Talk:Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative

It seems to me that this article only proves in full one direction of the implication, it doesn't prove that if A' is nicely normed and alternative, then A is nicely normed and associative. 06:18, 16 October 2011 201.114.146.174
 * Perhaps the last line:
 * "It follows from reversing the argument that if $A'$ is not a nicely normed and alternative algebra then $A$ will not be a nicely normed associative algebra."
 * doesn't do the job? Feel free to explain why. --prime mover 05:40, 16 October 2011 (CDT)
 * Well, it technically is valid, since "if $A'$ is not a nicely normed and alternative algebra then $A$ will not be a nicely normed associative algebra" is the contrapositive and thus reversing the argument of that would indeed complete the proof. It's just that I would imagine that the argument for the quoted proposition is the argument given in the article (adequately reversed, since it's the contrapositive), and it's not quite clear that the argument will be exactly reversed, even considering the contrapositive. I hope I was clear enough. Thanks for your time.


 * Not clear at all. "if $A'$ is not a nicely normed and alternative algebra then $A$ will not be a nicely normed associative algebra" is not the contrapositive.


 * The contrapositive is:
 * "if $A$ is not a nicely normed associative algebra then $A'$ is not a nicely normed and alternative algebra." --prime mover 11:22, 16 October 2011 (CDT)


 * I'm sorry for the confusion, what I meant is that "if $A^′$ is not a nicely normed and alternative algebra then $A$ will not be a nicely normed associative algebra" is the contrapositive of "if $A$ is a nicely normed and associative algebra then $A'$ is a nicely normed alternative algebra" (this one is proved in the article).


 * I beg your pardon, I think I see what you mean now. I'm being silly - I need to rethink this. --prime mover 11:50, 16 October 2011 (CDT)
 * ... better now.