The following was the original article.
A division algebra that is also a normed vector space.
The normed division algebras are:
- the Real numbers
- the Complex numbers
- the Quaternions
- the Octonions
These are the only possible normed division algebras.
The entry is clearly false as it stands, but it seems to have some hidden core of truth. How can we rescue the article? --AxelBoldt
(This is a belated response, as I didn't see this article in Recent Changes.) This result is given in John Baez's octonion article. I've never checked it myself, but I believe it's correct if you use the definition of "normed division algebra" that Baez uses: a finite-dimensional unital algebra over R that is also a normed vector space satisfying ||ab|| = ||a||.||b||. This definition appears unreasonably restrictive in disallowing infinite-dimensional algebras and algebras over C. It also conflicts with your definition of division algebra, which requires associativity. I'm not sure what to do about the article. --Zundark, 2002 Jan 9
Would it be better to omit associativity from our definition of division algebra? After all, we already have division ring. --AxelBoldt
I don't claim to understand what you are talking about, but if different authors use these terms with different definitions, we should note that in the article, even if we decide to adopt one of those definitions for Wikipedia's own use. -- SJK