In this talk, we explore the structure of some non-compact linear
operators and embeddings between function spaces.
While traditional tools such as approximation, Kolmogorov, and entropy
numbers effectively describe compact operators, they fail to provide
insights into the finer structural properties of non-compact maps. We
discuss the necessity of employing more refined notions, notably strict
singularity, finite strict singularity, and Bernstein numbers, to
characterize the degree of non-compactness adequately.

Focusing specifically on embeddings between Besov and Sobolev spaces, as
well as the Fourier and Hilbert transformations, we highlight recent
advancements in understanding these mappings through the lens of
Bernstein numbers and using concepts of maximal non-compactness and
strict singular operators. Our results demonstrate that Bernstein
numbers, initially introduced and later neglected, are indispensable for
revealing detailed structural differences between non-compact maps. In
particular, we provide conditions under which embeddings are strictly
singular, finitely strictly singular, or fail to exhibit strict
singularity entirely, thus enriching the classical theory of functional
analysis.

Lecter is based on series of joint papers with Chian Yeong Chuah, 
Liding Yao, and David E Edmunds.