Using the simplest definition of a complex Lie group, I want to show that $\text{GL}_n(\mathbb{C})$ is a complex Lie group by first showing that its product map (i.e. $(A,B)\mapsto AB$) is holomorphic.
I'm trying to teach myself about complex Lie groups, but I'm a bit unsure of how this sort of map can be shown to be holomorphic. I've taken complex analysis, but only in one variable so I'm a bit stuck. Any advice is much appreciated!
I've considered a partial derivative like in real multivariable calculus, but I am not sure if this is the correct approach.
The proof really works just about the same as in the real valued case, appropriately generalized, as with much of Complex Analysis. So you can really use the answer for this question linked here and make the necessary modifications. Essentially, ignoring the determinant condition, 'most basic properties' about matrix multiplication are 'nice' because each entry in the product can be considered as a polynomial functions (or rational functions) in entries from the factors in the product.