Reference for : a Fréchet nuclear space is Montel

Have a look at Proposition 50.2 in

F. Treves: Topological Vectors Spaces, Distributions and Kernels, Academic Press 1995 or Dover 2006

Statement (50.12) in that proposition is precisely what you need.

does Wikipedia qualify as a reference? for a historical overview, see A pedagogical history of compactness

Maybe not in a single theorem, but you can go for Cor1 in Section 33 and Cor3 in Section 50 in Treves book.