Some fundamentals of locally convex spaces.- Countably compact sets and the theorem of Eberlein - Grothendieck.- Bounding sets in the weak topology.- Sequential compactness and angelic spaces.- Pointwise and weak compactness in spaces of continuous functions.- Best approximations and the theorem of R.C.James.- Proof of the theorem of R.C.James.- Applications of the sup-theorem.- The topology related to Rainwater's theorem.