal basis $e_n$ is: $$ f = \sum_{n=-\infty}^{\infty}\langle f,e_n\rangle e_n. $$ That's essentially what you're doing with the Fourier transform with regard to a basis with a continuous index 's' that ranges over the real numbers: $$ f = \int_{\mathbb{R}}\left\langle f,\frac{e^{isx'}}{\sqrt{2\pi}}\right\rangle \frac{e^{isx}}{\sqrt{2\pi}}ds $$
You can write $f(x)=c_1\int_{-\infty}^\infty f(p)\delta(x-p)dp$ and $f(x)=c_2\int_{-\infty}^\infty \hat{f}(p)e^{ipx} dp$ [with constants chosen according to your convention for delta functions and Fourier transforms]. I think a physicist would say that you have expressed $f$ in a 'position basis' or a 'momentum basis' in the two cases, so the Fourier transform can be thought of as a change of basis as long as you imagine $f$ as originally expressed in the delta function 'position basis'.