Question: Consider the Hilbert space $L^2(\mathbb R/\mathbb Z)$ of periodic measurable functions. Recall that it has a Hilbert space basis consisting of functions $\chi_n(x)=e^{2\pi i n x}$ where $n\in > \mathbb Z$. Let $f\in L^2(\mathbb R/\mathbb Z)$ be the periodic function defined by $f(x)= x$, for $x\in [0,1)$. Compute $||f||^2$ in two ways, directly, and then using the Parseval's identity, i.e. using the Fourier series expansion of $f$.
My attempt: We have that $$\|f(x)\|^2 = \langle f, f \rangle = \int_0^1 |f(x)|^2 dx = \int_0^1 |x|^2 dx = \frac{|x|^3}{3}dx = \frac{1}{3}$$ since $x \in [0,1)$.
I am not sure if this is right though. I feel like the norm should be 1...
Now, I will find the norm using Parseval's identity using the Fourier series expansion of $f$. By Parseval's, given the basis $\chi_n$ such that $\langle \chi_n, \chi_m \rangle=\delta_{n,m}$ like in our problem, we have that for every $x\in L^2(\mathbb{R}/\mathbb{Z})$ $$\sum_n |\langle x,\chi_n \rangle|=\|x\|^2$$ For our problem, we have $f(x)=x$ for $x\in [0,1)$, thus we have that $$f(x)=x=\sum_n \langle x,\chi_n \rangle \chi_n$$ $$\Rightarrow \|f(x)\|^2=\|x\|^2=\bigg(\|\sum_n \langle x,\chi_n \rangle \chi_n\|\bigg)^2$$
I am not sure if this is right so far or how to continue from here. Any help would be much appreciated.