Proof of `The index of compactness is the growth rate of the compactness seminorm`
What this proof establishes
no assumptions
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
Each is bounded below by the compactness growth rate, and conversely a cover of the image of the unit ball of controlled cardinality bounds from above for large , as in the appendix of Lee (2024). The comparison of Bernstein and Gelfand numbers is in the spirit of Pietsch's theory of -numbers (Pietsch 1974).