Plancherel Theorem

Let \(f,g\in L^2(\mathbb{R}^n)\), then

\[ \langle f, g\rangle_{L^2(\mathbb{R}^n)}=\langle \hat{f}, \hat{g}\rangle_{L^2(\mathbb{R}^n)}, \]

where \(\hat{f}\) and \(\hat{g}\) are the Fourier transforms of \(f\) and \(g\).

This implies

\[ \lVert \hat{f}\rVert_{L^2(\mathbb{R}^n)}=\lVert f\rVert_{L^2(\mathbb{R}^n)}. \]