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I would like to know the historical reason why the letter H is used for Sobolev spaces. In particular, why not S? It would be interesting to know the same for the letter W.

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Note that (in my experience) $H$ is only used for Sobolev spaces with $p=2$, which are Hilbert spaces. Also, $S^n$ already means "sphere". (Granted, $H^n$ might mean "cohomology"...) – Nate Eldredge Mar 8 '12 at 2:49
Nate, $S$ is also already the Schwartz space. – B R Mar 8 '12 at 4:18
@Nate, $H$ is also used for the Bessel potential version of Sobolev spaces that are $L^p$-based (but this was probably inspired by the already existing notation for the $L^2$-based case). Actually, the Hardy space notation $H^p$ might predate the Sobolev spaces. – timur Mar 8 '12 at 4:26
@Nate. When I graduated, I found a humoristic picture about pure and applied mathematicians. One mathematical child is piling up cohomology groups (there are arrows), while another is lining up Sobolev spaces (with embedding arrows). The idea was that pure maths goes to the heavens, whereas applied maths are down-to-earth. – Denis Serre Mar 8 '12 at 8:09
According to Pietro Majer (comment to this answer…) the $H$ is a Russian en standing for S.M. Nikolsky. – Theo Buehler Mar 8 '12 at 8:14
up vote 9 down vote accepted

The history of both the name "Sobolev space" and the notation (which changed over the years), has been well described by J. Naumann [link to pdf file]. The history of the name is particularly amusing:

These spaces, at least in the particular case $p=2$, were known since the very beginning of this century, to the Italian mathematicians Beppo Levi and Guido Fubini who investigated the Dirichlet minimum principle for elliptic equations. Later on many mathematicians have used these spaces in their work. Some French mathematicians, at the beginning of the fifties, decided to invent a name for such spaces as, very often, French mathematicians like to do. They proposed the name Beppo Levi spaces. Although this name is not very exciting in the Italian language and it sounds because of the name ”Beppo”, somewhat peasant, the outcome in French must be gorgeous since the special French pronunciation of the names makes it to sound very impressive. Unfortunately this choice was deeply disliked by Beppo Levi, who at that time was still alive, and - as many elderly people - was strongly against the modern way of viewing mathematics. In a review of a paper of an Italian mathematician, who, imitating the Frenchman, had written something on ”Beppo Levi spaces”, he practically said that he did not want to leave his name mixed up with this kind of things. Thus the name had to be changed. A good choice was to name the spaces after S.L. Sobolev. Sobolev did not object and the name Sobolev spaces is nowdays universally accepted.

Concerning notation, in the 1950's $BL$ was used (for Beppo-Levi space), notably by Nikodym. Sobolev himself originally used $L$ before switching to $W$ in his 1950 book. With the demise of the Beppo-Levi name, $W$ became commonplace.

I agree with Nate that the notation $H$ for $p=2$ is a clear reference to Hilbert.

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Thanks for the link to J. Naumann's essay! – paul garrett Aug 27 '12 at 15:02
Thanks for the answer! Do you have a guess on why $W$ was used? My guess is for "weakly differentiable". – timur Aug 27 '12 at 16:12

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