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Topological vector space with a locally convex topology, i.e. induced by a system of seminorms.

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Some special sequence in $C(\mathbb{R})$

Yes. Take the countable set $P$ of all polynomials with rational coefficients and enumerate it somehow so that $P=\{p_1,p_2,\dots\}$. Given any $f\in C(\mathbb R)$, for each natural $k$ there exists s …
Iosif Pinelis's user avatar