F A S C I C U L I M A T H E M A T I C I

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1 F A S C I C U L I M A T H E M A T I C I Nr N Subramanian, BC Tripathy and C Murugesan THE DOUBLE SEQUENCE SPACE Γ Abstract Let Γ denote the space of all prime sense double entire sequences and Λ the space of all prime sense double analytic sequences This paper is devoted to the general properties of Γ Key words: entire sequence, analytic sequence, double sequence, dual AMS Mathematics Subject Classification: 40A05, 40C05, 40D05 1 Introduction Throughout w, Γ and Λ denote the classes of all, entire and analytic scalar valued single sequences respectively We write w for the set of all complex sequences (x mn ), where m, n N, the set of positive integers Then w is a linear space under the coordinatewise addition and scalar multiplication Some initial works on double sequence space is found in Bromwich[] Later on it was investigated by Hardy [3], Moricz [4], Moricz and Rhoades [5], Basarir and Solancan [1], Tripathy [6], Colak and Turkmenoglu [7] and many others We need the following inequality in the sequel of the paper For a, b 0 and 0 < p < 1, we have (1) (a + b) p a p + b p The double series x mn is convergent if and only if the double sequence (S mn ) is convergent, where S mn = m,n i,j=1 x ij (m, n = 1,, 3, )(see [9]) A sequence x = (x mn ) is said to be double analytic if sup x mn 1/m+n < m,n The vector space of all double analytic sequences will be denoted by Λ 1 A sequence x = (x mn ) is called double entire sequence if x mn m+n 0, as m, n The double entire sequences will be denoted by Γ Let

2 9 N Subramanian, BC Tripathy and C Murugesan φ = all finite sequences Consider a double sequence x = (x ij ) The (m, n) th section x [m,n] of the sequence is defined by x [m,n] = all m, n N 0, 0, 0, 0, 0, 0, 0, 0, δ mn = 0, 0, 1, 0, 0, 0, 0, 0, i,j=0 x ij δ ij, for with 1 in the (m, n) th position and zero otherwise An F K-space (or a metric space) X is said to have AK property if (δ mn ) is a Schauder basis for X Or equivalently x [m,n] x An F DK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = (x k ) (x mn ) (m, n N) are also continuous If X is a sequence space, we give the following definitions: (i) X = the continuous dual of X; (ii) X α = a = (a mn ) : a mn x mn <, for each x X; (iii) X β = a = (a mn ) : (iv) X γ = a = (a mn ) : sup m,n 1 a mn x mn, is convergent, for each x X; a mn x mn <, for each x X; (v) let X be an F K-space φ, then X f = f(δ mn ) : f X; (vi) X = a = (a mn ); sup m,n a mn x mn 1/m+n <, for each x X; X α, X β, X γ are called α (or Kothe-Toeplitz) dual of X, β-(generalized- Kothe-Toeplitz) dual of X, γ-dual of X and -dual of X respectively Definitions and preliminaries Let w denote the set of all complex double sequences A sequence x = sup (x mn ) is said to be analytic if (m,n) x mn 1/m+n < The vector space of all prime sense double analytic sequences will be denoted by Λ A sequence x = (x mn ) is called prime sense double entire sequence if x mn 1/m+n 0 as m, n The double entire sequences will be denoted by Γ The spaces and Γ are metric spaces with the metric () d(x, y) = sup x mn y mn 1/m+n : m, n = 1,, 3 m,n for all x = x mn and y = y mn in Γ

