INFINITE MATRICES AND ABSOLUTE ALMOST CONVERGENCE

In 1973, Stieglitz [5] introduced a notion of FB-Convergence which provided a wide generalization of the classical idea of almost convergence due to Lorentz [1]. The concept of strong almost convergence was introduced by Maddox [3] who later on generalized this concept analogous to Stieglitz's extension of almost convergence [4]. In the present paper we define absolute FB-Convergence which naturally emerges from the concept of FB-Convergence.


B
B I (B (I) i ), it is same as the space f of almost convergent sequences [i], where B(1).(b(1) (i)) with Maddox [4] generalized strong almost convergence by saying that x -s[F B] if P and only if _Dnp(i) IXp s 0 (n-oo, uniformly in i) assuming that the series in (i.i) converges for each n and i.
Let s be the space of all complex sequences and d B {x s: Lim Bx lira (BiX)n exists for each i} F B {x (dB ): lim t (i x) exists uniformly in i n and the limit is independent of i}, where and Let Therefore, we have (Bi) be a sequence of infinite matrices with B i (bnp(i)).
A sequence x is said to be absolutely FB-Convergent if 19 (i x) converges n=0 n uniformly for i >-0, and lim t (i,x) which must exist should take the same value for n->oo n all i.We denote the space of absolute FB-Convergent sequences by v(B).

THE MAIN RESULT.
In this note, we denote by (v,v(B)) the set of matrices which give new classes of absolute B-conservative matrices and absolute almost B-conservative matrices.
Let A be any infinite complex matrix for which the pth row-sum converges for a given x for all x in some class.Now, by (1.2) and (2.1), we have where gnk(i) [bnp(i) bn_l,p p=O 30p (i) apk p=O (i)]apk (n > i), (n 0).
HEOREM.Let B (B i) be a sequence of infinite matrices with SUPn p0= bnp (i) < o, for each i.
Let A be an infinite matrix.Then A: v--> v(B) if and only if (i) sup me < p,k =k (ii) there is an N such that for r,i 0,i,2,... r E gnk(i) < K (constant), n=N k=0 (iii) (apk)p>_0 e v(B) for each k, and Let A e (v,v(B)).For each k, let apk be FB-Convergent with limit k" And let k=OE apk be FB-Convergent with limit c. (In each case, limit is taken for p > 0).
We use the following lemma in the proof.
LEMMA.If either the necessity part or the sufficiency part of the theorem holds, then, for x v, p=0 bnp(i) K=0 apkxk kO Xk p0 bnp(i) apk'= PROOF.If either A: vv(B) or the conditions (i)-(iv) of the theorem hold, then by part_ i summation, for k_o np pk PROOF OF THEOREM.Necessity.Condition (i) follows from the fact that A: v Since ek,e v, necessity of (iii) and (iv) is obvious.
It is clear that, for fixed p and j, J x / apk x k k=0 is a continuous linear functional on v.We are given that, for all x e v, it tends to a limit as j (for fixed p) and hence, by the Banach-Steinhaus Theorem [2], this limit A x is also a continuous linear functional on v. P We observe that, although Y.I (i,Ax) is uniformly convergent in i it needs n=0 n (i,Ax) i and (i,Ax) o not be uniformly bounded in i.For example, if t n (n -> i and i), then l ln(i,Ax) is uniformly convergent in i -> o but not uniformly n-o bounded.Now, we can say that uniforn convergence bears only on the behaviour of (i,Ax) for sufficiently large n.Thus, by definition, there is an m such that n qm,i(x) [n(i'Ax)[" n=l For m >-o, i >-o, qm, i is a continuous seminorm on v, and there is an integer N such is pointwise bounded on v.Such an N exists.For suppose not.Then that {qN,i (x) for r 0,1,2 i>o r, i But this contradicts the fact that to each x e v there exists an integer N with x i->oSUp qNx,i(x < o.Now, by another application of the Banach-Steinhaus Theorem, there exists a con- stant M such that qN,i(x) -< M Ixll.
(2.3) Apply (2.3) with x (x k) defined by x k i for k -< r and o for k > r.Hence (ii) must hoid.
Sufficiency.Suppose that the conditions (i)-(iv) hold and that x v.We have defined v(B) as a subspace of % Thus, in order to show that Ax v(B), it is necessary to prove that Ax is bounded.By virtue of condition (i), this follows im- mediately.
Now, it follows from (iv) and the lemma that gnk (i)   k=O converges for all i,n.Hence, if we write hnk(i) gnu(i), %--k then hnk(i) is defined, also for fixed i,n, hnk (i) --> 0 (2.4) as k--.Now condition (iv) gives us that n=O (2.5) converges aniformly in i, and, for suitable chosen N, lhno (i) n=N (2.6) is bounded.By virtue of condition (iii), for fixed k, we get that n=O converges uniformly in i.Since k-i hnk(i) hn0(i) the interchange of order of summation can be justified (see lemma) Since condition (i) holds, dpk is bounded for all p,k.Thus INFINITE MATRICES AND ABSOLUTE ALMOST Xk-l) = bnp(i) ape' (where the inversion is justified by absolute convergence) principle of condensation of singularities [6], {x e v: sup qr (x) for r 0,1,2,.. i_>o i MURSALEEN is of second category in v and hence nonempty, i.e., there is x v with sup q , by virtue of (2.9), we have lim tn(i,Ax) tN_l(i,Ax) _ __-hnk(i (x k Xk_l) assertion being justified by absolute convergence because of the boundedness of (2.10).By(2.7),we have This completes the proof.