Density follows by applying the Carath�odory kernel theorem.
2.
Following it can be deduced from the parallel slit theorem using a variant of Carath�odory's kernel theorem and Brouwer's theorem on invariance of domain.
3.
I was hoping someone could help me with a couple of problems I'm having with Peano kernel theorems-firstly to fill in the missing step in the end of my proof : I can't quite seem to finish the problem off.
4.
No matter what dual topology is placed on D2 ( " U " ), a " sequence " of distributions converges in this topology if and only if it converges pointwise ( although this need not be true of a nuclear Montel space and it is with this topology that the kernels theorem of Schwartz holds.