A Structure Theorem for Compact Sets in a Banach Space

Let E be a normed space and A ⊂ E. Recall that A is:

bounded if there exists an λ > 0 such that A ⊂ λU_E;

totally bounded if for any ∊ > 0, A is covered by a finite ∊-net; namely, there is { a_l,⋯,a_n } ⊂ A such that A ⊂ ∪ i = 1 n ( a i + ϵ U E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq746.tif"/>