Master's Thesis

I finished chapter IV on friday and continued on Hilbert space theory. Unfortunately, when it came to connections with Fourier transformation (Sobolev and Schwartz spaces, distributions), my understanding began to slow down. Some questions occured:

1. Why is for and , i.e., why is
   
   
   
   for with compact support? Why do the border terms in the partial integration disappear in this case?
2. What’s so special about functions with compact support anyway?