Banach limits: some new thoughts and perspectives

Abstract

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1, 1, . . . , 1, . . .) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces.The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of 'simultaneous continuous linear' operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces.1.1. Existence of Banach limits. One of the common methods to prove the existence of Banach limits is via the use of the invariant version of the Hahn Banach theorem whereas the first proof by M.M. 0

Hausdorff dimension of the set of almost convergent sequences

“…As a method of summation, the almost convergence is not a matrix one [9, Section 7] and is fairly weak [9,Theorem 7]. For further studies of almost convergence and its applications, we refer to [10][11][12][13][14].…”

Banach Limit in the Stability Problem of a Linear Functional Equation

“…Moreover, Φ is given by (16) where LIM denotes the Banach limit and it is the only solution to (3) such that…”

“…is bounded. Then we can define a function Φ : X → R by (16) and 20shows that Φ is a solution to (3). On the other hand, (18) with ε(x) ≡ 0 shows that every solution of Φ : X → R of (3) fulfils the condition…”

“…There are known transfers of the concept of the Banach limit to the vector case (see Sofi [16]), but wishing to remain legible and to relate our results to some previously proven theorems, we remain in the scalar case.…”

Applications of Banach Limit in Ulam Stability

“…Let us also mention here that in the vector case, there are Banach spaces with no Banach limits (a typical example is c 0 , that is, the space of all null sequences [11]). Further information on the Banach limit and its applications can be found in surveys [12][13][14].…”