I am a beginner of Hodge Theory. When I read the notes, I find the following which makes me feel uncomfortable:
Below is an example:
First, from my understanding, $V$ is a vector space and $V^{*}$ is its dual space. Usually, we should use $dx, dy, dz$ to represent an element in dual. From my understanding, the example use $dx, dy, dz$ to represent elements in $V$. What's wrong here? Should I make confusion?
Second, I also want to ask if it is necessary to define an inner product in $V^{*}$. Usually, in a manifold, we can define a (0,2) tensor $g$ such that $g_{ij}=g(\partial_{i},\partial_{j})$. However, how to define an inner product for its dual? i.e. $g(dx^{i},dx^{j})$; is it simply the inverse of $(g_{ij})$? How to derive it?
Third, I do not understand the volume form. Why there are exactly two choices differing by sign?
I hope someone can answer the above questions to clear my mind about Hodge theory.