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I am wondering if there exists some known useful distribution spaces which are larger than tempered distributions, but that are defined from Banach test function spaces, as Schwartz space. For instance, one may think of the order $n$ norm $$ \sup_{\vert \alpha \vert \leq n , x \in \mathbb{R}^d} \vert \partial^\alpha \varphi (x) \vert \chi (\vert x \vert ) $$ where $\chi$ is a weighting function (say, exponential).

The advantage is obviously that order $n$ distributions are dual Banach spaces.

Have you any basic reference on this ?

Thanks.

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There is a well-developed theory of this kind based on Hilbert spaces. See

Berezanskiĭ, Yu.M. Expansions in eigenfunctions of selfadjoint operators. American Mathematical Society 1968.

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