Geometry of Banach limits and their applications

A Banach limit is a positive shift-invariant functional on $\ell_\infty$ which extends the functional
$$ (x_1,x_2,\dots)\mapsto\lim_{n\to\infty}x_n $$
from the set of convergent sequences to $\ell_\infty$. The history of Banach limits has its origins in classical papers by Banach and Mazur. The set of Banach limits has interesting properties which are useful in applications. This survey describes the current state of the theory of Banach limits and of the areas in analysis where they have found applications.
Bibliography: 137 titles.