The Fouriertransform is a change of bases in infinite dimensional space.

That sounds complicated but if you look at the integral the exponential kernel is essentially a continuous “matrix” and the function you are integrating over that kernel is a continuous vector.

This observation on can be a guide to better understand infinite dimensional Hilbert spaces ( inner product space + stuff) and is one of the core observations in quantum mechanics where it’s part of the particle-wave concept as it transforms location space -> momentum space.