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' $$ \Z / N\Z \cong \{0,1,\cdots, N-1\}.$$

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

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 $$ \| F\|_p^2 := \int_\R |F(p)|^2 dp $$

$$ \| f\|_x^2 := (1/N) \sum_{x=0}^{N-1} |f(x)|^2 $$

Let $F(p)$ be a complex valued function on $p \in \R$, we define $$ \| F\|_p^2 := \sum_{p=0}^{N-1} |F(p)|^2 $$