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It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like

$$ \dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \mathcal{B}'(R),\quad D_{L^p}(R),\quad D_{L^p}'(R) $$ etc.

I have never seen any English references on these spaces. Does anyone know one?


For example, $f\in D_{L^p}(R)$ if and only if $f$ is a smooth function such that it, together with all its derivatives, belongs to $L^p(R)$. $D_{L^p}'(R)$ is its dual.

$\mathcal{B}(R)=D_{L^\infty}(R)$, and $\mathcal{B}'(R)$ is its dual.

Thank you very much!


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Perhaps you or somebody else should at least give the definition, since notation changes often ... – Marc Palm Jul 26 '11 at 19:56
@pm, these notations are from Schwartz's book. I didn't find them at any other places. There is no wonder many people may not familiar with these spaces. – Anand Jul 26 '11 at 20:00
I have tried almost all functional analysis books in our library. I didn't find any other books covering most of the above spaces. I am wondering why this book has not been translated into English? – Anand Aug 8 '11 at 19:05
Do you need these specific names or do you just need the same space but perhaps denoted differently? There are plenty of modern books on distributions, including ones by Treves, Taylor, Hormander, and others. – Deane Yang Feb 9 '12 at 13:07
Anand, you're right. Based on your explanation of what these spaces are, they are not often used, except maybe when $p = 2$. Why do you need them? – Deane Yang Feb 9 '12 at 14:49

As far as I remember, the book "Topological vector spaces and distributions" by Juan Horvath discusses at least the $\mathcal{D}_{L^1}$ and the $\dot{\mathcal{B}}$-spaces.

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Thanks Vobo, I will have a look. :-) – Anand Feb 9 '12 at 14:11
Dear VoBo, I checked this book, especially Chapter 4. I didn't find these spaces you mentioned. However, I find two spaces akin to the above mentioned the spaces: (1) the space of integrable distributions which corresponds to $\mathcal{D}_{L^1}'$; (2) the space of smooth functions with bounded derivatives which corresponds to $\mathcal{D}_{L^\infty}$ in my case. Thank you for the reference. – Anand Feb 22 '12 at 14:41
Oh, I am sorry. I checked my old thesis, $\dot{\mathcal{B}}$ in Schwartz' book ist the subspace of $\mathcal{B}$ of functions with all their derivates vanishing at infinity. – VoBo 0 secs ago – VoBo Feb 22 '12 at 21:03

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