3 The double sequence space Γ 93 Proposition 1 Γ has monotone metric Proof We know that d(x, y) = sup m,n x mn y mn 1/m+n : m, n = 1,, 3, d(x n, y n ) = sup x nn y nn 1/n and n,n d(x m, y m ) = sup m,m x mm y mm 1/m Let m > n Then sup x mm y mm 1/m sup x nn y nn 1/n m,m n,n (3) d(x m, y m ) d(x n, y n ), m > n Also d(x n, x n ) : n = 1,, 3, is monotonically increasing bounded by d(x, y) For such a sequence (4) sup n,n x nn y nn 1/n = lim n d(xn, y n ) = d(x, y) From(3) and (4) it follows that d(x, y) = sup x mn y mn 1/m+n is a m,n monotone metric for Γ This completes the proof Proposition The dual space of Γ is Λ In other words (Γ ) = Λ Proof The proof is easy, so omitted Proposition 3 Γ is separable Proof The proof is easy, so omitted Proposition 4 Λ is not separable Proof Since x mn 1/m+n 0 as m, n, so it may so happen that first row or column may not be convergent, even may not be bounded Let S be the set that has double sequences such that the first row is built up of sequences of zeros and ones Then S will be uncountable Consider open balls of radius 3 1 units Then these open balls will not cover Λ Hence Λ is not separable This completes the proof Proposition 5 Γ is not reflexive

4 94 N Subramanian, BC Tripathy and C Murugesan Proof Γ is separable by Proposition 3 But (Γ ) = Λ, by Proposition Since Λ is not separable, by Proposition 4 Therefore Γ is not reflexive This completes the proof Proposition 6 Γ is not an inner product space and hence not a Hilbert space Proof Let us take 1 1/ / 0 0 x = (x mn ) = and y = (y mn ) = x /, x 1 0 1/3, d(x, θ) = sup x 1 0 1/, x 0 1/4, 1 0 1/, 1/ 0 1/3, = sup 0, 0, 1 1/, 1/ 1/3, 0, = sup 0, 0, 0, Here and later on in the paper sup will represent the supremum of the elements inside the matrix We get d(x, θ) = 1 Similarly d(y, θ) = 1 Hence d(x, θ) = d(y, θ) = 1 x + y = = 1 1/ / d(x + y, x y) = sup( x mn + y mn x mn y mn ) 1/m+n : m, n = 1,, 3,

5 The double sequence space Γ 95 ( ) x11 + y 11 1/, x 1 + y 1 1/3, d(x mn + y mn, θ) = sup ( ) /, 1/ 1/ 1/3, = sup 1/, 0, 1414, 0, = sup 0, 0, = sup 0, 0, = 1414 Therefore d(x + y, θ) = 1414 Similarly d(x y, θ) = 1 By parallelogram law, [d(x + y, θ)] + [d(x y, θ)] = [(d(x, θ)) + (d(θ, y)) ] = (1414) + 1 = [1 + 1 ] = = 4 Hence the parallelogram law is not satisfied Therefore Γ is not an inner product space Assume that Γ is a Hilbert space But then Γ would satisfy reflexivity condition [Theorem 466 [10]] Proposition 5, Γ is not reflexive Thus Γ is not a Hilbert space This completes the proof Proposition 7 Γ is not rotund Proof Let us take x = (x mn ) and y = (y mn ) defined by x mn = and y mn = Then x = (x mn ) and y = (y mn ) are in Γ Also d(x, y) x 11 y 11 1, x 1 y ,, x 1n y 1n 1+n, 0, = sup x m1 y m1 m+1, xm y m m+,, xmn y mn m+1, 0, 0,,, 0, Therefore 1, 0, 0, 0, 0, 0, 0, 0, d(x, θ) = sup 0, 0, 0, 0, d(θ, y) = 1 Obviously x = (x mn ) y = (y mn )

