Projecting on orthonormal basis

Hello everyone!

I have a couple of questions, but first if this is not the right forum to post, please redirect me to the correct forum.

(1) For a periodic function with period T, we can decompose this function over the the fourier basis where the coefficients of the projections are given by Now we say that the basis is infinite but countable. Now when the functions is a periodic, this Fourier series approaches a Fourier transform, but is calculating the Fourier transform projecting on a basis? If so, is this basis uncountable (because we have a continuous frequency spectrum?)

(2) Suppose i.e. we're projecting f on some basis, now, are two sides of the equation really equal for e.g. when the set is the Haar basis.