Let K denote either the field of real numbers or the complex fields. An analytic function over $K^n$ is a function that can be represented locally by a convergent power series in n variables with coefficients in K.

My question is that can we take K to be other fields? It seems that such a field K should satisfy some criteria:

- It is a metric space or at least a topological space.
- There should be complete, in the sense that Cauchy nets converge.
- The above 2 points probably force that K should have cardinality at least the size of the continuum.

Can there be other K where a reasonable theory of analytic functions can be developed? Say for cardinalities larger than the continuum? Probably this invokes some model theory.

Lie Algebras and Lie Groups. For geometric applications, yes, it's often better to have a more rigid collection of analytic functions: it depends on what you're trying to do. – Pete L. Clark Jun 1 '10 at 19:04