F-seminorms on generalized double sequence spaces defined by modulus functions

Using a double sequence of modulus functions and a solid double scalar sequence space, we determine F-seminorm and F-norm topologies for certain generalized linear spaces of double sequences. The main results are applied to the topologization of double sequence spaces related to 4-dimensional matrix methods of summability.


INTRODUCTION
Let N = {1, 2, . . .} and let K be the field of real numbers R or complex numbers C. We specify the domains of indices only if they are different from N. By the symbol ι we denote the identity mapping ι(z) = z.We also use the notation R + = [0, ∞).
Let e n = (e n k ) k∈N (n ∈ N) be the sequences, where e n k = 1 if k = n and e n k = 0 otherwise.We also consider the corresponding double sequences e n (2) = (e n ki ) (n ∈ N) such that, for all i ∈ N, e n ki = 1 if k = n and e n ki = 0 if k = n.
Unlike the module | • |, following also [8], the seminorm of an element x ∈ E is often denoted by |x |.It is known (see [8,Remark 1]) that F-seminorms coincide with paranorms satisfying (N3).Let X 2 be a double sequence of seminormed linear spaces X ki , | • |ki (k, i ∈ N).Then the set s 2 (X 2 ) of all double sequences x 2 = (x ki ), x ki ∈ X ki (k, i ∈ N), together with coordinatewise addition and scalar multiplication, is a linear space (over K).Any linear subspace of s 2 (X 2 ) is called a generalized double sequence space In the case X = K we omit the symbol X 2 in notation.So, for example, s 2 denotes the linear space of all K-valued double sequences u 2 = (u ki ).By s we denote the linear space of all single K-valued sequences u = (u k ).As usual, linear subspaces of s Double sequence spaces are also the sets c 2 and c 2 0 of all double scalar sequences which, respectively, converge and converge to zero in the Pringsheim sense.Recall that a sequence (u ki ) is said to be Pringsheim convergent to a number L if for every ε > 0 there exists an index n 0 such that |u ki − L| < ε whenever k, i > n 0 (see [12] or [18,Chapter 8]).In this case we write P-lim k,i u ki = L.
For example, the function ι p (t) = t p is an unbounded modulus for p ≤ 1 and the function φ (t) = t/(1+t) is a bounded modulus. Since , the moduli are continuous everywhere on R + .We also remark that the modulus functions are the same as the moduli of continuity (see [5, p. 866 and Λ(Φ Φ, X 2 )∩ s2 (X 2 ) are solid GDS spaces if Λ ⊂ s 2 is a solid DS space and Φ Φ = (φ ki ) is a double sequence of moduli.
Our aim is to determine F-seminorm topologies for GDS spaces of sequences , where Y 2 is another double sequence of seminormed linear spaces, T : ) is a linear operator defined on a linear subspace s 2 T (X 2 ) of s 2 (X 2 ) and the solid DS space Λ is topologized by an absolutely monotone F-seminorm.Similar theorems have been proved earlier in [7,10,13,16] for analogical sets of single number sequences in the case T = ι.The results of this paper are applied to the topologization of GDS spaces related to 4-dimensional matrix methods of summability.Some special cases of such spaces are considered, for example, in [1,3,4,15,17].

