Yesterday I defined the Hilbert space of square-integrable 1-forms on a Riemann surface . Today I will discuss the decomposition of it. Here are the three components:

1) is the closure of 1-forms where is a smooth function with compact support.

2) is the closure of 1-forms where is a smooth function with compact support.

3) is the space of square-integrable harmonic forms.

Today’s goal is:

Theorem 1As Hilbert spaces,

The proof will be divided into several steps. (more…)