- Banach limit
In
mathematical analysis , a Banach limit is a continuouslinear functional phi: ell_infty o mathbb{R} defined on theBanach space ell_infty of all bounded complex-valuedsequence s such that for any sequences x=(x_n) and y=(y_n), the following conditions are satisfied:
# phi(alpha x+eta y)=alphaphi(x)+eta phi(y)(linearity);
# if x_ngeq 0 for all nge1, then phi(x)geq 0;
# phi(x)=phi(Sx), where S is theshift operator defined by Sx)_n=x_{n+1}.
# If x is aconvergent sequence , then phi(x)=lim x. Hence,phi is an extension of the continuous functional lim x:c_0mapsto mathbb C.In other words, a Banach limit extends the usual limits, is 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.
The existence of Banach limits is usually proved using the
Hahn-Banach theorem (analyst's approach) or usingultrafilter s (this approach is more frequent in set-theoretical expositions). It is worth mentioning, that these proofs useAxiom of choice (so called non-effective proof).Almost convergence
There are non-convergent sequences which have uniquely determined Banach limits. For example, if x=(1,0,1,0,ldots),then x+S(x)=(1,1,1,ldots) is a constant sequence, and it holds2phi(x)=phi(x)+phi(Sx)=1. Thus for any Banach limit this sequence has limit frac 12.
A sequence x with the property, that for every Banach limit phi the value phi(x) is the same, is called almost convergent.
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