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Wikipedia / Banach limit

In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences:

  1. (linearity);

  2. (positivity);

  3. is the shift operator (shift-invariance);

  4. is a convergent sequence.

is the complex vector space.

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:

The existence of Banach limits is usually proved using the Hahn–Banach theorem (analyst's approach), or using ultrafilterss (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.

Almost convergence

is a constant sequence, and

.

is the same is called almost convergent.

Banach spaces

is the continuous dual space is known as the ba space, and consists of all (signed) finitely additive measures on the sigma-algebra of all subsets of the natural numbers, or equivalently, all (signed) Borel measures on the Stone–Čech compactification of the natural numbers.

References