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I am putting together an exposition on Lie theory; maths research is not my day job, let alone real maths history, so apologies in advance for any ignorance shown by these questions.

I am attempting a translation (with explanatory notes) of B. L. van der Waerden's 1932 papper "Stetigkeitssätze für halbeinfache Liesche Gruppen" (Continuity Theorem for Semisimple Lie Groups), see here. In this paper, an arbitrary sequence $\left\{C_{\nu, \mu}\right\}$ of elements $C_{\nu, \mu}$ belonging to the identity-connected component of a compact Lie group is considered, and a convergent subsequence $\left\{C_\nu\right\}$ picked out.

Clearly there is no problem with any of this, but a couple of details I would fill in are: (1) in compact connected Lie groups the exponential map is surjective, so that sequences thereof are equivalent to sequences in the Lie algebra, whence (2) Bolzano-Weierstrass is readily applied.

So I'm curious as to whether (1) van der Waerden is just assuming these details would be known to and filled in by one of his readers or (2) whether instead he had a slightly different concept of "compact lie group" to our modern one, perhaps something like a group wherein an obvious generalisation to Bolzano-Weierstrass applies. What would someone in 1932 understand by the idea of a compact group, or indeed of compactness in general; would it have been exactly the same as what we would understand today?

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Every Lie group is in particular a manifold so can be embedded in Euclidean space. So a compact Lie group is homeomorphic to a compact subset of Euclidean space, thus Bolzano-Weierstrass applies. Is there more to it than this? – Pete L. Clark May 9 '11 at 2:08
By the way, so far as I know, in 1932 a compact space was a topological space in which every open cover has a finite subcover. In other words, the same as today modulo the Bourbakistes who sometimes use "quasi-compact" for this (including me!). But anyway we are talking about Hausdorff, even metrizable spaces... – Pete L. Clark May 9 '11 at 2:12
Hello Pete. Many thanks for your comments. As for your first one, I was under the impression that the Whitney embedding theorems were not around in 1932 - I guess though that they were pretty strongly believed conjectures though at the time? (See Wikipedia page – WetSavannaAnimal aka Rod Vance May 9 '11 at 2:32
@Qiaochu: thanks for that -- Whitney embedding theorems were proved circa 1936. But this was not exactly my point: before Whitney embedding, by a "manifold" most people meant things that could a priori be embedded into Euclidean space. This is certainly true for a semisimple Lie group, since it is a subgroup of a general linear group. Right? – Pete L. Clark May 9 '11 at 3:17
@LSpice: Hmm. To me semisimple groups are necessarily linear, so the metaplectic groups do not qualify. But maybe that's just me...Can we get a ruling on this one? I'm starting to think I should quit while I'm behind. – Pete L. Clark May 9 '11 at 4:42
up vote 2 down vote accepted

Any compact manifold is sequentially compact, if by "manifold" we understand "second-countable Hausdorff etc." This is because such spaces are metrizable by Urysohn's metrization theorem, and for metrizable spaces compactness and sequential compactness are equivalent. Wikipedia tells me this result was published in 1925.

Given the time period, though, I would not be surprised if "Lie group" meant "matrix Lie group."

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Concerning your last speculation: "Lie group" meant what Lie intended it to be, namely a local transformation group with Euclidean coordinate patches such that the action map is differentiable in these patches, as made precise by Hilbert in the statement of his fifth problem: (p.451f). Van der Waerden assumes a global group structure and no space acted upon and refers to Schreier's paper for the foundations he doesn't make explicit in his article. – Theo Buehler May 9 '11 at 7:03

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