Are there any infinite dimensional division algebras? Appart from the finite dimensional division algebras like $\mathbb{R, C, H, O}$
Are there any infinite dimensional division algebras? (Especially any "exceptional" ones?)
I was thinking maybe the ring over polynomials might be a division algebra if you include negative exponents and allow infinite series. But I'm not sure if every series gives a unique member of the algebra. You might have $(1+x)^{-1} = 1-x+x^2-...$
Well I guess the space of functions is a division algebra since you can add and divide them $f(x)g(x)$ and $f(x)/g(x)$ and has an identity element $1$.
What about the ring over polynomials with rational or irrational exponents?
Or ones based on lattices?

The rational functions (in one variable) provide an example of such a ring ... after we mod out by the appropriate equivalence relation, namely "equality off a finite set" (so that e.g. "$x+3$" and "${(x-2)(x+3)\over x-2}$" are the same thing). And the rational functions are only the tip of a much larger iceberg of "well-behaved" functions, especially when we work over $\mathbb{C}$ instead of $\mathbb{R}$ (see the notion of analytic function).
However, this idea isn't so straightforward for arbitrary collections of functions. Rational functions are extremely well behaved, and when we lose that good behavior things get quite difficult. For example, let $f(x)=1$ if $x>0$ and $0$ if $x\le 0$, and let $g(x)=1-f(x)$. Then the product of $f$ and $g$ is the always-zero function, but both $f$ and $g$ are nonzero a lot of the time so there's no clear picture of which one "ought to be" the zero element of our algebra. In general, finding a natural equivalence relation making a division algebra out of a given class of functions can be quite hard.
