Some Sequence Spaces and Almost Convergence

In this paper we investigate some new sequence spaces which naturally emerge from the concept of almost convergence. Just as ordinary, absolute and strong summability, it is expected that almost convergence must give rise to almost, absolutely almost and strongly almost summability. Almost and absolutely almost summable sequences have been discussed by several authors. The object of this paper is to introduce the spaces of strongly almost summable sequences which happen to be complete paranormed spaces under certain conditions. Some topological results, characterisation of strongly almost regular matrices, uniqueness of generalized limits and inclusion relations of such sequences have been discussed. Introduction Let S be the set of all sequences real or complex and L denote the Banach space of bounded sequences x = {xk}k-0 normed by \\x || = supks0|xt |. Let D be the shift operator on S, that is, Dx = {xtK-i, D x = {xk}°Z-2 and so on. It may be recalled that [see Banach (1932)] Banach limit L is a nonnegative linear functional on L such that L is invariant under the shift operator (that is, L(Dx)= L(x)Vx e L) and that L{e)=\ where e = {1,1, • • •}. A sequence x G L is called almost convergent [Lorentz (1948)] if all Banach limits of x coincide. Let c denote the set of all almost convergent sequences. Lorentz (1948) proved that { 1 m 1 x: lim r y\ xn+i exists uniformly in n\. ».-» m+\ fr'o J The summability methods of real or complex sequences by infinite matrices are of three types [see Maddox (1970), p. 185]—ordinary, absolute and strong. In the same vein, it is expected that the concept of almost convergence must give rise to three types of summability methods-almost, absolutely almost and strongly almost. The almost summable sequences have been discussed by King (1966), Schaefer (1969) and some others. More recently Das and Nanda 446 of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870001630X Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 12 Sep 2017 at 19:38:06, subject to the Cambridge Core terms I [2] Almost convergence 447 (unpublished) have considered absolute almost convergent and absolute almost summable sequences. Therefore our only concern is the strongly almost summable sequences, which naturally come up for investigation. The strongly summable sequences have been systematically investigated by Hamilton and Hill (1938), Kuttner (1946) and some others. The spaces of strongly summable sequences were introduced and studied by Maddox (1967, 1970). The purpose of this paper is to introduce the spaces of strongly almost summable sequences, which will fill up a gap in the existing literature. Let A = (ank) be an infinite matrix of nonnegative real numbers and p = {pk} be a sequence such that pk >0. (These assumptions are made throughout.) We write Ax = {An(x)j if An(x) = S t ank | xk \ " converges for each n (Here and afterwards summation without limits runs over 1 to °°) and then we write '""" {X} = ~r^+l % An+i (X} = ? a "' ' m} ' '"" where We define [see Maddox (1967)], [A,p]0 = {jc: An(x)^>0}; [A,p] = {x: An(x le)-*0 for some /}; and [A,p]«, = {x: sup An(x)<<*>}. The spaces [A,p]0, [A,p] and [A,p]«, are respectively called the spaces of strongly summable to zero, strongly summable and strongly bounded sequences. We now write [A,p]0 = {x: fmn(JC) —>0 uniformly in n}; [A,p) = {x: tm.n(x -/e)—>0 for some / uniformly in n}; and