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Peano is typically credited with giving the first abstract definition of a vector space (1888):

http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_linear_spaces.html

Apparently, Peano credits Möbius, Grassmann, and Hamilton for inspiring the idea. However, I recall reading somewhere that there was an earlier and independent definition, not mentioned by Peano. (Maybe it was Dedekind?)

Does anyone know what I'm talking about?

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@Drew: I usually use the html characters ä (ä), ë (ë), ï (ï), ö (ö), ü (ü) for umlauts. These will work in questions and answers, but apparently not in comments. Alternatively, you can just input ä, ë, ï, ö, ü directly (if you are able to input those; otherwise, you can copy and paste from somewhere else), and that should work in questions, answers, and comments on mathoverflow. –  Ricardo Andrade Jun 1 '13 at 23:50

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Before Peano (1888), a more limited notion of vector space over the reals was axiomatized by Gaston Darboux (1875), in Sur la composition des forces en statiques.

This early history is discussed by Gregory Moore, The axiomatization of linear algebra: 1875-1940

An earlier approach to axiomatizing vectors emerged from the work of Gaston Darboux. In 1875 he published an article analyzing various proofs of the composition of forces in statics (i.e., the parallelogram law), beginning with one due to Daniel Bernoulli in 1726. Darboux set himself the task of treating this matter in pure geometry and then determining which assumptions are necessary.

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Very interesting. I would be curious then to learn what is Darboux' "more limited notion". –  Pietro Majer Jun 2 '13 at 13:08
    
@Carlos: Thanks. That paper of Gregory Moore is extremely helpful. –  Drew Armstrong Jun 3 '13 at 16:11
    
P.S. I believe I was mistakenly thinking of Weyl's (1918) <b>re-discovery</b> when I asked the question. I still don't remember where I read about it... –  Drew Armstrong Jun 3 '13 at 16:14

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