I heard a non-mathematician professor saying that "every function can be represented by sines and cosines". I want it to be more precise.
Let the set $$\{C_m:m=0,1,2,\cdots\}\cup\{S_n:n=1,2,\cdots\}$$ be the Fourier basis, where $C_m(t)=\cos mt$ and $S_n(t)=\sin nt$. With the standard inner product $\langle f,g\rangle=\int_{-\infty}^\infty f(t)g(t)\,dt$, the Fourier basis is orthogonal and thus is linearly dependent.
Question) What is the span of the Fourier basis? More precisely, what does the set $$\left\{f:\mathbb R\to\mathbb R\mid f=c_0+\sum_{n=1}^\infty a_nC_n+b_nS_n,\quad a_n\in\mathbb R,\: b_n\in\mathbb R\right\}$$ mean?
Surely, the above definition of the set, say, $\text{span}(\mathcal F)$, involves a limit of a function, and it requires a metric on the function space and I'm assuing the $\mathcal L^2$ space. Does the Fourier basis span the whole $\mathcal L^2$ space? (I think the wikipedia document is stating it quite explicitly. And I want to know if it's right or not.)