# 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 [19]. Matrix transformation involving [chi] were characterized by Sridhar [20] and Sirajiudeen [21]. 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 [4] used the idea of Orlicz function to construct the space ([L.sup.M]). Lindenstrauss and Tzafriri [5] 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 [6], Mursaleen et al.[7], Bektas and Altin [8], Tripathy et al. [9], Rao and Subramanian [10] and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in Ref [11].

Recall ([4], [11]) 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 [18] and further discussed by Ruckle [12] and Maddox [13] 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 [5] 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 [15]].

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 [15]] 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 [14].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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[15]). Snyder and Wilansky[1] introduced the concept of semi conservative spaces. Snyder[3] studied the properties of semi conservative spaces. In the year 1996 the semi replete spaces were introduced by Chandrasekhara Rao and Srinivasalu [2]. K. Chandrasekhara Rao and N. Subramanian [17] and [22] 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 [23] 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 [10]].

[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 [15]. 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

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

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

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

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

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

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[8] 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.

[9] 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.

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

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

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

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

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

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

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

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

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

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

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

[21] 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.

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

[23] 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

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 [19]. Matrix transformation involving [chi] were characterized by Sridhar [20] and Sirajiudeen [21]. 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 [4] used the idea of Orlicz function to construct the space ([L.sup.M]). Lindenstrauss and Tzafriri [5] 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 [6], Mursaleen et al.[7], Bektas and Altin [8], Tripathy et al. [9], Rao and Subramanian [10] and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in Ref [11].

Recall ([4], [11]) 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 [18] and further discussed by Ruckle [12] and Maddox [13] 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 [5] 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 [15]].

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 [15]] 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 [14].

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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[15]). Snyder and Wilansky[1] introduced the concept of semi conservative spaces. Snyder[3] studied the properties of semi conservative spaces. In the year 1996 the semi replete spaces were introduced by Chandrasekhara Rao and Srinivasalu [2]. K. Chandrasekhara Rao and N. Subramanian [17] and [22] 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 [23] 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 [10]].

[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 [15]. 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

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[23] 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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Author: | Subramanian, N.; Krishnamoorthy, S.; Balasubramanian, S. |
---|---|

Publication: | Global Journal of Pure and Applied Mathematics |

Article Type: | Report |

Geographic Code: | 9INDI |

Date: | Dec 1, 2009 |

Words: | 3147 |

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