Let's say a **normed division algebra** is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$. 


There are only four finite-dimensional normed division algebras: the real numbers, the complex numbers, the quaternions and the octonions. This was proved by Hurwitz in 1898:


* Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, *Nachr. Ges. Wiss. Göttingen* (1898), 309-316.


Are there any infinite-dimensional normed division algebras? If so, how many are there?



Infinite-dimensional normed division algebras
A MathSciNet search reveals a paper by Urbanik and Wright ([Absolute-valued algebras. *Proc. Amer. Math. Soc.* **11** (1960), 861–866](https://doi.org/10.1090/S0002-9939-1960-0120264-6)) where it is proved that an arbitrary real normed algebra (with unit) is in fact a finite-dimensional division algebra, hence is one of the four mentioned in the OP. A key piece of the argument (Theorem 1) is to show that such an algebra $A$ is **algebraic**, in the sense that if $x \in A$, then the subalgebra of $A$ generated by $x$ is finite-dimensional. The authors then invoke a theorem of A. A. Albert stating that a unital algebraic algebra is a finite-dimensional division algebra.


