We have following definition of the Fourier transform $\mathcal{F} f$ of a function $f \in L_1(\mathbb{R})$: $(\mathcal{F}f)(y) := \dfrac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} f(x) e^{-ixy} dx , y\in \mathbb{R}$.
Furthermore we have:
1) For $y \in \mathbb{R}$ define the function $e_y:\mathbb{R} \rightarrow \mathbb{C}$ by $e_y(x) := \dfrac{1}{\sqrt{2 \pi}} e^{ixy}$. Then $\{ e_k: k \in \mathbb{Z}$} is a orthomal system in $L_2(-\pi, \pi)$.
2) $||\mathcal{F}f||_2^2 = \int_\mathbb{R} | \langle f, e_y \rangle |^2 dy$.
Now let space $L_2(-\pi, \pi)$ be a closed subspace of $L_2(\mathbb{R})$ (by setting $f \in L_2(\mathbb{R})$ to zero on $\mathbb{R}$ \ $(-\pi, \pi)$.
We know that for $f \in L_2(-\pi, \pi)$ it is $||\mathcal{F}f||_2^2 = ||f||_2^2$.
I need to show that for each $n \in \mathbb{N}$ and $f \in L_2 (-n \pi, n\pi)$ equation $||\mathcal{F}f||_2 = ||f||_2$ holds. To show this i should use isometric isomorphism between $L_2 (-n \pi, n\pi)$ and $L_2 (\pi, \pi)$.
Futhermore we need to show that $span \{L_2(-n \pi, n \pi): n \in \mathbb{N} \}$ is dense in $L_2({\mathbb{R}})$ and we should use it to extend $\mathcal{F} $ by continuity to an isometric operator $L_2(\mathbb{R})$
I would be grateful for any advice :-)