While Weierstrass’ Approximation Theorem guarantees that continuous functions can be uniformly approximated by polynomials, it provides no information about the rate of this convergence. Bernstein’s Lethargy Theorem (BLT) classically addresses this gap by proving that the error of best polynomial approximation can decay at an arbitrarily slow, prescribed rate. This talk explores the evolution of BLT from its roots in classical approximation theory to its broad applications in functional analysis. We will discuss extensions of BLT to abstract Banach spaces and Frechet spaces. Building on this framework, we will investigate the deep connections between lethargy phenomena and operator ideals, the influence of Banach space reflexivity on the existence of lethargic convergence, and the interplay between BLT and interpolation theory via the Peetre K-functional.

In this talk, a selection of results from my thesis will be presented. The results will concern the application of ultraproduct methods in operator algebras and surrounding fields. They will be organized around three themes: applications to the structure theory of operator algebras, continuous model theory of tracial von Neumann algebras, and approximation theory of groups and group actions.