6 96 N Subramanian, BC Tripathy and C Murugesan But (x mn ) + (y mn ) = = ( xmn + y mn d = sup ), θ ( x11 +y 11 ( xm1 +y m1 ) 1, ( ) 1 x1 +y 1 3,, ( ) 1 x1n +y 1n n+1, 0, 0,, 0, 0, ) 1 ( ) 1 ( ) 1 m+1, xm +y m m+,, xmn+ymn m+1,,,,,, d( x mn + y mn 1, 0, 0, 0,, θ) = sup 0, 0, 0, 0, = 1 Therefore Γ is not rotund This completes the proof Proposition 8 Weak convergence and strong convergence are equivalent in Γ Proof Step 1 Always strong convergence implies weak convergence Step So it is enough to show that weakly convergence implies strongly convergence in Γ y (η) tends to y weakly in Γ, where (y mn) (η) = y (η) and y = (y mn ) Take any x = (x mn ) Γ and (5) f(z) = z mn x mn 1/m+n, for each z = (z mn ) Γ Then f (Γ ) by Proposition η By hypothesis f(y (η) ) f(y) as (6) f(y (η) y) 0, as η

7 ( y (η) and (6) we get The double sequence space Γ 97 mn y mn 1/m+n x mn 1/m+n ) 0, as η By using (5) Since x = (x mn ) Λ we have ( y (η) mn y mn 1/m+n ) 0 as η sup( (y mn (η) y mn ), 0 1/m+n ) 0 as η mn sup( (y mn (η) y mn ) 1/m+n ) 0, as η mn d((y (η) y), 0) 0, as η d((y (η) y) 0, as η This completes the proof x mn 1/m+n <, for all x Λ Proposition 9 We shall construct an infinite matrix A for which Γ A = Γ Example Consider the matrix y 11 y 1 y 1n 0 0 y 1 y y n 0 0 y m1 y m y mn x 11 x 1 x 1n x 1 x x n = x m1 x m x mn

8 98 N Subramanian, BC Tripathy and C Murugesan x 11 x 1 x 1n 0 0 x 11 x 1 x 1n 0 0 x 1 x x n 0 0 x 1 x x n 0 0 y 11 y 1 y 1n 0 0 x 1 x x n 0 0 y 1 y y n 0 0 x 1 x x n 0 0 x 31 x 3 x 3n 0 0 y m1 y m y mn 0 0 = x 31 x 3 x 3n 0 0 x x 3 x 3n 0 0 x 31 x 3 x 3n 0 0 x 31 x 3 x 3n 0 0 x 31 x 3 x 3n 0 0 x 31 x 3 x 3n 0 0 x 31 x 3 x 3n 0 0 y 11, y 1,, y 1n = x 11, x 1,, x 1n y 1, y,, y n = x 11, x 1,, x 1n y 31, y 3,, y 3n = x 1, x,, x n y 41, y 4,, y 4n = x 1, x,, x n y 51, y 5,, y 5n = x 1, x,, x n y 61, y 6,, y 6n = x 1, x,, x n and so on For any x = (x mn ) Γ (Ax) mn = d(x, 0), where metric defined on Γ is given by lim x mn 1/m+n m,n (7) [d(x, θ)] Γ A [d(x, θ)] Γ Conversely Given x [d(x, θ)] Γ A fix any m, n then, (Ax) mn lim x mn 1/m+n m,n [d(x, θ)] Γ A lim x mn 1/m+n m,n (8) [d(x, θ)] Γ [d(x, θ)] Γ A Therefore the matrix A = (x mnlk ) for which the summability field [d(x, θ)] Γ = [d(x, θ)] Γ is given by A

9 //Program for generalization: #include iostreamh #include conioh #include mathh #include f streamh void main() clrscr() ; int m,n,i,nn=0,j,count=1,k,1pp,abc; ofstream fout,fout 1; foutopen( aa1txt ); fout1open( aatxt ); cout << enter the value of m: ; cin>> m; for(i=1;i<=m;i++) nn=nn+pow(,i); while(count<=nn) cout<< ; fout<< ; for(abc=1;abc<=m+3;abc++) The double sequence space Γ A =

