Some estimations of Banach limits

In the first part of this paper, we introduce the notions of upper weight, lower weight and weight of subsequences of natural numbers and investigate some new estimations about Banach limits by using some results from Sucheston [L. Sucheston, On existence of finite invariant measures, Math. Z. 86 (1964) 327–336; L. Sucheston, Banach limit, Amer. Math. Monthly 74 (1967) 308–311]. In the second part of this paper, we study the connections between weights and densities of subsequences of natural numbers, and give a familiar formula to find some values of Banach limits on almost convergent sequences.