# Erratum to: Cesàro summable difference sequence space

The Original Article was published on 05 July 2013

## 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 [1], Bhardwaj and Gupta have introduced the Cesàro summable difference sequence space ${C}_{1}\left(\mathrm{\Delta }\right)$ as the set of all complex sequences $x=\left({x}_{k}\right)$ with $\left({x}_{k}-{x}_{k+1}\right)\in {C}_{1}$, where ${C}_{1}$ is the linear space of all $\left(C,1\right)$ summable sequences.

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

It is well known that ${C}_{1}$ is separable (see, for example, Theorem 4 of [2]). In view of the fact [[3], Theorem 3] that ‘if a normed space X is separable, then so is $X\left(\mathrm{\Delta }\right)$’, it follows that Theorem 3.7 of [1] 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 ${C}_{1}\left(\mathrm{\Delta }\right)$ and hence negates Corollary 3.8 of [1].

Theorem 1 ${C}_{1}\left(\mathrm{\Delta }\right)$ has Schauder basis namely $\left\{\overline{e},e,{e}_{1},{e}_{2},\dots \right\}$, where $\overline{e}=\left(0,1,2,3,\dots \right)$, $e=\left(1,1,1,\dots \right)$ and ${e}_{k}=\left(0,0,0,\dots ,1,0,0,\dots \right)$, 1 is in the kth place and 0 elsewhere for $k=1,2,\dots$ .

Proof Let $x=\left({x}_{k}\right)\in {C}_{1}\left(\mathrm{\Delta }\right)$ with $\frac{1}{k}{\sum }_{i=1}^{k}\mathrm{\Delta }{x}_{i}\to \ell$, i.e., ${lim}_{k}\frac{1}{k}\left({x}_{1}-{x}_{k+1}\right)=\ell$. We have

so that $x={x}_{1}e-\ell \overline{e}+{\sum }_{k}\left({x}_{k}-{x}_{1}+\left(k-1\right)\ell \right){e}_{k}$. If also we had $x=ae+b\overline{e}+{\sum }_{k}{a}_{k}{e}_{k}$, then

But for all $n\in \mathbb{N}$, $|{x}_{1}-a-{a}_{1}|\le {\parallel {s}_{n}\parallel }_{\mathrm{\Delta }}$, $|\frac{kb-{x}_{k+1}+{x}_{1}+{a}_{k+1}-{a}_{1}}{k}|\le {\parallel {s}_{n}\parallel }_{\mathrm{\Delta }}$ for $1\le k\le n-1$ and $|\frac{-{a}_{1}+k\left(\ell +b\right)}{k}|\le {\parallel {s}_{n}\parallel }_{\mathrm{\Delta }}$ for all $k\ge n$. Letting $n\to \mathrm{\infty }$, we see that ${x}_{1}=a$, $b=-\ell$, ${a}_{1}=0$ and ${a}_{k+1}={x}_{k+1}-kb-{x}_{1}+{a}_{1}=k\ell -{x}_{1}+{x}_{k+1}$, for $k\ge 1$, so that the representation $x={x}_{1}e-\ell \overline{e}+{\sum }_{k}\left({x}_{k}-{x}_{1}+\left(k-1\right)\ell \right){e}_{k}$ is unique. □

The following is a correction of Theorem 3.7 of [1].

Corollary 2 ${C}_{1}\left(\mathrm{\Delta }\right)$ 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 [1].

Theorem 3 ${C}_{1}\left(\mathrm{\Delta }\right)$ does not have the AK property.

Proof Let $x=\left({x}_{k}\right)=\left(1,2,3,\dots \right)\in {C}_{1}\left(\mathrm{\Delta }\right)$. Consider the n th section of the sequence $\left({x}_{k}\right)$ written as ${x}^{\left[n\right]}=\left(1,2,3,\dots ,n,0,0,\dots \right)$. Then

$\begin{array}{rcl}{\parallel x-{x}^{\left[n\right]}\parallel }_{\mathrm{\Delta }}& =& {\parallel \left(0,0,0,\dots ,n+1,n+2,\dots \right)\parallel }_{\mathrm{\Delta }}\\ =& \underset{k\ge n}{sup}|\frac{0-\left(k+1\right)}{k}|\\ =& 1+\frac{1}{n}\end{array}$

which does not tend to 0 as $n\to \mathrm{\infty }$. □

## References

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

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

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