# On the Statistical Convergence of Order α in Paranormed Space

## Abstract

**:**

## 1. Introduction

- (i)
- $g\left(x\right)=0$ if $x=\theta ,$
- (ii)
- $g\left(-x\right)=g\left(x\right),$
- (iii)
- $g\left(x+y\right)\le g\left(x\right)+g\left(y\right),$
- (iv)
- if $\left({\alpha}_{n}\right)$ is a sequence of scalars with ${\alpha}_{n}\to {\alpha}_{0}$ $\left(n\to \infty \right)$ and ${x}_{n}$, $a\in X$ with ${x}_{n}\to a$ $\left(n\to \infty \right)$ in the sense that $g\left({x}_{n}-a\right)\to 0$ $\left(n\to \infty \right)$, then ${\alpha}_{n}{x}_{n}\to {\alpha}_{0}a$ $\left(n\to \infty \right)$, in the sense that $g\left({\alpha}_{n}{x}_{n}-{\alpha}_{0}a\right)\to 0$ $\left(n\to \infty \right)$.

**Definition**

**1.**

## 2. Main Results

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

- (i)
- If $\alpha =\beta $, then ${S}^{\alpha}\left(g\right)={S}^{\beta}\left(g\right)$.
- (ii)
- If $\alpha =1$, then ${S}^{\alpha}\left(g\right)=S\left(g\right)$.

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Remark**

**4.**

**Theorem**

**6.**

**Proof.**

**Remark**

**5.**

- (i)
- If $\alpha =\beta $, then ${w}_{p}^{\alpha}\left(g\right)={w}_{p}^{\beta}\left(g\right)$.
- (ii)
- ${w}_{p}^{\alpha}\left(g\right)\subseteq {w}_{p}^{\beta}\left(g\right)$ for each $\alpha \in \left(0,1\right]$ and $0<p<\infty $.

**Theorem**

**7.**

**Proof.**

**Remark**

**6.**

**Theorem**

**8.**

**Proof.**

**Remark**

**7.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Remark**

**8.**

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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ERCAN, S.
On the Statistical Convergence of Order *α* in Paranormed Space. *Symmetry* **2018**, *10*, 483.
https://doi.org/10.3390/sym10100483

**AMA Style**

ERCAN S.
On the Statistical Convergence of Order *α* in Paranormed Space. *Symmetry*. 2018; 10(10):483.
https://doi.org/10.3390/sym10100483

**Chicago/Turabian Style**

ERCAN, Sinan.
2018. "On the Statistical Convergence of Order *α* in Paranormed Space" *Symmetry* 10, no. 10: 483.
https://doi.org/10.3390/sym10100483