I've seen an "hand-waving" proof showing that the set of harmonics form a basis to all periodic signals. since:
$$b_0 + \cos(t) + \cos(2t) + \cos(3t) + \ldots$$
at time $t=0$ the series diverges and for any other $t\ne 0$ the harmonics cancel each other to zero.
First question: Why is it showing that the harmonics are a basis? It isn't clear to me. The author claims that we just built an impulse, but at time $t=0$ it diverges to infinity.
Next, I've seen that the set $$v_0(t) = \sqrt \frac{1}{t} \\ v_{2k+1}(t) = \sqrt\frac{2}{T} \cos \left (\frac{2\pi k t}{T}\right) \\ v_{2k}(t) = \sqrt\frac{2}{T} \sin \left (\frac{2\pi k t}{T}\right )$$
Is orthonormal and hence linearly independent.
Second question: Why is it also spanning the set of periodic signals?