A pre-compactness criterion of subsets in subspaces spanned by compactly supported smooth functions

Fréchet and Kolmogorov characterized pre-compact sets in Lebesgue spaces. Since then, many characterizations of various function spaces have been developed. Although Morrey spaces are extensions of Lebesgue spaces, characterizing pre-compact sets within Morrey spaces remains open. This paper suggests that certain closed subspaces of Morrey spaces are more manageable compared to the Morrey spaces themselves. It provides a characterization of pre-compactness for subsets in the closure of compactly supported smooth functions in Banach lattices. This finding refines and broadens the characterization of pre-compact subsets in Morrey spaces.