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Let the Schwartz function space on $\mathbb R^n$ be endowed with its usual metric structure. Is this space separable? Can anyone give a reference?

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  • $\begingroup$ I'd say the analytic Schwartz functions are dense. $\endgroup$
    – reuns
    May 6, 2017 at 13:51
  • $\begingroup$ Thanks! But what do you mean by analytic Schwartz functions? Do you mean real analytic ones or complex analytic ones? Could you give a link to the references? $\endgroup$ May 6, 2017 at 15:49
  • $\begingroup$ $f \ast m^n e^{-\pi |m x|^2} \to f$ and it is (real) analytic $\endgroup$
    – reuns
    May 6, 2017 at 15:57
  • $\begingroup$ Thank you for your comment! So how is the set of analytic Schwartz functions related to the countability? $\endgroup$ May 6, 2017 at 16:32
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    $\begingroup$ An analytic function is fully determined by the sequence of its derivatives at $0$. $\endgroup$
    – reuns
    May 6, 2017 at 16:35

1 Answer 1

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Schwartz space is separable as a metric space.

Indeed it is isomorphic to $\mathfrak{s}$, the space of sequences $(x_n)_{n\ge 0}$ of fast decay with topology defined by the seminorms $$ ||x||_k=\sup_n\ (1+n)^k |x_n| $$ In $\mathfrak{s}$ take almost finite sequences of rationals and that gives a countable dense subset.

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    $\begingroup$ May I have a reference to your answer? $\endgroup$
    – Taro Tokyo
    Mar 4, 2019 at 11:10
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    $\begingroup$ You mean, more precisely, a reference for the isomorphism of $S$ with $\mathfrak{s}$? In that case you can see aip.scitation.org/doi/abs/10.1063/1.1665472 but the proof is not very well written. Simon gave a much better presentation in section 6.4 on Hermite expansions of his book "Real Analysis : A Comprehensive Course in Analysis, Part 1" bookstore.ams.org/simon-1 $\endgroup$ Mar 4, 2019 at 13:59

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