MAIN THEOREMS
Let λ ⊂ s be a sequence space, Λ ⊂ s 2 be a DS space, and let e k , e k (2) (k ∈ N) be sequences defined above.Recall that an F-seminormed space (λ , g) is called an AK-space, if λ contains the sequences e k (k ∈ N) and for any u = (u k ) ∈ λ we have lim n u [n] = u, where u [n] = ∑ n k=1 u k e k .Generalizing this definition, we say that an F-seminormed DS space (Λ, g) is an AK-space if Λ contains the sequences e k (2) (k ∈ N) and for any u 2 = (u ki ) ∈ Λ we have lim n u 2[n] = u 2 , where u 2[n] = ∑ n k=1 u k e k (2) with u k = (u ki ) i∈N and u k e k(2) = (u ki e k ji ) j,i∈N .It is not difficult to see that c2 0 is the AK-space with respect to norm u 2 ∞ = sup ki |u ki |.An F-seminorm g on a sequence space λ ⊂ s is said to be absolutely monotone if for all u = (u k ) and ) is said to be absolutely monotone if for all x 2 = (x ki ) and Soomer [16] and Kolk [7] proved that if a solid sequence space λ ⊂ s is topologized by an absolutely monotone F-seminorm (or paranorm) g and Φ = (φ k ) is a sequence of moduli, then the solid sequence space may be topologized by the absolutely monotone F-seminorm (paranorm) whenever either (λ , g) is an AK-space or the sequence Φ satisfies one of the two equivalent conditions (M5) there exist a function ν and a number δ > 0 such that lim u→0+ ν(u This result was generalized in [10] and [13] to the sequence space defined by means of a solid DS space Λ and a double sequence of moduli Φ Φ = (φ ki ).Thereby, in the case of AK-space (Λ, g) it is assumed that Φ Φ satisfies the conditions (M7) φk In the following we extend these results to the generalized double sequence spaces defined by means of a linear operator T : , and by means of the set s2 (Y 2 ), where Y 2 is another double sequence of seminormed spaces Theorem 1.Let Λ ⊂ s 2 be a solid DS space which is topologized by an absolutely monotone F-seminorm g.If the double sequence of moduli Φ Φ = (φ ki ) satisfies the condition (M5') there exist a function ν and a number δ > 0 such that lim u→0+ ν(u Thereby, if g is an F-norm in Λ, the spaces Y ki are normed and T satisfies the condition The F-seminorm g Φ Φ,T is absolutely monotone if Proof.Similarly to the proof of Theorem 2.2 [10], using also the linearity of T , it is not difficult to show that the functional g Φ Φ,T satisfies the axioms (N1)-(N3).To prove (N4), let lim n α n = 0. Then there exists an index n 0 with |α n | < δ for n ≥ n 0 .Since by (M5') we have and g is absolutely monotone, Now, let g be an F-norm on Λ and let the spaces Y ki be normed by the norms • ki .If g Φ Φ,T (x 2 ) = 0, then, using also (M1), we have and since g is absolutely monotone, Consequently, F-seminorm (F-norm) g Φ Φ,T is absolutely monotone if (2) holds.Remark 1.It is easy to see that the condition (M5') in Theorem 1 may be replaced by the equivalent condition Theorem 2. Let Λ ⊂ s 2 be a solid AK-space with respect to an absolutely monotone F-seminorm g.If the double sequence of moduli Φ Φ = (φ ki ) satisfies (M7) and (M8), then the GDS space may be topologized by the F-seminorm g Φ Φ,T .Thereby, if g is an F-norm in Λ, the spaces Y ki are normed and T satisfies (1), Proof.The functional g Φ Φ,T : Λ(Φ Φ, T , X 2 , Y 2 ) → K obviously satisfies the axioms (N1)-(N3).To prove (N4), let lim n α n = 0 and let x 2 be an arbitrary element from the space Λ(Φ Φ, T , X 2 , Y 2 ).Then Φ Φ(T x 2 ) ∈ Λ and since Λ is an AK-space, lim Using the equality by (3) we can find, for fixed ε > 0, an index m such that The double sequence M8), and g satisfies (N4), we have that Further, since g satisfies (N2) and it is absolutely monotone, we may write 2) .

