The pre-limit of a real-valued function

Abstract

In [1] S. Banach shown the existence of very known Banach linear shift-invariant functionals defined on the real vector space of all bounded realvalued functions on the semi-axis t ≥ 0 and especially on the space of all real bounded sequences. In [2] G.G. Lorentz defined, by Banach shift-invariant functionals, the class of almost convergent sequences. In [3] almost convergence was extended to real-valued functions on the semi-axis t ≥ 0. In [4] almost convergence was extended to bounded sequences in a real normed space. This paper is devoted to a class of functions defined in the semi-axis t ≥ 0 which are near to the functions ƒ having lim t → ∞ ƒ (t). The paper is organized as follows. First, for a sufficiently large a (written a > a 0 for some a 0) by Ω we denote the real vector space of all functions defined on [0,+∞> and bounded on [a,+ ∞>. Next, we will show the existence of a family of functionals defined on the space Ω. By these functionals we define the notion of pre-limit of a function ƒ Є Ω and investigate the family of all these functions. Further, we will show a theorem characterizing a function having the pre-limit. Also we show another theorem which is very applicable, though it contains a new restrictive condition. Finally, to make the idea of pre-limit a little clearer, we give several example functions having pre-limit.