10 100 N Subramanian, BC Tripathy and C Murugesan cout<< ; fout<< ; cout<< \n ; fout<< \n; for(j = 1; j <= m; j + +) for(k=1;k<=pow(,j);k++) for(pp=1;pp<=;pp++) fout1<< Y << count <<, << pp << ; fout1<< Y << count <<, n = ; cout<< ; fout<< ; for(int q=1;q<=m+1;q++) if(q==j) cout<< 1 ; fout<< 1 ; else cout<< 0 ; fout<< 0 ; for(l=1;l<=;l++) foutl<< X << j <<, <<l<< ; fout1<< X << j << n ; cout<< \n ; fout<< \n ; fout1<< \n ; count++;

11 cout<< \n \n \n ; fout<< \n \n \n ; cout<< ; fout<< ; for(abc=1;abc<<=m+1;abc++) cout<< ; fout<< ; cout<< ; fout<< ; fout1<< \n\n ; foutclose(); fout1close(); getch(); SAMPLE INPUT/OUTPUT The double sequence space Γ 101 enter the value of m : Y 1,1, Y 1,,, Y 1,n = X 1,1, X 1,,, X 1,n X,1, X,,, X,n = X 1,1, X 1,,, X 1,n Y 3,1, Y 3,,, Y 3,n = X,1, X,,, X,n

12 10 N Subramanian, BC Tripathy and C Murugesan Y 4,1, Y 4,,, Y 4,n = X,1, X,,, X,n Y 5,1, Y 5,,, Y 5,n = X,1, X,,, X,n Y 6,1, Y 6,,, Y 6,n = X,1, X,,, X,n Y 7,1, Y 7,,, Y 7,n = X 3,1, X 3,,, X 3,n Y 8,1, Y 8,,, Y 8,n = X 3,1, X 3,,, X 3,n Y 9,1, Y 9,,, Y 9,n = X 3,1, X 3,,, X 3,n Y 10,1, Y 10,,, Y 10,n = X 3,1, X 3,,, X 3,n Y 11,1, Y 11,,, Y 11,n = X 3,1, X 3,,, X 3,n Y 1,1, Y 1,,, Y 1,n = X 3,1, X 3,,, X 3,n Y 13,1, Y 13,,, Y 13,n = X 3,1, X 3,,, X 3,n Y 14,1, Y 14,,, Y 14,n = X 3,1, X 3,,, X 3,n References [1] Basarir M, Sonalcan O, On Some Double Sequence Spaces, J Indian Acad Math, 1()(1999), [] Bromwich TJI, An Introduction to the Theory of Infinite Series, MacMillan and Co Ltd, New York 1965 [3] Hardy GH, On the convergence of certain multiple series, Proc Camb Phil Soc, 19(1917), [4] Moricz F, Extension of the spaces c and c 0 from single to double sequences, Acta Math Hungerica, 57(1-)(1991), [5] Moricz F, Rhoades BE, Almost convergence of double sequences and strong regularity of summability matrices, Math Proc Camb Phil Soc, 104(1988), [6] Tripathy BC, On statistically convergent double sequences, Tamkang J Math, 34(3)(003), [7] Colak R, Turkmenoglu A, The double sequence spaces l (p), c 0(p) and c (p), (to appear) [8] Turkmenoglu A, Matrix transformtion between some classes of double sequences, Jour Ins of Math Comp Sci (MathSer), 1(1)(1999), 3-31 [9] Apostol T, Mathematical Analysis, Addison-Wesley, London 1978

13 The double sequence space Γ 103 [10] Kreyszig E, Introductory Functional Analysis with Applications, by John wiley & Sons Inc, 1978 N Subramanian Department of Mathematics Sastra University, Tanjore , India nsmaths@yahoocom Binod Chandra Tripathy Institute of Advanced Study in Science and Technology Pachim Boragaon, Garchuk, Guwahati , India tripathybc@yahoocom or tripathybc@rediffmailcom C Murugesan Department of Mathematics Sathyabama University Chennai , India murugaa3@sifycom Received on and, in revised form, on 50008