This yields lim
because of (5).Thus there exists an index n 0 such that, for all n ≥ n 0 , Now, by ( 4) and ( 6) we get for n ≥ n 0 .Hence lim n g Φ Φ,T (α n x 2 ) = 0, i.e., (N4) is true for g Φ Φ,T .Similarly to the proof of Theorem 1 we can see that the F-seminorm g Φ Φ,T is absolutely monotone whenever (2) holds, and g Φ Φ,T is an F-norm if g is an F-norm, the spaces X ki are normed and (1) is true.Remark 2. The investigations of Basu and Srivastava [1] contain, for one modulus φ and for a sequence p 2 = (p ki ) of positive numbers p ki ≤ 1, the GDS space Λ(Φ Φ, X 2 ), where φ ki (t) = [φ (t)] p ki .They assert (see [1,Theorem 3.2]) that if Λ is topologized by an absolutely monotone paranorm g, then is a paranorm on Λ(Φ Φ, X 2 ) whenever inf k,i p ki > 0. But this is not true in general.Indeed, if φ is a bounded modulus, p ki = 1, and the solid sequence space ˜ 2 ∞ is topologized by the absolutely monotone norm g(u 2 ) = sup k,i |u ki |, then the set ) contains an unbounded sequence z 2 = (z ki ) with z ki = 0 such that for a subsequence of indices (k j ) the equality lim j |z k j ,k j | = ∞ holds.Then, defining we get the sequence (α n ) with lim n α n = 0. Since we have that lim Thus g Φ Φ does not satisfy the axiom (N4) and, consequently, it is not a paranorm on the GDS space ˜ 2 ∞ (φ , X 2 ) if the modulus φ is bounded.Theorem 1 (for T = ι) shows that if the solid double sequence space Λ is topologized by an absolutely monotone F-seminorm (or a paranorm with (N3)) g, then is an absolutely monotone F-seminorm (paranorm) on the GDS space whenever the modulus φ satisfies the condition (M5 • ) there exist a function ν and a number δ > 0 such that lim u→0+ ν(u) = 0 and φ (ut These conditions clearly fail if φ is bounded, since by sup t>0 φ for any fixed u > 0.
It should be noted that the same remark is true concerning [2, Theorem 3.1].

SOME APPLICATIONS
Let A = (a mnki ) be a non-negative 4-dimensional matrix, i.e., a mnki ≥ 0 (m, n, k, i ∈ N).By I we denote the 4-dimensional unit matrix.We say that A is essentially positive if for any k, i ∈ N there exist indices m k and n i such that a m k ,n i ,k,i > 0. A sequence and bc 2 0 [A ] p is related to an arbitrary solid F-seminormed (or seminormed) sequence space (Λ, g Λ ).It is easy to see that the set  is a solid linear subspace of s 2 .In addition, if the F-seminorm (seminorm) g Λ is absolutely monotone, then the functional A is an F-norm (a norm) whenever the space Λ is F-normed (normed).
Let φ be a modulus function and let p 2 = (p ki ) ∈ ˜ 2 ∞ with r = max{1, sup k,i p ki }.Some sets of sequences [1,3,4,15].These investigations lead us to the following, more general, notion of GDS spaces.For an arbitrary 4-dimensional matrix B = (b kilj ) let s 2 B (X 2 ) be the set of all sequences x 2 = (x ki ) ∈ s 2 (X 2 ) such that the series B ki x 2 = ∑ l j b kilj x lj converge.Let Bx 2 = (B ki x 2 ).Using also a double sequence of moduli Φ Φ = (φ ki ) and a solid DS space Λ, we consider the sets where .
The following representations of these sets are useful.Using the equality p ki = (p ki /r)r and denoting by Φ Φ p 2 /r the sequence of moduli φ p 2 /r ki (t) = (φ ki (t)) p ki /r , we may write Since the DS space Λ[A ] r is solid and the summability operator B is linear, on the ground of ( 7) and ( 8) it is not difficult to verify the linearity of Equalities (7) and ( 8) are applicable also to the topologization of the GDS spaces Proposition 1.Let Φ Φ = (φ ki ) be a double sequence of moduli and p 2 = (p ki ) be a bounded sequence of positive numbers with r = max{1 , sup k,i p ki }.Let A = (a mnki ) be a non-negative infinite matrix and let B = (b kilj ) be an infinite matrix of scalars.Suppose that (X, | • |) is a seminormed space and Λ ⊂ s 2 is a solid DS space which is topologized by an absolutely monotone F-seminorm g Λ .a) If the sequence of moduli Φ Φ p 2 /r satisfies the condition (M5'), then the GDS space Λ[A 1/r , Φ Φ, p 2 , B, X 2 ] may be topologized by the F-seminorm ) is an AK-space and the sequence of moduli Φ Φ p 2 /r satisfies the conditions (M7) and (M8), If, in a) and b), g Λ is an F-norm, X is normed, A is essentially positive, and Proof.Since by (7) we have with T = B and Y = X, statement a) follows from Theorem 1 because of Analogously, we deduce statement b) from Theorem 2 in view of (8).
