Fourier basis of $L^2([-\pi,\pi])$

Question

I have read that the Hilbert space $L^2([-\pi,\pi])$ has a Hilbert basis: $$\{e^{inx}|n\in\Bbb{Z}\}$$ This to me indicates that we can only represent a function $u(x)$ by a Fourier Series iff $u(x)\in L^2([-\pi,\pi])$. However, I believe this is not a necessary condition and you may have a function $g(x)\notin L^2([-\pi,\pi])$ that can be represented by a Fourier series. To me this makes no sense since it means that: $$g(x) \in span(e^{inx}|n\in\Bbb{Z})$$ But not in $L^2([-\pi,\pi])$ which is also equal to $span(e^{inx}|n\in\Bbb{Z})$. Please can someone explain this to me.