Extraction of Convergent Subsequences
Jacques Simon · 2022
This chapter is devoted to the extraction of convergent subsequences of space which is an important tool for solving partial differential equations by the compactness method. If every bounded sequence in space E has a subsequence converging in E-weak, this is a Neumann space. The chapter characterizes bounded sets of distributions, which will be useful since every convergent or Cauchy sequence is bounded. It characterizes convergent sequences of distributions. The chapter then characterizes the relatively sequentially compact sets of distributions that is those in which every sequence has a convergent subsequence. It shows that if the space E of values is included in a “larger” space, the same holds for the corresponding spaces of distributions. The chapter focuses on the case of an inclusion that is only “sequentially continuous”.