Plancherel Theorem

Theorem
Let $n \in \Z_{>0}$.

Let $\map \BB {\R^n}$ be the Borel $\sigma$-algebra on $\R^n$.

Let $\lambda^n$ be the Lebesgue measure on $\R^n$.

For each $p \in \closedint 1 \infty$, let:
 * $\map {L^p} {\R^n} := \map {L^p} {\R^n, \map \BB {\R^n}, \lambda ^n}$

be the $L^p$ space.

For all $f \in \map {L^1} {\R^n} \cap \map {L^2} {\R^n}$, we have:
 * $\norm {\map \FF f}_2 = \norm {\map {\FF ^{-1} } f}_2 = \norm f_2$

where:
 * $\map \FF f$ is the Fourier transform of $f$
 * $\map {\FF ^{-1} } f$ is the inverse Fourier transform of $f$
 * $\norm \cdot_2$ denotes the $L^2$ norm