The semi Orlicz space of [chi] of analytic.

1. Introduction

A complex sequence, whose [k.sup.th] terms is [x.sub.k] is denoted by {[x.sub.k]} or simply x Let w be the set of all sequences x = ([x.sub.k]) and [phi] be the set of all finite sequences. Let [l.sub.[infinity]], c, [c.sub.0] be the sequence spaces of bounded, convergent and null sequences x = ([x.sub.k]) respectively. In respect of [l.sub.[infinity]], c, [c.sub.0] we have [parallel]x[parallel] = [sup.sup.sub.k][absolute value of [x.sub.k]], where x = ([x.sub.k]) [member of] [c.sub.0] [subset] c [subset] [l.sub.[infinity]]. A sequence x = {[x.sub.k]} is said to be analytic if [sup.sub.k][[absolute value [x.sub.k]]].sup.l/k] < [infinity]. The vector space of all analytic sequences will be denoted by [LAMBDA]. A sequence x is called entire sequence if [lim.sub.k[right arrow][infinity]] [[absolute value of [x.sub.k]].sup.1/k] = 0. The vector space of all entire sequences will be denoted by [[GAMMA].sub.[chi]]. was discussed in kamthan . Matrix transformation involving [chi] were characterized by Sridhar  and Sirajiudeen . Let [chi] be the set of all those sequences x = ([x.sub.k]) such that [(k![absolute value of [x.sub.k]]).sup.l/k] [right arrow] 0 as k [right arrow] [infinity]. Then [chi] is a metric space with the metric

d(x, y) = [sup.sub.k]{[(k![[x.sub.k] - [y.sub.k]]).sup.l/k] : k = 1,2,3,...}

Orlicz  used the idea of Orlicz function to construct the space ([L.sup.M]). Lindenstrauss and Tzafriri  investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space [l.sub.M] contains a subspace isomorphic to [l.sub.p](1 [less than or equal to] p < [infinity]). Subsequently different classes of sequence spaces defined by Parashar and Choudhary , Mursaleen et al., Bektas and Altin , Tripathy et al. , Rao and Subramanian  and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in Ref .

Recall (, ) an Orlicz function is a function M :[0, [infinity]) [right arrow] [0, [infinity]) which is continuous, non-decreasing and convex with M(0) = 0, M(x) > 0, for x > 0 and M(x) [right arrow] [infinity] and x [right arrow] [infinity]. If convexity of Orlicz funtion M is replaced by M(x + y) [less than or equal to] M(x) + M(y) then this function is called modulus function, introduced by Nakano  and further discussed by Ruckle  and Maddox  and many others.

An Orlicz function M is said to satisfy [[DELTA].sub.2]--condition for all values of u, if there exists a constant K > 0, such that M(2u) [less than or equal to] KM(u)(u [greater than or equal to] 0). The [[DELTA].sub.2]--condition is equivalent to M(lu), for all values of u and for l > 1. Lindenstrauss and Tzafriri  used the idea of Orlicz function to construct Orlicz sequence space

[l.sub.M] = {x [member of] w: [[infinity].summation over (k=1)]M([[x.sub.k]]/[rho]) < [infinity], for some [rho] > 0}. (1)

The space [l.sub.M] with the norm

[parallel]x[parallel] = inf {p > 0 : [[infinity].summation over (k=1)]M([[x.sub.k]/[rho]]) [less than or equal to] 1} (2)

becomes a Banach space which is called an Orlicz sequence space. For M(t) = [t.sup.p], 1 [less than or equal to] p < [infinity], the space [l.sub.M] coincide with the classical sequence space [l.sub.p]. Given a sequence x = {[x.sub.k]} its nth section is the sequence [x.sup.(n)] = {[x.sub.1], [x.sub.2],...,[x.sub.n],0,0,...}[[delta].sup.(n)] = (0,0,...,1,0,0,...), 1 in the nth place and zero's else where; and [s.sup.(k)] = (0,0,...,1,-1,0,...), 1 in the nth place, -1 in the (n +1)th place and zero's else where. An FK-space (Frechet coordinate space) is a Frechet space which is made up of numerical sequences and has the property that the coordinate functionals [p.sub.k](x) = [x.sub.k](k = 1,2,3,...) are continuous. We recall the following definitions [see ].

