A metric formula on a quotient space which is related to the sequence space $\Sigma _{2}$

Mustafa Saltan,Nisa Aslan

doi:10.31801/cfsuasmas.729550

Abstract

In this paper, we first define an equivalence relation on the sequence space $\Sigma _{2}$. Then we equip the quotient set $\Sigma _{2}/_{\sim}$ with a metric $d_1$. We also determine an isometry map between the metric spaces $(\Sigma _{2}/_{\sim},d_1)$ and $([0,1],d_{eucl})$. Finally, we investigate the symmetry conditions with respect to some points on the metric space $(\Sigma _{2}/_{\sim},d_1)$ and we compare truncation errors for the computations which is obtained by the metrics $d_{eucl}$ and $d_1$.

Highlights

Let n denote the set of all in...nite sequences of 0’s, 1’s, : : :, n 1’s
We give a metric formula on the quotient space 2=
Metric formulas on quotient spaces, which are related to the sequence spaces n for every natural n, can be de...ned (for an example see ( [6])

Summary

Introduction

Let n denote the set of all in...nite sequences of 0’s, 1’s, : : :, n 1’s. That is, n = fs1s2s3 : : : jsi 2 f0; 1; : : : ; n 1gg (see [1, 2, 8]). We determine an isometry map between the metric spaces ( 2= ; d1) and ([0; 1]; deucl). We investigate the symmetry conditions with respect to some points on the metric space ( 2= ; d1) and we compare truncation errors for the computations which is obtained by the metrics deucl and d1. To obtain an injective mapping, an equivalence relation on 2 is de...ned by s0 s00 , s0 = s00 or there are si; ; 2 f0; 1g such that s0 = s1s2 : : : sn ; s00 = s1s2 : : : sn for an integer n: (3)

Results

Conclusion