Definition talk:Unital Normed Algebra

As defined here, the underlying field is unnecessarily restrictive to $\R$ or $\C$, whereas in Definition:Normed Unital Algebra there is no such restriction, and the underlying structure is a general (normed) division ring.

Suggest we merge these two pages and ignore the Allan and Dales restriction, as they appear to be working within a particular agenda.

What we would of course do is configure this page as a second definition of Definition:Normed Unital Algebra, with all the refactoring and rework accordingly. --prime mover (talk) 11:29, 23 August 2023 (UTC)

They are slightly different, this page here insists that the norm of the identity should be $1$, and the linked page doesn't. The naming seems ok, this page is talking about what it means for a normed algebra to be unital (which is for the underlying algebra to be unital and $\norm { {\mathbf 1}_A} = 1$), and the linked page is just talking about a unital algebra with a norm smf no extra restriction. I don't think there's any issue with changing things over from $\GF \in \set {\R, \C}$ to a normed division ring. Caliburn (talk) 17:11, 23 August 2023 (UTC)

It may end up that the norm of the identity has to be $1$ if there is such a unit in the underlying algebra, but I haven't stopped to think about it and I may well be wrong.

It the norm is not multiplicative, it did not have to be. If $\norm \cdot$ is a norm, then $c \norm \cdot$ for $\forall c > 0$ is also a norm. --Usagiop (talk) 17:37, 23 August 2023 (UTC)

If they are indeed completely different beasts then we would do well to link between them with a warning page explaining the differences. --prime mover (talk) 17:29, 23 August 2023 (UTC)

Nope I'm talking crap sorry, the definition of "norm" cited builds in both $\norm 1 = 1$ through Definition:Norm on Unital Algebra and submultiplicativity through Definition:Norm on Algebra. This is not particularly clear on the page since the link is just to "norm". Caliburn (talk) 18:32, 23 August 2023 (UTC)