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F-Seminorms on Generalized Double Sequence Spaces Defined By Proceedings of the Estonian Academy of Sciences, 2014, 63, 2, 121–132 doi: 10.3176/proc.2014.2.02 Available online at www.eap.ee/proceedings F-seminorms on generalized double sequence spaces defined by modulus functions Enno Kolka and Annemai Raidjoe˜ b¤ a Institute of Mathematics, University of Tartu, 50090 Tartu, Estonia; [email protected] b Institute of Mathematics, Tallinn University of Technology, 19086 Tallinn, Estonia Received 15 November 2013, revised 28 November 2013, accepted 11 February 2014, available online 20 May 2014 Abstract. 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. Key words: double sequence, F-seminorm, F-norm, matrix method, modulus function, paranorm, sequence space. 1. INTRODUCTION Let N = f1;2;:::g 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 i we denote the identity mapping i(z) = z. We also use the notation R+ = [0;¥). n n n n Let e = (ek)k2N (n 2 N) be the sequences, where ek = 1 if k = n and ek = 0 otherwise. We also consider n(2) n n n the corresponding double sequences e = (eki)(n 2 N) such that, for all i 2 N, eki = 1 if k = n and eki = 0 if k 6= n. In all definitions which contain infinite series we tacitly assume the convergence of these series. An F-space is usually understood as a complete metrizable topological vector space over K. It is known that the topology of an F-space E can be given by an F-norm, i.e., by a functional g : E ! R+ with the axioms (see [6, p. 13]) (N1) g(0) = 0, (N2) g(x + y) · g(x) + g(y)(x;y 2 E), (N3) jaj · 1 (a 2 K); x 2 E =) g(ax) · g(x), (N4) limn an = 0 (an 2 K); x 2 E =) limn g(anx) = 0, (N5) g(x) = 0 =) x = 0. A functional g with the axioms (N1)–(N4) is called an F-seminorm.A paranorm on E is defined as a functional g : E ! R+ satisfying the axioms (N1), (N2), and (N6) g(¡x) = g(x)(x 2 E), (N7) limn an = a (an;a 2 K); limn g(xn ¡ x) = 0 (xn;x 2 E) =) limn g(anxn ¡ ax) = 0. A seminorm on E is a functional g : E ! R with the axioms (N1), (N2), and (N8) g(ax) = jajg(x)(a 2 K; x 2 E). ¤ Corresponding author, [email protected] 122 Proceedings of the Estonian Academy of Sciences, 2014, 63, 2, 121–132 An F-seminorm (paranorm, seminorm) g is called total if (N5) holds. So, an F-norm (norm) is a total F-seminorm (seminorm). Unlike the module j ¢ j, following also [8], the seminorm of an element x 2 E is often denoted by ˙jx˙j. It is known (see [8, Remark 1]) that F-seminorms coincide with³ paranorms´ satisfying (N3). 2 ˙ ˙ 2 2 Let X be a double sequence of seminormed linear spaces Xki;j ¢ jki (k;i 2 N). Then the set s (X ) 2 of all double sequences x = (xki), xki 2 Xki (k;i 2 N), together with coordinatewise addition and scalar 2 2 multiplication, is a linear space (over³ K).´ Any³ linear´ subspace of s (X ) is called a generalized double ˙ ˙ ˙ ˙ 2 2 sequence space (GDS space). If Xki;j ¢ jki = X;j ¢ j (k;i 2 N), then we write X instead of X . In the case X = K we omit the symbol X2 in notation. So, for example, s2 denotes the linear space of all K-valued 2 double sequences u = (uki). By s we denote the linear space of all single K-valued sequences u = (uk). As usual, linear subspaces of s2 are called double sequence spaces (DS spaces) and linear subspaces of s are called sequence spaces. Well-known sequence spaces are the sets `¥, c, c0, and `p (p > 0) of all bounded, convergent, convergent to zero, and absolutely p-summable number sequences, respectively. Examples of DS spaces are 2 2 2 s˜ = fu 2 s :u ˜k = supjukij < ¥ (k 2 N)g i and ˜ 2 2 2 l = fu 2 s˜ : u˜ = (u˜k) 2 lg 2 2 with