11 Entropy numbers versus s-numbers

Among the s-number sequences, exactly two detect compactness on every Banach space: the Gelfand and the Kolmogorov numbers. The approximation numbers do not — a compact operator into a space without the approximation property is not a limit of finite-rank operators (Enflo) — so the equivalence \(a_n(S) \to 0 \iff S\) compact is available only on Hilbert spaces, where all s-numbers agree with \(a_n\).

This chapter compares \(c_n\), \(d_n\) and \(h_n\) with the entropy numbers of Chapter 10, in both directions: upper bounds for \(c_n, d_n\) in terms of \(e_n\) give “compact \(\Rightarrow \) decay”, and a lower bound for \(e_n\) gives the sharp comparison. The comparisons are formalised in EntropyBounds.lean, the compactness criteria in AddOns/Compact.lean, both over \(\mathbb {K} \in \{ \mathbb {R}, \mathbb {C}\} \).