The Fourier Transform is isometric

I.e. $⟨\mathcal{F}f,\mathcal{F}g⟩=⟨f,g⟩\forall f,g$, where $\mathcal{F}$ is the Fourier Transform and $f,g$ are arbitrary functions in ${\mathcal{L}}^2(\mathbb{F})$ for $\mathbb{F}$ being either the Field of reals or complex numbers.

Proof

Since Fourier tranform’s inverse is it’s adjoint, we have $⟨\mathcal{F}f,\mathcal{F}g⟩=⟨{\mathcal{F}}^{†}\mathcal{F}f,g⟩=⟨f,g⟩$. Since all Inner Products are preserved, so are Distances, showing that the Fourier Transform is an Isometry.