Banach limit

In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb{C} defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\infty, and complex numbers \alpha:

1. \phi(\alpha x+y) = \alpha\phi(x) + \phi(y) (linearity);
2. if x_n\geq 0 for all n \in \mathbb{N}, then \phi(x) \geq 0 (positivity);
3. \phi(x) = \phi(Sx), where S is the shift operator defined by (Sx)n=x{n+1} (shift-invariance);
4. if x is a convergent sequence, then \phi(x) = \lim x . Hence, \phi is an extension of the continuous functional \lim: c \to \mathbb C where c \subset\ell^\infty is the complex vector space of all sequences which converge to a (usual) limit in \mathbb C.

In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely determined in this case.

As a consequence of the above properties, a real-valued Banach limit also satisfies:

: \liminf_ {n\to\infty} x_n \le \phi(x) \le \limsup_{n\to\infty} x_n.

The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach), or using ultrafilters (this approach is more frequent in set-theoretical expositions). These proofs are inherently non-constructive, as they use the axiom of choice or similar concepts and are not provable in only Zermelo–Fraenkel set theory.