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