# Erratum to: Cesàro summable difference sequence space

• The original article was published in Journal of Inequalities and Applications 2013 2013:315

## Abstract

Theorem 3.7 of Bhardwaj and Gupta, Cesàro summable difference sequence space, J. Inequal. Appl. 2013:315, 2013, is incorrect as it stands. The corrected version of this theorem is given here.

MSC:40C05, 40A05, 46A45.

In , Bhardwaj and Gupta have introduced the Cesàro summable difference sequence space $C 1 (Δ)$ as the set of all complex sequences $x=( x k )$ with $( x k − x k + 1 )∈ C 1$, where $C 1$ is the linear space of all $(C,1)$ summable sequences.

Unfortunately, Theorem 3.7 of  is incorrect, as it stands. Consequently the assertions of Corollaries 3.8 and 3.9 of  remain actually open. The corrected version of Theorem 3.7 of  is obtained here as Corollary 2 to Theorem 1, which is itself a negation of Corollary 3.8 of . Finally Corollary 3.9 of  is proved as Theorem 3.

It is well known that $C 1$ is separable (see, for example, Theorem 4 of ). In view of the fact [, Theorem 3] that ‘if a normed space X is separable, then so is $X(Δ)$’, it follows that Theorem 3.7 of  is untrue. The mistake lies in the third line of the proof where it is claimed that A is uncountable. In fact, A is countable.

The following theorem provides a Schauder basis for $C 1 (Δ)$ and hence negates Corollary 3.8 of .

Theorem 1 $C 1 (Δ)$ has Schauder basis namely ${ e ¯ ,e, e 1 , e 2 ,…}$, where $e ¯ =(0,1,2,3,…)$, $e=(1,1,1,…)$ and $e k =(0,0,0,…,1,0,0,…)$, 1 is in the kth place and 0 elsewhere for $k=1,2,…$ .

Proof Let $x=( x k )∈ C 1 (Δ)$ with $1 k ∑ i = 1 k Δ x i →ℓ$, i.e., $lim k 1 k ( x 1 − x k + 1 )=ℓ$. We have

so that $x= x 1 e−ℓ e ¯ + ∑ k ( x k − x 1 +(k−1)ℓ) e k$. If also we had $x=ae+b e ¯ + ∑ k a k e k$, then

But for all $n∈N$, $| x 1 −a− a 1 |≤ ∥ s n ∥ Δ$, $| k b − x k + 1 + x 1 + a k + 1 − a 1 k |≤ ∥ s n ∥ Δ$ for $1≤k≤n−1$ and $| − a 1 + k ( ℓ + b ) k |≤ ∥ s n ∥ Δ$ for all $k≥n$. Letting $n→∞$, we see that $x 1 =a$, $b=−ℓ$, $a 1 =0$ and $a k + 1 = x k + 1 −kb− x 1 + a 1 =kℓ− x 1 + x k + 1$, for $k≥1$, so that the representation $x= x 1 e−ℓ e ¯ + ∑ k ( x k − x 1 +(k−1)ℓ) e k$ is unique. □

The following is a correction of Theorem 3.7 of .

Corollary 2 $C 1 (Δ)$ is separable.

The result follows from the fact that if a normed space has a Schauder basis, then it is separable.

Finally, we prove a theorem which is in fact Corollary 3.9 of .

Theorem 3 $C 1 (Δ)$ does not have the AK property.

Proof Let $x=( x k )=(1,2,3,…)∈ C 1 (Δ)$. Consider the n th section of the sequence $( x k )$ written as $x [ n ] =(1,2,3,…,n,0,0,…)$. Then

$∥ x − x [ n ] ∥ Δ = ∥ ( 0 , 0 , 0 , … , n + 1 , n + 2 , … ) ∥ Δ = sup k ≥ n | 0 − ( k + 1 ) k | = 1 + 1 n$

which does not tend to 0 as $n→∞$. □

## References

1. 1.

Bhardwaj VK, Gupta S: Cesàro summable difference sequence space. J. Inequal. Appl. 2013., 2013: Article ID 315

2. 2.

Bennet G: A representation theorem for summability domains. Proc. Lond. Math. Soc. 1972, 24: 193-203.

3. 3.

Çolak R: On some generalized sequence spaces. Commun. Fac. Sci. Univ. Ank. Ser. 1989, 38: 35-46.

## Author information

Authors

### Corresponding author

Correspondence to Vinod K Bhardwaj.

The online version of the original article can be found at 10.1186/1029-242X-2013-315

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Bhardwaj, V.K., Gupta, S. Erratum to: Cesàro summable difference sequence space. J Inequal Appl 2014, 11 (2014). https://doi.org/10.1186/1029-242X-2014-11 