Banach limit

There are non-convergent sequences which have a uniquely determined Banach limit. For example, if x=(1,0,1,0,\ldots), then x+S(x) = (1,1,1,\ldots) is a constant sequence, and :2\phi(x) = \phi(x)+\phi(x) = \phi(x)+\phi(Sx) = \phi(x+Sx) = \phi((1,1,1,\ldots)) = \lim((1,1,1,\ldots)) = 1 holds. Thus, for any Banach limit, this sequence has limit 1/2. A bounded sequence x with the property that for every Banach limit \phi the value \phi(x) is the same is called almost convergent. ==Banach spaces==

Given a convergent sequence x=(x_n) in c \subset\ell^\infty, the ordinary limit of x does not arise from an element of \ell^1, if the duality \langle\ell^1,\ell^\infty\rangle is considered. The latter means \ell^\infty is the continuous dual space (dual Banach space) of \ell^1, and consequently, \ell^1 induces continuous linear functionals on \ell^\infty, but not all. Any Banach limit on \ell^\infty is an example of an element of the dual Banach space of \ell^\infty which is not in \ell^1. The dual of \ell^\infty 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. ==External links==