normed division algebra Can we prove that every division algebra over $R$ or $C$ is a normed division algebra?
Or is there any example of division algebra in which it is not possible to define a norm?
Definition of normed division algebra is in here. Thanks!

Frobenius theorem for (finite-dimensional) asscoative real division algebras states there are only $\mathbb{R},\mathbb{C},\mathbb{H}$, and  the proof is elementary (it is given on Wikipedia in fact).
If you don't care about finite-dimensional, then the transcendental field extension $\mathbb{R}(T)/\mathbb{R}$, where here $\mathbb{R}(T)$ is the field of real-coefficient rational functions in the variable $T$, is a divison algebra (it is a commutative field) but cannot carry a norm. Indeed, there are no infinite-dimensional normed division algebras.
Finally, there are real division algebras that are not $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$ (which are the only normed ones), which means there are division algebras that cannot carry a norm. It is still true they all must have dimension $1,2,4$ or $8$ (see section 1.1 of Baez's The Octonions), but there are (for example) uncountably many isomorphism classes of two-dimensional real division algebras (but unfortunately I don't have a reference for this handy).
