Ideal convergence and other generalized limits
نویسنده
چکیده
This talk is mainly concerned with two generalizations of convergence of sequences called I-convergence and I∗-convergence. We will mention some other generalizations of limit which are related to I-convergence, e.g Banach limit and statistical convergence. 1 Generalizations of limit The notion of limit is one of the central notions in mathematical analysis. No wonder it was generalized by mathematicians in various ways. One of natural generalizations of limit is to define an operator extending the usual limit which assigns a value to some non-convergent sequences too. For example if we want define an extended limit in a such way that the sequence (1, 0, 1, 0, . . .) has a limit, one would expect that this limit to be 12 . Example of an operator satisfying this condition is the Cesàro mean. The Cesàro mean of the sequence (an) is the sequence bn = a1 + . . . + an n of arithmetic means of first n elements. It can be verified that if (an) is convergent then (bn) converges to the same limit. The limit of Cesàro mean of the sequence (1, 0, 1, 0, . . .) is 12 , as we expected. We see, that the operator φ(an) = lim n→∞ bn is an operator which extends the usual limit. (This is also known as (C, 1)-convergence. Cesàro summability or (C, 1)-summability is analogous generalization of a sum of a series. Summability methods like this are studied in summability theory.) Stefan Banach proved in [4] that limit can be extended to an operator on all bounded sequences. He proved the existence of so called Banach limit, i.e., a continuous linear functional φ : `∞ → R defined on the set `∞ of all bounded real sequences such that for any real sequences x = (xn), y = (yn) it holds: • φ(c.x + d.y) = c.φ(x) + d.φ(y); (linearity) • if x ≥ 0, then φ(x) ≥ 0; (positivity) • φ(x) = φ(Sx), where Sx is the shift operator defined by S(xn) = (xn+1). (shift-invariance)
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