The semi normed space defined by entire rate sequences.

[section]1. Introduction and preliminaries

A complex sequence, whose k-th term is [x.sub.k], is denoted by {[x.sup.k]} or simply x. Let [phi] be the set of all finite sequences. A sequence x = {[x.sub.k]} is said to be analytic rate if [sup.sub.k] [absolute value of [[x.sub.k] /[[pi].sub.k]].sup.1/k] < [infinity]. The vector space of all analytic sequences will be denoted by [[LAMBDA].sub.[pi]]. A sequence x is called entire rate sequence if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The vector space of all entire rate sequences will be denoted by [[GAMMA].sub.[pi]]. Let [sigma] be a one-one mapping of the set of positive integers into itself such that [[sigma].sup.m](n) = [sigma]([[sigma].sup.m-1](n)), m = 1,2,3, ...

A continuous linear functional [phi] on [[LAMBDA].sub.[pi]] is said to be an invariant mean or a [sigma]-mean if and only if

(1) [phi](x) [greater than or equal to] 0 when the sequence x = ([x.sub.n]) has [x.sub.n] [greater than or equal to] 0 for all n,

(2) [phi](e) = 1 where e = (1, 1, 1, ...) and,

(3) [phi]({[x.sub.[sigma]](n)}) = [phi]({[x.sub.n]}) for all x [member of] [[LAMBDA].sub.[pi]].

For certain kinds of mappings [sigma], every invariant mean [phi] extends the limit functional on the space C of all real convergent sequences in the sense that [phi](x) = lim x for all x [member of] C. Consequently C [subset] [V.sub.[sigma]], where [V.sub.[sigma]] is the set of analytic sequences all of those [sigma]- means are equal.

If x = ([x.sub.n]), set Tx = [(Tx).sup.1/n] = ([x.sub.[sigma]](n)). It can be shown that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[t.sub.mn](x) = ([x.sub.n] + T[x.sub.n] + (x)(x) + [T.sup.m][x.sub.n][).sup.1/n]/m + 1. (1)

Given a sequence x = {[x.sub.k]} its n-th section is the sequence [x.sup.(n)] = {[x.sub.1], [x.sup.2], ..., [x.sub.n], 0, 0, ...}, [[delta].sup.(n)] = (0, 0, ..., 1, 0, 0, ...), 1 in the n-th place and zeros elsewhere. 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.sup.k](k = 1, 2, ...) are continuous.

[section]2. Definition and properties

Definition 2.1. The space consisting of all those sequences x in w such that ([[absolute value of [x.sub.k]/[[pi].sub.k]].sup.1/k]) [right arrow] 0 as k [right arrow] [infinity] is denoted by [[GAMMA].sub.[pi]]. In other words ([[absolute value of [x.sub.k]/[[pi].sub.k]].sup.1/k]) is a null sequence. [[GAMMA].sub.[pi]] is called the space of entire rate sequences. The space [[GAMMA].sub.[pi]] is a metric space with the metric d(x, y) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x = {[x.sup.k]} and y = {[y.sub.k]} in [[GAMMA].sub.[pi]].

Definition 2.2. The space consisting of all those sequences x in w such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is denoted by [[LAMBDA].sub.[pi]]. In other words [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a bounded sequence.

Definition 2.3. Let p, q be semi norms on a vector space X. Then p is said to be stronger than q if whenever ([x.sub.n]) is a sequence such that p([x.sub.n]) [right arrow] 0, then also q([x.sub.n]) [right arrow] 0. If each is stronger than the other, then p and q are said to be equivalent.

Lemma 2.1. Let p and q be semi norms on a linear space X. Then p is stronger than q if and only if there exists a constant M such that q(x) [less than or equal to] Mp(x) for all x [member of] X.

Definition 2.4. A sequence space E is said to be solid or normal if ([[alpha].sub.k][x.sup.k]) [member of] E whenever ([x.sup.k]) [member of] E and for all sequences of scalars ([[alpha].sub.k]) with [absolute value of [[alpha].sub.k]] [less than or equal to] 1, for all k [member of] N.

