If $k\in\mathbb N$, then the discrete Fourier transform (DFT) of $x\in\mathbb C^k$ can be defined as $$\hat x_\omega:=\frac1k\sum_{i=0}^{k-1}e^{-{\rm i}2\pi\omega\frac ik}x_i\;\;\;\text{for }\omega\in\{0,\ldots,k-1\}$$ (I'm slightly abusing notation by starting indexiation of a $k$-dimensional vector at $0$).
Now, the Plancherel therorem can be written as $$\|x\|=\|\hat x\|,\tag1$$ where $\|\;\cdot\;\|$ is the usual norm on $\mathbb C^k$.
Now I wonder how $(1)$ can generalized to the multi-dimensional DFT. Let $x\in\mathbb C^{k_1\times\cdots\times k_d}$ and (using vector notation) $$\hat x_\omega:=\frac1k\sum_{i=0}^{k-1}e^{-{\rm i}2\pi\left\langle\omega,\:\frac ik\right\rangle}x_i$$ for $\omega\in\{0,\ldots,k_1-1\}\times\cdots\times\{0,\ldots,k_d-1\}$. Does $(1)$ still hold, when $\|\;\cdot\;\|$ is defined in the obvious (square-root of squared sum of array components) way?
Let $S = \{0,\dots,k_1-1\}\times\cdots\times\{0,\dots,k_d-1\}$. Let $x:S\rightarrow \Bbb{C}$ be some function. Then $\hat{x}:S\rightarrow \Bbb{C}$ is defined by $$\hat{x}(\omega) = \frac{1}{\prod_{m=1}^d \sqrt{k_m}} \sum_{\tau\in S} x(\tau)\exp\left(-2\pi i\sum_{m=1}^d \frac{\omega(m)\tau(m)}{k_m}\right)$$ The inversion formula is $$x(\tau) = \frac{1}{\prod_{m=1}^d \sqrt{k_m}} \sum_{\omega\in S} \hat{x}(\omega)\exp\left(2\pi i\sum_{m=1}^d \frac{\omega(m)\tau(m)}{k_m}\right)$$ and with this normalization $$\|x\|_2 = \|\hat{x}\|_2$$ where $$\|x\|_2 = \sqrt{\sum_{\tau\in S} |x(\tau)|^2}$$ and similarly for $\|\hat{x}\|_2$.