An FK-space is a locally convex Frechet space which is made up of sequences and has the property that coordinate projections are continuous. An metric-space (X,d) is said to have AK (or sectional convergence) if and only if d([x.sup.(n)], x) [right arrow] x as n [right arrow] [infinity]. [see ] The space is said to have AD (or) be an AD space if [phi] is dense in X. We note that AK implies AD by .

If X is a sequence space, we define

(i) X' = the continuous dual of X.

(ii) [X.sup.[alpha]] = {a = ([a.sub.k]) : [[summation].sup.[infinity].sub.k=1][absolute value of [a.sub.k][x.sub.k]]< [infinity], for each x [member of] X};

(iii) [X.sup.[beta]] = {a = ([a.sub.k]) : [[summation].sup.[infinity].sub.k=1] [a.sub.k][x.sub.k] is convergent, for each x [member of] X};

(iv) [X.sup.[gamma]] = {a = ([a.sub.k]) : [sup.sup.sub.n][absolute value of [[summation].sup.n.sub.k=1] [a.sub.k][x.sub.k]]< [infinity], for each x [member of] x};

(v) Let X be an FK-space [contains] 0. Then [X.sup.f] = {f([[delta].sup.(n)]): f [member of] X'}.

[X.sup.[alpha]], [X.sup.[beta]], [X.sup.[gamma]] are called the [alpha]- (or Kothe-Toeplitz) dual of X, [beta]-(or generalized Ko the-T oeplitz) dual of X, [gamma]-dual of X. Note that [X.sup.[alpha]] [subset] [X.sup.[beta]] [subset] [X.sup.[gamma]]. If X [subset] Y then [Y.sup.[micro]] [subset], for [micro] = [alpha], [beta], or [gamma].

1.1 Lemma

(See (15, Theorem 7.27)). Let X be an FK-space [contains] [phi]. Then

(i) [X.sup.[gamma]] [subset] [X.sup.f]. (ii) If X has AK, X[beta] = [X.sup.f]. (iii) If X has AD, [X.sup.[beta]] = [X.sup.[gamma]].

2. Definitions and Prelimiaries

Let w denote the set of all complex double sequences x = [([x.sub.k]).sup.[infinity].sub.k=1] and M : [0, [infinity]) [right arrow] [0, [infinity]) be an Orlicz function, or a modulus function. Let

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The space [[chi].sub.M] is a metric space with the metric

d(x, y) = in{[rho] > 0 : [sup.sub.k](M[(k! [absolute value of [x.sub.k] - [y.sub.k]]).sup.1/k]/[rho])) [less than or equal to] 1} (3)

The space [[GAMMA].sub.M] and [[LAMBDA].sub.M] is a metric space with the metric

d{x, y) = inf{[rho] > 0 : [sup.sub.k](M[(k! [absolute value of [x.sub.k] - [y.sub.k]]).sup.1/k]/[rho])) [less than or equal to] 1}

Because of the historical roots of summability in convergence, conservative space and matrices play a special role in its theory. However, the results seem mainly to depend on a weaker assumption, that the spaces be semi conservative. (See). Snyder and Wilansky introduced the concept of semi conservative spaces. Snyder studied the properties of semi conservative spaces. In the year 1996 the semi replete spaces were introduced by Chandrasekhara Rao and Srinivasalu . K. Chandrasekhara Rao and N. Subramanian  and  introduced the concept of semi analytic spaces and the semi Orlicz space of analytic sequences. Recently N. Subramanian, B.C. Tripathy and C. Murugesan has  introduced the concept of the semi Orlicz space of cs [intersection] [d.sub.1].

In a similar way, in this paper we define semi Orlicz space of of analytic, and hence show that [[GAMMA].sub.M] is smallest semi Orlicz space of [chi] of analytic.

3. Main Results

3.1 Proposition

[[chi].sub.M] [subset] [[GAMMA].sub.M]

Proof. It is easy. Therefore omit the proof.

