What does $\left< dz_j , \frac{\partial}{\partial z_j}\right>$ mean?

Question

Here, in page 2 of Steven Krantz's book Function Theory of Several Complex Variables, what do those angle brackets mean? What kind of product is that between differentials and partial derivatives?

Thank you.

Answer 1

This comes, essentially, from differential geometry. Here's how you think of it: at each point in $\mathbb C$ you have a basis of the "tangent space" that is $d/dx$ and $d/dy$, and changing basis you get $d/dz$ and $d/d\bar z$. Then $dz$ and $d\bar z$ are a basis of the dual space. The bracket is notation for the covectors acting on the vectors.

You can take this a bit further and develop the machinery of "differential forms," which makes sense of the $dz$ you see in integrals. For more information, consult any book on smooth (or in this case, complex) manifold theory. I don't think you will need any of this to read Krantz's book, though.