Schwartz, Infra-Schwartz Locally Convex Space

I've been writing a paper based on Chapter 1 of the book Nuclear and Conuclear Spaces by H Hogbe-Nlend. Its about Schwartz and infra-Schwartz spaces.

View my blog, http://greentea28a.wordpress.com/, section 2-2, section 2-3.

Theorem 1's and 3's proof needs simplification.

For motivation, I think the aforementioned concepts give way to Clay Math's Millennium problem. (Navier-Stokes existence and smoothness)

Can someone help?