3.2 Proposition

[[chi].sub.M] has AK, where M is a modulus function.

Proof. Let x = {[x.sub.k]} [member of] [[chi].sub.M], but then {M([(k! [absolute value of [x.sub.k]]).sup.1/k]/p)} [member of] [chi], and hence

[sup.sub.k[greater than or equal to]n+1](M([(k! [absolute value of [x.sub.k]]).sup.1/k]/p) [right arrow] as n [right arrow] [infinity]. Therefore

d(x, [x.sup.[n]]) = inf{p > 0 : [sup.sub.k[greater than or equal to]n+1](M([(k! [absolute value of [x.sub.k]]).sup.1/k]/[rho])) [less than or equal to] 1} [right arrow] 0asn [right arrow] [infinity] (5)

[??] [x.sup.[n]] [right arrow] x as n [right arrow] [infinity], implying that [[chi].sub.M] has AK. This completes the proof.

3.3 Proposition

[([[chi].sub.M]).sup.[beta]] = [LAMBDA].

Proof. Step 1: [[chi].sub.M] [subset] [[GAMMA].sub.M] by Proposition 3.1.

[??] [([[GAMMA].sub.M]).sup.[beta]] [subset] [([[chi].sup.M]).sup.[beta]]. But [([[GAMMA].sub.M]).sup.[beta]] = [LAMBDA]. [see ].

[lambda] [subset] [([[chi].sup.M]).sup.[beta]] (6)

Step 2: Let y [member of] [([[chi].sup.M]).sup.[beta]]. But f(x) = [[summation].sup.[infinity].sub.[k=1]] [x.sub.k][y.sub.k], with x [member of] [[chi].sub.M] we recall that [s.sup.k] has

(1/k!) in the kth place and zero's elsewhere, with x = [s.sup.k], {M [(k! [absolute value of [x.sub.k]]).sup.1/k]/p)} = {0,0,...,(M([(0).sup.1/k]/P),0,...)} which converges to zero. Hence [s.sup.k] [member of] [[chi].sub.M]. Hence d([s.sup.k], 0) = 1. But [absolute value of [y.sub.k]] [less than or equal to] [parallel]f[parallel] d([s.sup.k], 0) < [infinity] for all k. Thus ([y.sub.k]) is a bounded sequence and hence an analytic sequence. In other words y [member of] [LAMBDA].

[([[chi].sub.m]).sup.[beta]] [subset] [LAMBDA] (7)

From (6) and (7) we obtain [([[chi].sub.M]).sup.[beta]] = [LAMBDA]. This completes the proof.

3.4 Lemma

[15, Theorem 8.6.1] Y [contains] X [??] [Y.sup.f] [subset] [X.sup.f] where X is an AD-space and 7 an FK-space.

3.5 Proposition

Let be any FK-space [contains] [phi]. Then [Y] [contains] [[chi].sub.M] if and only if the sequence [s.sup.(k)] is weakly analytic

Proof: The following implications establish the result.

Y [contains] [[chi].sup.M] [??] [Y.sup.f] [subset] [([[chi].sub.M]).sup.f], since [[chi].sub.M] has AD by Lemma 3.4.

[??] [Y.sup.f] [subset] [LAMBDA], since [([[chi].sub.M]).sup.f] = A.

[??] for each f [member of] Y', the topological dual of Y. f([s.sup.(k)]) [member of] [LAMBDA].

[??] f([s.sup.(k)]) is analytic.

[??] [s.sup.(k)] is weakly analytic. This completes the proof.

4. Properties of Semi Orlicz Space of [chi] of Analytic

4.1 Definition

An FK-space X is called "semi Orlicz space of [chi] of analytic" if its dual [(X).sup.f] [subset] A. In other words X is semi Orlicz space of [chi] of analytic if f([s.sup.(k)]) [member of] [LAMBDA] for all f [member of] (X)' for each fixed k.

4.2 Example

[[chi].sub.M] is semi Orlicz space of [chi] of analytic. Indeed, if [[chi].sub.M] is the space of all Orlicz sequence of [chi], then by Lemma 4.3 [([[chi].sub.M]).sup.f] = A.

