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 $\displaystyle f(t)$ with period T, we can decompose this function over the the fourier basis where the coefficients of the projections are given by $\displaystyle \int _{-T/2} ^{T/2} f(t)e^{-i2\pi nf_0 t} dt$ 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 $\displaystyle \displaystyle f(t) = \Sigma _{i=0} ^{N} \alpha _i \psi _i (t)$ i.e. we're projecting f on some basis, now, are two sides of the equation really equal for e.g. when the set $\displaystyle \{\psi _i\} _{i=0} ^N$ is the Haar basis.