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Consider $L^2(\mathbb R^n, \mathbb R^m)$. There should be a Fourier transform for these functions, like in the case $L^2( \mathbb R^n, \mathbb R )$. I wonder how these can be defined.

The application I have in mind is defining a Fourier transform for differential forms on $\mathbb R^n$.

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    Related : https://math.stackexchange.com/q/2332005/29872017-06-22

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