Parseval Equality says, Fourier transformation, as a linear map from one function space (function on x), to another function space (function on p), preserves 'norm'. Norm is just a fancy way of saying 'length of a vector'.

What do we mean by the length of a function?

Continuous Fourier transformation (OK, I switched to Boas convention) $f(x) = \int_\R F(p) e^{ipx} dp.$ $F(p) = (1/2\pi) \int_\R f(x) e^{-ipx} dx.$

Discrete Fourier transformation

Fix a positive integer $N$. $x,p$ are valued in the 'discretized circle' $\Z / N\Z \cong \{0,1,\cdots, N-1\}.$

$f(x) = \sum_{p \in \Z / N\Z} F(p) F(p) e^{2\pi i \cdot px/N}.$ $F(p) = (1/N) \sum_{p \in \Z / N\Z} f(x) e^{-2\pi i \cdot px/N}.$

Let $f(x)$ be a complex valued function on $x \in \R$, we define $\| f\|_x^2 := (1/2\pi) \int_\R |f(x)|^2 dx$

Let $F(p)$ be a complex valued function on $p \in \R$, we define $\| F\|_p^2 := \int_\R |F(p)|^2 dp$

$\| f\|_x^2 := (1/N) \sum_x |f(x)|^2$

Let $F(p)$ be a complex valued function on $p \in \R$, we define $\| F\|_p^2 := \sum_p |F(p)|^2$