Compact Sets

This chapter defines some notions of compactness related to covering properties and convergence of subsequences. In a Frechet space, and hence in a Banach space, sequential compactness is equivalent to compactness, and relative sequential compactness is equivalent to relative compactness, which in turn is equivalent to precompactness. The chapter characterizes the relatively compact subsets of Lebesgue space, L p (Ω;E), when E is a Frechet space. This is referred to as the Riesz-Kolmogorov theorem. The chapter also shows that a bounded sequence in a space E 0 , which converges to a limit in a space E 1 , also converges in some space E, which is said to be intermediate between E 0 and E 1 .