l 2 f`¥; c0; `pg. Double sequence spaces are also the sets c and c0 of all double scalar sequences which, respectively, converge and converge to zero in the Pringsheim sense. Recall that a sequence (uki) is said to be Pringsheim convergent to a number L if for every e > 0 there exists an index n0 such that juki ¡ Lj < e whenever k;i > n0 (see [12] or [18, Chapter 8]). In this case we write P-limk;i uki = L. The idea of a modulus function was structured by Nakano [11]. Following Ruckle [14] and Maddox [9], we say that a function f : R+ ! R+ is a modulus function (or, simply, a modulus), if (M1) f(t) = 0 () t = 0, (M2) f(t + u) · f(t) + f(u), (M3) f is non-decreasing, (M4) f is continuous from the right at 0. For example, the function i p(t) = t p is an unbounded modulus for p · 1 and the function f(t) = t=(1+t) is a bounded modulus. Since jf(t) ¡ f(u)j · f(jt ¡ uj) by (M1)–(M3), 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]). 2 2 2 2 2 ˙ ˙ A GDS space L(X ) ½ s (X ) is called solid if (yki) 2 L(X ) whenever (xki) 2 L(X ) and jykijki · ˙ ˙ jxkijki (k;i 2 N). For example, it is not difficult to see that the sets ½ ¾ ¡ ¢ 2 2 2 2 2 ˙ ˙ s˜ X = x 2 s (X ) : supjxkijki < ¥ (k 2 N) ; i ¡ ¢ n ³ ³ ´´ o 2 2 2 2 2 ˙ ˙ L FF;X = x 2 s (X ) : FF(x ) = fki jxkijki 2 L ; 2 2 2 2 and L(FF;X )\s˜ (X ) are solid GDS spaces if L ½ s is a solid DS space and FF = (fki) is a double sequence of moduli. 2 2 2 2 Our aim is to determine F-seminorm topologies for GDS spaces of sequences x 2 sT (X ) with Tx in L(FF;Y2) or in L(FF;Y2) \ s˜2(Y2), where Y2 is another double sequence of seminormed linear spaces, 2 2 2 2 2 2 2 2 T : sT (X ) ! s (Y ) is a linear operator defined on a linear subspace sT (X ) of s (X ) and the solid DS space L 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 = i. 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]. E. Kolk and A. Raidjoe:˜ F-seminorms on double sequence spaces 123 2. MAIN THEOREMS Let l ½ s be a sequence space, L ½ s2 be a DS space, and let ek; ek(2) (k 2 N) be sequences defined above. Recall that an F-seminormed space (l;g) is called an AK-space, if l contains the sequences ek (k 2 N) [n] [n] n k and for any u = (uk) 2 l we have limn u = u, where u = ∑k=1 uke . Generalizing this definition, we say that an F-seminormed DS space (L;g) is an AK-space if L contains the sequences ek(2) (k 2 N) and 2 2[n] 2 2[n] n k(2) k(2) for any u = (uki) 2 L we have limn u = u , where u = ∑k=1 uke with uk = (uki)i2N and uke = k 2 2 (ukie ji) j;i2N. It is not difficult to see that c˜0 is the AK-space with respect to norm ku k¥ = supki jukij. An F-seminorm g on a sequence space l ½ s is said to be absolutely monotone if for all u = (uk) and v = (vk) from l with jvkj · jukj (k 2 N), we have g(v) · g(u). Analogously, an F-seminorm g on a GDS 2 2 2 2 2 2 space L(X ) ½ s (X ) is said to be absolutely monotone if for all x = (xki) and y = (yki) from L(X ) with ˙ ˙ ˙ ˙ 2 2 jykijki · jxkijki (k;i 2 N) we have g(y ) · g(x ). Soomer [16] and Kolk [7] proved that if a solid sequence space l ½ s is topologized by an absolutely monotone F-seminorm (or paranorm) g and F = (fk) is a sequence of moduli, then the solid sequence space l(F) = fu = (uk) 2 s : F(u) = (fk(jukj)) 2 lg may be topologized by the absolutely monotone F-seminorm (paranorm) gF (u) = g(F(u)) (u 2 l(F)) whenever either (l;g) is an AK-space or the sequence F satisfies one of the two equivalent conditions (M5) there exist a function n and a number d > 0 such that limu!0+ n(u) = 0 and fk(ut) · n(u)fk(t) (0 · u < d; t ¸ 0; k 2 N), f (ut) (M6) lim sup sup k = 0.
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