Definition 2.5. A sequence space E is said to be monotone if it contains the canonical pre-images of all its step spaces.

Remark 2.1. From the above two definitions, it is clear that a sequence space E is solid implies that E is monotone.

Definition 2.6. A sequence E is said to be convergence free if ([y.sub.k]) [member of] E whenever ([x.sup.k]) [member of] E and [x.sup.k] = 0 implies that [y.sub.k] = 0.

Let p = ([p.sub.k]) be a sequence of positive real numbers with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let D = max(1,[2.sup.G - 1]). Then for [a.sub.k], [b.sub.k] [member of] C, the set of complex numbers for all k [member of] N we have

[[absolute value of [a.sub.k] + [b.sub.k]].sup.1/k] [less than or equal to] D{[[absolute value of [a.sub.k]].sup.1/k] + [[absolute value of [b.sub.k]].sup.1/k]}. (2)

Let (X, q) be a semi normed space over the field C of complex numbers with the semi norm q. The symbol [LAMBDA](X) denotes the space of all analytic sequences defined over X. We define the following sequence spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[section]3. Main results

Theorem 3.1. [[GAMMA].sub.[pi]](p, [sigma], q, s) is a linear space over the set of complex numbers.

The proof is easy, so omitted.

Theorem 3.2. [[GAMMA].sub.[pi]](p, [sigma], q, s) is a paranormed space with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Clearly g(x) = g(-x) and g([theta]) = 0, where [theta] is the zero sequence. It can be easily verified that g(x + y) [less than or equal to] g(x) + g(y). Next x [right arrow] [theta], [lambda] fixed implies g([lambda]x) [right arrow] 0. Also x [right arrow] [theta] and [lambda] [right arrow] 0 implies g([lambda]x) [right arrow] 0. The case [lambda] [right arrow] 0 and x fixed implies that g([lambda]x) [right arrow] 0 follows from the following expressions.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where r = 1/[absolute value of [lambda]]. Hence [[GAMMA].sub.[pi]](p, [sigma], q, s) is a paranormed space. This completes the proof.

Theorem 3.3. [[GAMMA].sub.[pi]](p, [sigma], q, s)[Intersection][[LAMBDA].sub.[pi]](p, [sigma], q, s)[subset or equal to] [[GAMMA].sub.[pi]](p, [sigma], q, s).

The proof is easy, so omitted.

Theorem 3.4. [[GAMMA].sub.[pi]](p, [sigma], q, s)[subset] [[LAMBDA].sub.[pi]](p, [sigma], q, s).

The proof is easy, so omitted.

Remark 3.1. Let [q.sub.1] and [q.sub.2] be two semi norms on X, we have

(i) [[GAMMA].sub.[pi]](p, [sigma], [q.sub.1], s)[Intersection] [[GAMMA].sub.[pi]](p, [sigma], [q.sub.2], s) [subset or equal to] [[GAMMA].sub.[pi]](p, [sigma], [q.sub.1] + [q.sub.2], s),

(ii) If [q.sub.1] is stronger than [q.sub.2], then [[GAMMA].sub.[pi]](p, [sigma], [q.sub.1], s) [subset or equal to] [[GAMMA].sub.[pi]](p, [sigma], [q.sub.2], s),

(iii) If [q.sub.1] is equivalent to [q.sub.2], then [[GAMMA].sub.[pi]](p, [sigma], [q.sub.1], s) = [[GAMMA].sub.[pi]](p, [sigma], [q.sub.2], s).

Theorem 3.5.

(i) Let 0 [less than or equal to] [p.sub.k] [less than or equal to] [r.sub.k] and {[r.sub.k]/[p.sub.k]} be bounded. Then [[GAMMA].sub.[pi]](r, [sigma], q, s) [subset] [[GAMMA].sub.[pi]](p, [sigma], q, s),

(ii) [s.sub.1] [less than or equal to] [s.sub.2] implies [[GAMMA].sub.[pi]](p, [sigma], q, [s.sub.1]) [subset] [[GAMMA].sub.[pi]](p, [sigma], q, [s.sub.2]).

