IARJSET
International Advanced Research Journal in Science, Engineering and Technology
ISO 3297:2007 Certified Impact Factor 7.105 Vol. 9, Issue 8, August 2022
DOI: 10.17148/IARJSET.2022.9813
© IARJSET This work is licensed under a Creative Commons Attribution 4.0 International License 91
ISSN (O) 2393-8021, ISSN (P) 2394-1588
Some Generalized Normed Spaces
Characteristics
Jayashree Patil
1
, Basel Hardan
2*
, Alaa A. Abdalla
3
Department of Mathematics, Vasantrao Naik Mahavidyalaya, Cidco, Aurangabad, India
1
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India
2,3
*
Corresponding author
Abstract: In this paper, some characteristics of generalized metric spaces are presented, which will help many
researchers to prove the results when working within groups volume.
Key words: normed space, inner product space, characteristic.
MSC2010: 46A40; 46J10, 60E10.
1. INTRODUCTION
In this work, we will focus on n-normed and n -inner generated spaces, two topics that are central to the field of
functional analysis, which was first developed at the turn of the 19th century and finally established in the 1920s and
1930s.
Gahler offered a fascinating n-norm theory on linear spaces in [4]. Numerous authors, including Kim et al. [9],
Malceski [11], Misiak [12], and Gunawan [5], have developed linear n-normed spaces systematically. As of late,
Kritiantoo et al. [10] have researched the equivalence of n-norms in n-normed spaces. In a linear n-Banach space, n-
norms must satisfy certain conditions in order to be fully equivalent, and that is what this work aims to achieve. current
research on the functional analysis parts we're referring to ( [1,2], [6-8], [13,15]).
According to Chen et al. [3], the extended parallelogram law is a necessary and sufficient condition for an n-normed
space (n-Ns) to be an n-inner product space as follows:
1
2
‖ + ,
2
,…,
‖
2
+ ‖ − ,
2
,…,
‖
2
= (‖,
2
,…,
‖
2
+ ‖,
2
,…,
‖
2
) (1.1) such that the
–inner product (n-Es) space for all , ,
2
,…,
is introduced by
4⟨, |
2
,…,
⟩ = (‖ + ,
2
,…,
‖ − ‖ − ,
2
,…,
‖). (1.2)
Soenjaya [16] consider a relation (1.2) as the attribution of n-Es
We require the following definitions for this work:
Definition 1.1. [2] A real-valued function ⟨∙,∙ | ∙, … ,∙⟩ on
+1
satisfied the following properties:
I 1: ⟨
1
,
1
|
2
,…,
⟩ ≥ 0 and ⟨
1
,
1
|
2
,…,
⟩=0,
if and only if
1
,
2
…,
are linearly dependent.
I 2: ⟨
1
,
1
|
2
,…,
⟩ = ⟨
1
,
1
|
2
,…,
⟩, for any permutation (
1
,…,
) of (1, … , ).
I 3: ⟨
1
́ ,
1
|
2
,…,
⟩ = ⟨
1
,
1
́ |
2
,…,
⟩,