Banach limit

Summary

A Banach limit is a positive, shift-invariant linear functional on the space $\ell^\infty$ of bounded real sequences. It gives the value $1$ to the constant sequence $1$. It extends the ordinary limit from convergent sequences to all bounded sequences. Banach limits exist by the Hahn–Banach theorem, but none can be written down explicitly, and there are many of them. They define almost convergence and are a basic example of an invariant mean.