What Is the Fourier Transform?

You are being confused with #samples needed for perfect reconstruction, i.e. Nyquist sampling frequency. Fourier series/transforms work regardless of the bandwidth of the signal, as long as the integral exists, i.e. it must vanish at infinity.

Essentially it's just projection in infinite-dimensional vector spaces.

That's what is commonly understood by reconstruction: perfect reconstruction. And for that you need a band-limited signal. Otherwise he would have said approximate- or lossy reconstruction.

> And for that you need a band-limited signal.

Luckily, we live in a physical universe, where such mathematical oddities, like infinite bandwidth signals, cannot exist, so this isn't an actual issue. Any signal that that contains infinite bandwidths only exists because it has sampling artifacts. You would, necessarily, be attempting to reconstruct errors. There are many "tricks" around dealing with such flawed signals. But yes, you can't fully reconstruct impossible signals with FFT.