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The completion $\overline{L^2_w(\mathbb{R}^d)}$ of $L^2_w(\mathbb{R}^d)$ (i.e. the completion of $L^2(\mathbb{R}^d)$ endowed with the $\sigma(L^2(\mathbb{R}^d),L^2(\mathbb{R}^d))$ topology) is homeomorphic to the algebraic dual $L^2(\mathbb{R}^d)^*$ endowed with the $\sigma(L^2(\mathbb{R}^d)^*,L^2(\mathbb{R}^d))$ topology (a standard result that can be found in Bourbaki TVS book for example).

Do we have some additional information that would help characterize the space $\overline{L^2_w(\mathbb{R}^d)}$?

In particular I would be interested to know whether $\mathscr{S}'(\mathbb{R}^d)$ (or a subset of it larger than just $L^2$) can be injected in $\overline{L^2_w(\mathbb{R}^d)}$.

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  • $\begingroup$ Probably this is not what you want, but there is a standard operator from $H^{-1}$ to $L^2$, which is then embedded continuously in the algebraic dual of $L^2$. $\endgroup$
    – Fan Zheng
    Nov 29, 2016 at 17:08
  • $\begingroup$ @FanZheng Let's say that I would like to have some additional properties. If $i_{-1}: H^{-1}\to (L^2)^*$ is the injection we are considering, and $i_0: L^2\to (L^2)^*$ is the natural injection of $L^2$ into its (continuous) dual, I would like that $i_{-1}(L^2)=i_0(L^2)$ and that $i_{-1}(H^{-1}\setminus L^2)\cap i_0(L^2)=\emptyset$ (i.e. $i_{-1}$ extends the injection $i_0$). In that way, the relation between $L^2$ and $H^{-1}$ as subspaces of $\mathscr{S}'$ are preserved by the mapping to $(L^2)^*$. $\endgroup$
    – yuggib
    Nov 29, 2016 at 21:19
  • $\begingroup$ Probably again not what you want, but you can extend$i_0$ by finitely many dimensions however like. $\endgroup$
    – Fan Zheng
    Nov 29, 2016 at 21:36

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