The Eberlein–Šmulian and Eberlein–Grothendieck Theorems

The well-known Eberlein–Smulian theorem states the equivalence of several versions of weak compactness for subsets of a Banach space. The proof will be a consequence of properties of subsets H ⊆ C(X) with respect to the product topology on Open image in new window, where X is a suitable topological space. These considerations will also yield results for more general locally convex spaces.