Proof. (i) Let

x [member of] [[GAMMA].sub.[pi]](r, [sigma], q, s), (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[lambda].sub.k] = [p.sub.k]/[r.sub.k]. Since [p.sub.k] [less than or equal to] [r.sub.k], we have 0 [less than or equal to] [[lambda].sub.k] [less than or equal to] 1. Take 0 < [lambda] > [[lambda].sub.k]. Define [u.sub.k] = [t.sub.k]([t.sub.k] [greater than or equal to] 1), [u.sub.k] = 0([t.sub.k] < 1) and [v.sub.k] = 0([t.sub.k] [greater than or equal to] 1), [v.sub.k] = [t.sub.k]([t.sub.k] < 1), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by (5).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

x [member of] [[GAMMA].sub.[pi]](p, [sigma], q, s). (6)

From (3) and (6) we get [[GAMMA].sub.[pi]](r, [sigma], q, s) [subset] [[GAMMA].sub.[pi]](p, [sigma], q, s). This completes the proof.

(ii) The proof is easy, so omitted.

Theorem 3.6. The space [[GAMMA].sub.[pi]](p, [sigma], q, s) is solid and as such is monotone.

Proof. Let ([x.sub.k]/[[pi].sub.k]) [member of] [[GAMMA].sub.[pi]](p, [sigma], q, s) and ([[alpha].sub.k]) be a sequence of scalars such that [absolute value of [[alpha].sub.k]] [less than or equal to] 1 for all k [member of] N. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This completes the proof.

Theorem 3.7. The space [[GAMMA].sub.[pi]](p, [sigma], q, s) are not convergence free in general.

The proof follows from the following example.

Example 3.1. Let s = 0; [p.sub.k] = 1 for k even and [p.sub.k] = 2 for k odd. Let X = C, q(x) = [absolute value of x] and [sigma](n) = n + 1 for all n [member of] N. Then we have [[sigma].sup.2](n) = [sigma]([sigma](n)) = [sigma](n + 1) = (n + 1) + 1 = n + 2 and [[sigma].sup.3](n) = [sigma]([[sigma].sup.2](n)) = [sigma](n + 2) = (n + 2) + 1 = n + 3. Therefore, [[sigma].sup.k](n) = (n + k) for all n, k [member of] N. Consider the sequences ([x.sub.k]) and ([y.sub.k]) defined as [x.sup.k] = [(1/k).sup.k][[pi].sub.k] and [y.sub.k] = [k.sup.k][[pi].sub.k] for all k [member of] N, i.e. [[absolute value of [x.sub.k]/[pi]].sup.1/k] = 1/k and [[absolute value of [y.sub.k]/[[pi].sub.k]].sup.1/k] = k, for all k [member of] N.

Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore ([x.sub.k]/[[pi].sub.k]) [member of] [[GAMMA].sub.[pi]](p, [sigma]). But [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence ([y.sub.k]/[[pi].sub.k]) [not member of] [[GAMMA].sub.[pi]](p, [sigma]). Hence the space [[GAMMA].sub.[pi]](p, [sigma], q, s) are not convergence free in general. This completes the proof.

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N. Subramanian ([dagger]), P. Thirunavukarasu ([double dagger]) and P. Anbalagan #

([dagger]) Department of Mathematics, SASTRA University, Thanjavur-613401, India

([double dagger]) P. G. and Research Department of Mathematics, Periyar E. V. R. College (Autonomous), Trichy-620023, India

# Department of Mathematics, Government ArtsCollege, Trichy-620022, Tamil Nadu, India

E-mail: nsmaths@yahoo.com ptavinash1967@gmail.com anbalaganmaths@gmail.com
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