I am trying to understand the definition of $d^\ast$ of $d$ where $d$ denotes the exterior derivative as given in these lecture notes. (please see page 3)
Here are my thoughts so far:
Let us restrict for simplicity to $\mathbb R^n$.
Then the inner product on $\mathcal A^0 $ (=differential zero forms) is defined as (or so I understand):
$$ \langle f,g \rangle = \int_{\mathbb R^n}\int_{\mathbb R^n} |f(x) - g(x)|dx_i d x_j$$
for some $i,j$ and $d^\ast$ is the unique linear map such that
$$ \langle df , g\rangle = \langle f, d^\ast g\rangle$$
But it doesn't work because one of the arguments is a zero form and one is a one form.
What is it that I misunderstand?
My goal is to find $d^\ast$.