What is a(n uncountable) basis in the topological vector space $\mathcal{S}' \left(\mathbb{R}^n\right)$ ? How can any tempered distribution be expanded in terms of such a basis?
2026-03-25 01:37:06.1774402626
A basis in the space of all tempered distributions over $\mathbb{R}^n$
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You said "uncountable" which suggests you are talking about a Hamel basis (only allowed finite linear combinations to get all vectors). This is a useless notion in the present context. What you might need rather is a Schauder basis (where you are allowed infinite sums, with suitable notion of convergence). There is a countable Schauder basis given by Hermite functions. See this article by B. Simon.