Prove that the $L_2$ norm of a function is equal to the sum of the squared Fourier series coefficients?

Question

I am new to this sort of analysis, so corrections to my terminology or understanding are very welcome. In one of my courses, an equivalence is drawn between $l^2$ and $L^2$ spaces. For $(c_1, c_2, ...) \in l^2$, the functions $f \in L^2$ are related by $c_k = \langle f, \phi_k \rangle$ for basis functions $\phi_k$, making the series $\sum_{k=1}^{\infty} c_k \phi_k$ the Fourier series of $f$ with respect to $\{ \phi_k \}$.

This makes sense to me, but it then defines the $L^2$ norm of $f$ as being equal to the sum of squared coefficients:

$\| f \|_{L_2}^2 = \sum_{k=1}^{\infty} c_k^2$

My question is, how does one get from the original definition of the $L^2$ norm (below) to the result shown above, in the case where $f(x) = \sum_{k=1}^{\infty} c_k e^{ikx}$ (i.e. a Fourier series)? Additionally, is there any significance to the fact that this $L^2$ norm is the same as the $l^2$ norm in the space spanned by the $c_k$ coefficients?

$\| f \|_{L_2}^2 = \int_{S} |f(x)|^2 dx$

Answer 1

Simply expand $|f(x)|^2=f(x)^*\cdot f(x)$ in terms of the Fourier basis functions, and then use the orthogonality relations to cancel all cross-terms. $$\sum_{ij}c_i^* c_j\int_S \phi_i(x)^*\cdot\phi_j(x)~\mathrm{d}x = \sum_{ij}c_i^* c_j \delta_{ij} = \sum_{i}|c_i|^2 $$

Answer 2

You have $ f = \sum _ k c _ k \phi _ k $, yes? And you know that $ \int \phi _ i ^ * \phi _ j = \delta _ { i , j } $ (where the star indicates complex conjugation and $ \delta _ { i , j } $ is $ 1 $ if $ i = j $ but $ 0 $ if $ i \ne j $), which you can work out directly from the explicit formula $ \phi _ k ( x ) = e ^ { i k x } $. So calculate: $$ \int \lvert f \rvert ^ 2 = \int f ^ * f = \int \Big ( \sum _ k c _ k \phi _ k \Big ) ^ * \Big ( \sum _ k c _ k \phi _ k \Big ) = \int \Big ( \sum _ k c _ k ^ * \phi _ k ^ * \Big ) \Big ( \sum _ k c _ k \phi _ k \Big ) = \int \sum _ { i , j } c _ i ^ * \phi _ i ^ * c _ j \phi _ j \text . $$ Now you need a theorem, that you can reverse the order of an integral and an infinite sereis if they both converge (in the Lebesgue sense for the integral and absolutely for the series), which we have here. So: $$ \int \lvert f \rvert ^ 2 = \sum _ { i , j } \int c _ i ^ * c _ j \phi _ i ^ * \phi _ j = \sum _ { i , j } c _ i ^ * c _ j \int \phi _ i ^ * \phi _ j = \sum _ { i , j } c _ i ^ * c _ j \delta _ { i , j } = \sum _ k c _ k ^ * c _ k 1 = \sum _ k \lvert c _ k \rvert ^ 2 \text . $$ (Note that you get absolute values around the $ c $; otherwise it's not correct.)

As for the significance of this, this shows that $ L ^ 2 $ and $ l ^ 2 $ are isometrically isomorphic: there's a linear map (although you should check that it's linear) from either space to the other, the two maps are inverses of each other, and both maps preserve the norm (that's what we checked just now). So the two spaces are equivalent as Banach spaces and have all the same properties. (For example, they're both separable Hilbert spaces.)