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Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?

Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as $$ H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
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