ABSTRACT

This chapter shows that closure in a metric space generally amounts to measuring the size of the sets which can be approximated in the topology in question. It gives closed subspaces in terms of closure for actual approximations in Morrey spaces as well as canonical approximations. The chapter considers various subspaces. There are several standard approximations of functions; truncation of functions using compact sets, truncation of functions using level sets, mollifications and so on. The chapter investigates how we can approximate Morrey functions by using other function spaces. Zorko explained that the set of smooth functions in Morrey spaces cannot approximate Morrey spaces. Adams pointed out that any element in diamond subspaces can be approximated by the smooth functions in his textbook.