4.3 Lemma

[([[chi].sub.M]).sup.f] = A.

Proof: [([[chi].sub.M]).sup.[beta]] = [LAMBDA] by Theorem 3.3. But ([[chi].sub.M]) has AK by Proposition 3.2. Hence [([[chi].sub.M]).sup.[beta]] = [([[chi].sub.M]).sup.f]. Therefore [([[chi].sub.M]).sup.f] = [LAMBDA] This completes the proof. We recall

4.4 Lemma

(See 15, Theorem 4.3.7) Let z be a sequence. Then ([z.sup.[beta]], P) is an AK space with P = ([P.sub.k] :k = 0,1,2,...), where [P.sub.0](x) = [sup.sup.sub.m][absolute value of [[summation].sup.m.sub.k=1][z.sub.k][x.sub.k]], [P.sub.n](x) = [absolute value of [x.sub.n]]. For any k such that [z.sub.k] [not equal to] 0, [P.sub.k] may be omitted. If z [member of] [phi], [P.sub.0] may be omitted.

4.5 Proposition

Let z be a sequence [z.sup.[beta]] is semi Orlicz space of [chi] of analytic if and only if z is [lambda].

Proof: Step 1: Suppose that [z.sup.[beta]] is semi Orlicz space of [chi] of analytic. [z.sup.[beta]] has AK by Lemma 4.4. Therefore [Z.sup.[beta][beta]] = [([z.sup.[beta]]).sup.f] by Theorem 7.2.7 of Wilansky . So [Z.sup.[beta]] is semi Orlicz space of [chi] of analytic if and only if [z.sup.[beta][beta]] [subset] [lambda]. But then zk [member of] [z.sup.[beta][beta]] [subset] [lambda]. Hence z is [lambda].

Step 2: Conversely, suppose that z is [lambda]. Then [z.sup.[beta][beta]] [contain] [{[lambda]}.sup.[beta]] and [z.sup.[beta][beta]] [subset] [{[lambda]}.sup.[beta][beta]] = [[GAMMA].sup.[beta]] = [lambda], because [[lambda].sup.[beta]] = [GAMMA]. But [([z.sup.[beta]]).sup.f] = [z.sup.[beta][beta]]. Hence [([z.sup.[beta]]).sup.f] [subset] [lambda]. Therefore [z.sup.[beta]] is semi Orlicz space of [chi] of analytic. This completes the proof.

4.6 Proposition

Every semi Orlicz space of [chi] of analytic contained [[GAMMA].sub.M].

Proof: Let X be any semi Orlicz space of [chi] of analytic. Hence [(X).sup.f] [subset] [lambda]. Therefore f([s.sup.(k)]) [member of] [LAMBDA] forall f [member of] (X)'. So, {[s.sup.(k)]} is weakly analytic with respect to X. Hence X [contains] [[chi].sub.M] by Proposition 3.5. But [[chi].sub.M] [subaet] [[GAMMA].sub.M]. Hence X [subset] [[GAMMA].sub.M] This completes the proof.

4.7 Proposition

The intersection of all semi Orlicz space of [chi] of analytic {[X.sub.n] :n = 1,2,...} is semi Orlicz space of x of analytic.

Proof: Let X = [[intersection].sup.[infinity].sub.n=1][X.sub.n]. Then X is an FK-space which contains [phi]. Also every f [member of] (X)' can be written as f = [g.sub.1] + [g.sub.2] +... + [g.sub.m], where [g.sub.k] [member of] ([X.sub.n])' for some n and for 1 [less than or equal to] k [less than or equal to] m. But then f([s.sup.k]) = [g.sub.1]([x.sup.k]) + [g.sub.2]([s.sup.k]) +... + [g.sub.m]([s.sup.k]). Since [X.sub.n] (n = 1,2,...) are semi Orlicz space of [chi] of analytic, it follows that [g.sub.i] ([s.sup.k]) [member of] [LAMBDA] for all i = 1,2,...m .

Therefore f([s.sup.k]) [member of] [LAMBDA] forall k and for all f. Hence X is semi Orlicz space of [chi] of analytic. This completes the proof.