Corollary 1.Let Φ Φ, p 2 , A , B, and X be the same as in Proposition 1.
whenever the sequence of moduli Φ Φ p 2 /r satisfies the condition (M5').Thereby, if X is normed, A is essentially positive, and condition (9) holds, then h ].The proof of Proposition 1 shows that in the case B = I statements of Proposition 1 and Corollary 1 remain true if X 2 is replaced by X 2 .Moreover, condition ( 9) is automatically satisfied for B = I .Thus the following is true.Proposition 2. Let Φ Φ, p 2 , A , and (Λ, g Λ ) be the same as in Proposition 1. Then the following statements hold.a) The GDS space whenever the sequence of moduli Φ Φ p 2 /r satisfies the condition (M5').b) ) is an AK-space and the sequence of moduli Φ Φ p 2 /r satisfies the conditions (M7) and (M8), If, in a) and b), g Λ is an F-norm, the spaces X ki are normed and A is essentially positive, then h Corollary 2. Let Φ Φ, p 2 , and A be the same as in Proposition 1.
whenever the sequence of moduli Φ Φ p 2 /r satisfies the condition (M5').Thereby, if g Λ is an F-norm in Λ, the spaces X ki are normed and A is essentially positive, then h Proposition 2 (see also Remark 2) generalizes and corrects a theorem of Basu and Srivastava (see [1, Theorem 3.2]).Savas and Patterson [15] consider the space  b) If (Λ[A ] r , g r Λ,A ) is an AK-space, then h p 2 Λ,A ,B is an F-seminorm on If, in a) and b), g Λ is an F-norm, the space X is normed, A is essentially positive and condition (9) holds, then h p 2 ∞,A ,B is an F-norm on Λ[A 1/r , p 2 , B, X 2 ] and Λ[A
2are called double sequence spaces (DS spaces) and linear subspaces of s are called sequence spaces.Well-known sequence spaces are the sets ∞ , c, c 0 , and p (p > 0) of all bounded, convergent, convergent to zero, and absolutely p-summable number sequences, respectively.Examples of DS spaces are s2 = {u 2 ∈ s 2 called strongly A -summable with index p ≥ 1 to a number L if P-lim m,n ∑ k,i a mnki |u ki − L| p = 0, and strongly A -bounded with index p if sup m,n ∑ k,i a mnki |u ki | p < ∞.It is clear that the set c 2 0 [A ] p of all strongly A -summable with index p to zero sequences and the set ˜ 2 ∞ [A ] p of all strongly A -bounded with index p sequences are solid linear spaces.Since the Pringsheim convergent double sequences are not necessarily bounded, c 2 0 [15]0 .Because c 2 and c 2 o are not contained in ˜ 2 ∞ , Theorems 3.3 and 3.6 of[15]may not be true in general.Corollary 2 allows us to say that the spaces Λ[A , φ ] with (|u ki |) whenever φ satisfies the condition (M5 • ) (or the condition (M6 • )).Another special form of Proposition 1 is related to the modulus functions φ ki (t) = t (k, i ∈ N, t ∈ R + ).p ki /r t p ki /r = u p ki /rand thus, by Remark 1, (M5') holds if and only if inf k,i p ki > 0. The condition inf k,i p ki > 0 also guarantees that the sequence of moduli Φ p 2 /r satisfies the conditions (M7) and (M8) if φ ki (t) = t.These facts permit us to formulate the following proposition and its corollary.
1/r , p 2 , B, X 2 ].Moreover, for B = I all previous statements remain true with X 2 instead of X 2 .Corollary 3. Let p 2 , A , B, and X be the same as in Proposition 1. Suppose that inf k,i p ki > 0 and Λ ∈ ˜ 2 ∞ , c2 0 , bc 2 o with g Λ = • ∞ .Then the GDS space Λ[A , p 2 , B, X 2 ] may be topologized by the F-seminorm h p 2 ∞,A ,B (x 2 ) = sup (9)ereby, if X is normed, A is essentially positive and condition(9)holds, then h p 2 ∞,A ,B is an F-norm on Λ[A , p 2 , B, X 2 ].