Ratio Mathematica                                                                            Volume 44, 2022 

 

 
 

 

 

Relationship Between Weight Function and  

1 – Norm 
 

 

M. Melna Frincy* 

J. Robert Victor Edward† 

 

 
Abstract 

The  function on a subset  of  is the function defined by  

For 𝑥  , we define  . The Hamming 

weight   of  is the number of non – zero coordinates of , where  From 

this one could see that  , where  is the 1 – norm of  given 

by where  . This gives a relationship between the 

weight function and the 1 – norm.In this paper we establish certain properties of the 

weight function using the properties of norms. 

 

Keywords:mininorm, mininormed space, 1-norm, weight function. 

 

2010 AMS subject classification: 90B06‡ 
 

 

 

 

 

 

 
*Research Scholar, Department of Mathematics, Scott Christian College(Autonomous) 

Nagercoil- 629003, TamilNadu, India. Affiliated to ManonmaniumSundaranar University, Tirunelveli – 

627012, Tamilnadu, India.E- mail: melnabensigar84@gmail.com. 
†Department of Mathematics, Scott Christian 

College(Autonomous)Nagercoil629003,TamilNadu,IndiaAffiliated to ManonmaniumSundaranar 

University, Tirunelveli – 627012, Tamilnadu, India.E- mail: jrvedward@gmail.com. 
‡Received on June 24, 2022. Accepted on Aug 10th , 2022. Published on Nov 30th, 2022. doi: 

10.23755/rm.v44i0.914. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published 

under the CC-BY licence agreement. 

 

268

mailto:melnabensigar84@gmail.com
mailto:jrvedward@gmail.com


M. Melna Frincy and Dr. J. R. V. Edward 

 

1. Introduction 
Let , where is the set of real numbers.  Then,  is a vector space over  of 

dimension   The Hamming weight function on  is the function  given 

by  number of non –zero co-ordinates of  

For . 

Thus  satisfies the conditions: 

 for all  and  if and only if (1) 

 for all  and .           (2) 

 for all (3) 

 

A norm on  is a function   satisfying for all and if and 

only if (4) 

 for all  and .   (5) 

  and          for all (6) 

We see that  satisfies the condition of a norm except the condition (5).  Instead,  it 

satisfies (2). We may call such a function a mininorm.  Let us formalize the definition. 

 

Definition:1.1Let  be a vector space over or   A mininorm on  is a function 

  satisfying the following conditions: 

for all  and   if and only if (7) 

 for all  and (8) 

 for all (9) 

 a vector space with a mininorm defined on it is called a mininormed spaces.  

It is clear that  is a mininorm on  

 

2.The weight function and the 1- norm 

The  1- norm or on   is defined by  

 , where (10) 

We cannot connect the weight function with  the 1- norm using the  - function. 

 The  – function on   is defined by  (11) 

The  – function can be extended to  in the following way: 

, (12) 

 where   

This  – function satisfies the following  

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Relationship  between Weight Function and 1- Norm 

 

 

for all  and   if and only if (13) 

  for all  and (14) 

 and      for all (15) 

Hence the partial order relation  on  is defined as follows: 

For  and  in  

 if and only if (16) 

Now let  

Then,  

Now,  

Hence  = number of non- zero components of  

Thus, (17) 

This gives the connection between the Hamming weight function and the 1- norm, via 

the 

 – function. 

 

3.Topological Properties of the  – function 

Proposition:3.1The  – function on  is bounded. 

Proof:Let  

 

  

 since for all  

Hence  is bounded. 

 

Proposition:3.2The  – function on  is not continuous. 

Proof:First we show that  is not continuous, 

Let  

Then  as  

That is,  in  with . 

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M. Melna Frincy and Dr. J. R. V. Edward 

 

But  for all n. 

So,  

Hence  is not continuous. 

Now for all  

Thus,  where  denotes the composition of functions. 

 is continuous . 

Suppose  is continuous. 

Hence  is  continuous, since the composition of two continuous  functions is 

continuous. 

This is not possible. 

Hence is  not continuous  

 

Acknowledgements 

The authors thank the referees for their valuable suggestions and comments.  

 

References 

 
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[4] B.V.Limaye – Functional Analysis, New Age International Publishers, New Delhi, 

1996. 

[5] G. F. Simmons – Introduction to Topology and Modern Analysis. Mc – Graw Hill, 

Tokyo, 1963. 

 

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