4.8 Proposition

The intersection of all semi Orlicz space [chi] of analytic is [[GAMMA].sub.M].

Proof: Let I be the intersection of all semi Orlicz space of [chi] of analytic. By Proposition 4.5 we see that the intersection

I [subset] [intersection] {z[beta] : z [member of] [LAMBDA]} = [{[lambda]}.sup.[beta]] = [GAMMA] = [[GAMMA].sub.M]. (8)

By Proposition 4.7 it follows that I is semi Orlicz space of [chi] of analytic. Consequently

[[GAMMA].sup.M] [subset] I (by Proposition 4.6) (9)

From (8) and (9) we get I = [[GAMMA].sub.M]. This completes the proof.

4.9 Corollary

The smallest semi Orlicz space of [chi] of analytic is [[chi].sub.M].

References

 A.K. Snyder and A. Wilansky, Inclusion Theorems and semi conservative FK- spaces, Rocky Mountain Journal of Math., 2(1972), 595-603.

 K. Chandrasekhara Rao and T.G. Srinivasalu, The Hahn sequence space-II, Journal of Faculty of Education, 1(2)(1996), 43-45.

 A.K. Snyder, Consistency theory in semi conservative spaces, Studia Math., 5(1982), 1-13.

 W.Orlicz, Uber Raume (LM) Bull. Int. Acad. Polon. Sci. A, (1936), 93-107.

 J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10 (1971), 379-390.

 S.D. Parashar and B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math. , 25(4)(1994), 419-428.

 M. Mursaleen, M.A. Khan and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Math. , Vol. XXXII (1999), 145-150.

 C. Bektas and Y. Altin, The sequence space [l.sub.M] (p; q; s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4) (2003), 529-534.

 B.C. Tripathy, M. Etand Y. Altin, Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Analysis and Applications, 1(3)(2003), 175-192.

 K. Chandrasekhara Rao and N. Subramanian, The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68(2004), 3755-3764.

 M.A. Krasnoselskii and Y.B. Rutickii, Convex functions and Orlicz spaces, Gorningen, Netherlands, 1961.

 W.H. Ruckle, FK-spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25(1973), 973-978.

 I.J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1) (1986), 161-166.

 H.I. Brown, The summability field of a perfect l-l method of summation, J. Anal. Math., 20(1967), 281-287.

 Wilansky, Summability through Functional Analysis, North-Holland Mathematical Studies, North-Holland Publishing, Amsterdam, Vol.85(1984).

 P.K. Kamthan and M. Gupta, Sequence spaces and series. Lecture Notes in Pure and Applied Mathematics, Marcel Dekker Inc. New York, 65(1981).

 K. Chandrasekhara Rao and N. Subramanian, Semi analytic spaces, Science Letters, 26(2003), 146-149.

 Nakano, Concave modulars, J. Math. Soc. Japan, 5(1953), 29-49.

 P.K. Kamthan, Bases in a certain class of Frechet space, Tamkang J. Math., 1976, 41-49.

 S. Sridhar, A matrix transformation between some sequence spaces, Acta Ciencia Indica, 5 (1979), 194-197.

 S.M. Sirajiudeen, Matrix transformations of [c.sub.0](p), [l.sub.[infinity]](p), [l.sup.p] and l into X, Indian J. Pure Appl. Math., 12(9) (1981), 1106-1113.

 K. Chandrasekhara Rao and N. Subramanian, The semi Orlicz space of analytic sequences, Thai Journal of Mathematics, Vol.2 (No.1), (2004), 197-201.

 N. Subramanian, B.C. Tripathy and C. Murugesan, The semi Orlicz space of cs [intersection] [d.sub.1], Communications of the Korean Mathematical Society, in press.

(1) N. Subramanian, (2) S. Krishnamoorthy and (3) S. Balasubramanian

(1) Dept. of Mathematics, SASTRA University, Tanjore, India

(2&3) Dept. of Mathematics, Govt. Arts College (Autonomus), Kumbakonam, India E-mail: nsmaths@yahoo.com, drsk 01@yahoo.com, sbalasubramanian2009@yahoo.com
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