A NEW PROOF OF A THEOREM OF TOEPLITZ IN SUMMABILITY THEORY

The object of this paper is to give a new proof of a Theorem in Summability, as an application of a result of Antosik.

necessary and sufficient conditions for an infinite matrix A (a of real (or m,n complex numbers) numbers to be regular.The first proof, entirely analytical is given in Hardy [i].The second pr6of is obtained by using the uniform boundedness principle of Functional analysis.
In this paper in section 2, we obtain a new proof of Toeplitz's theorem based on a result of Antosik given in his paper [2].The above method has other applications also.(2 4) n+= Sn) n+= n P. ANTOSIK and V. K. SRINIVASAN We now state the diagonal theorem of Antoslk [2] used in this paper.
THEOREM A (Diagonal Theorem for Non-negatlve Matrices).Let A (xl, matrix of non-negatlve real numbers such that be a 311 (xi,j) 0 (for all i E N) (2.5a) (xi,j) 0 (for all j N) then there exists an increasing sequence of positive integers {pl} such that E x < i,j Pi 'Pj We now state and prove the theorem of Toeplitz.THEOREM i.Let A (a be an infinte matrix.
The necessary and sufficient m,n conditions for A to be regular are that the following three conditions must hold.
(b) ml a 0 for each n.m,n (c) ml I a n I. n=l m, PROOF.We now prove the sufficiency part of the theorem.
Let A be an infinite matrix satisfying the conditions (a), (b) and (c).Let {s n be a sequence such that s s as n .Then t converges for each m.We will n m now show that t s as m .m Given > 0, there is an integer N such that Is s < for n > N. By (c) (2.9) But by (b), there is an integer M' > M such that lam < /N ,n (2.10) for M' < m.If L sp(Is-Snl) we have for m > M' (2.11) (2.11) shows that t s as m =.
We now prove the necessity part.We first set We use (2.17 (2.24) The apparent contradiction in steps (2.23) and (2.24) shows that condition (a) must be valid.This now completes the proof of Theorem I.
REMARK I.The Diagonal theorem is mainly used to show the necessity of condition (a).
The remaining parts of the proof are quite standard and we presented them for the sake of making the proof complete.
The authors would like to express their thanks to the referee for his suggestions that resulted in the present form of this paper.
It is also possible to prove the non-archimedean analogue of Toeplitz's theorem, proved by Monna [3], who derived it by using the analogue of uniform boundedness principle over complete non-archimedean valued fields.As the proof is a trivial modification of the same techniques used in the classical case, this is left out.

( 2 .
7a)N being the set of natural numbers.Hence there exists {pl} such that i I x 0 and i I x j=l PlPj i=l Pl PJ 0(2.7b)

( 2 (
to (c).The necessity of (b) in Theorem is obtained by noting that A is convergence preserving and henceIn (e (k)) 0 A (in en(k))Thus we need to demonstrate the validity of condition (a).Suppose (a) does not hold.Then there exists a sequence of finite subsets {o m of positive integers such that max (o) < min (o N is the set of positive integers) with the additional property the columns and rows of the matrix (b converge to zero.Further it m)n is not difficult to show that there is a submatrix (See [2] diagonal theorem).(2.17)We now define a sequence [s as follows.