1) The document discusses concepts related to 2-normed spaces and 2-pre-Hilbert spaces, which generalize norms, inner products, and Banach spaces to two dimensions. 2) It defines key concepts such as 2-norms, 2-inner products, convergent sequences, and Cauchy sequences in 2-normed and 2-pre-Hilbert spaces. 3) Theorems and corollaries are presented characterizing properties of convergent sequences in 2-normed and 2-pre-Hilbert spaces, including their relationship to weakly convergent sequences.

1. International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 5| May. 2014 | 28|
About the 2-Banach Spaces
Risto Malčeski1
, Katerina Anevska2
1, 2
(Faculty of informatics/ FON University, Skopje, Macedonia)
I. INTRODUCTION
Let L be a real vector space with dimension greater than 1 and, || , ||  be a real function defined on
L L which satisfies the following:
a) || , || 0x y  , for each ,x y L и || , || 0x y  if and only if the set { , }x y is linearly dependent,
b) || , || || , ||,x y y x for each ,x y L ,
c) || , || | | || , ||,x y x y   for each ,x y L and for each , R
d) || , || || , || || , ||,x y z x z y z   for each , , .x y z L
Function || , ||  is called as 2-norm of L, and ( ,|| , ||)L   is called as vector 2-normed space ([1]).
Let 1n  be a natural number, L be a real vector space, dimL n and ( , | )   be real function on
L L L  such that:
i) ( , | ) 0x x y  , for each ,x y L и ( , | ) 0x x y  if and only if a and b are linearly dependent,
ii) ( , | ) ( , | )x y z y x z , for each , ,x y z L ,
iii) ( , | ) ( , | )x x y y y x , for each ,x y L ,
iv) ( , | ) ( , | )x y z x y z  , for each , ,x y z L and for each  R , and
v) 1 1( , | ) ( , | ) ( , | )x x y z x y z x y z   , for each 1, , ,x x y z L .
Function ( , | )   is called as 2-inner product, and ( ,( , | ))L    is called as 2-pre-Hilbert space ([2]).
Concepts of a 2-norm and a 2-inner product are two dimensional analogies of concepts of a norm and
an inner product. R. Ehret ([3]) proved that, if ( ,( , | ))L    be 2-pre-Hilbert space, than
1/2
|| , || ( , | )x y x x y , for each ,x y L (1)
defines 2-norm. So, we get 2-normed vector space ( ,|| , ||)L   .
II. CONVERGENT AND CAUCHY SEQUENCES IN 2-PRE-HILBERT SPACE
The term convergent sequence in 2-normed space is given by A. White, and he also proved a few
properties according to this term. The sequence 1{ }n nx 
 into vector 2-normed space is called as convergent if
there exists x L such that lim || , || 0n
n
x x y

  , for every y L . Vector x L is called as bound of the
sequence 1{ }n nx 
 and we note lim n
n
x x

 or nx x , n  , ([4]).
Theorem 1 ([4]). Let ( ,|| , ||)L   be a 2-normed space over L .
а) If lim n
n
x x

 , lim n
n
y y

 , lim n
n
 

 and lim n
n
 

 , then
Abstract: In this paper are proved a few properties about convergent sequences into a real 2-normed
space ( ,|| , ||)L   and into a 2-pre-Hilbert space ( ,( , | ))L    , which are actually generalization of
appropriate properties of convergent sequences into a pre-Hilbert space. Also, are given two
characterization of a 2-Banach spaces. These characterization in fact are generalization of appropriate
results in Banach spaces.
2010 Mathematics Subject Classification: Primary 46B20; Secondary 46C05.
Keywords: convergent sequence, Cauchy sequence, 2-Banach space, weakly convergent sequence,
absolutely summable series.

2. About the 2-Banach Spaces
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 5| May. 2014 | 29|
lim ( )n n n n
n
x y x y   

   .
б) If lim n
n
x x

 and lim n
n
x y

 , then x y . ■
Theorem 2 ([4]). а) Let L be a vector 2-normed space, 1{ }n nx 
 be the sequence in L and y L is
such that lim || , || 0n m
n
x x y

  . Then the real sequence 1{|| , ||}n nx x y 
 is convergent for every x L .
b) Let L be a vector 2-normed space, 1{ }n nx 
 be a sequence in L and y L is such that
lim || , || 0n
n
x x y

  . Then lim || , || || , ||n
n
x y x y

 . ■
In [7] H. Gunawan was focused on convergent sequences in 2-pre-Hilbert space ( ,( , | ))L    , and he
gave the term weakly convergent sequence in 2-pre-Hilbert space. According to this he proved a few
corollaries.
Definition 1 ([5]). The sequence 1{ }n nx 
 in the 2-pre-Hilbert space ( ,( , | ))L    is called as weakly
convergent if there exists x L such that lim ( , | ) 0n
n
x x y z

  , for every ,y z L . The vector x L is
called as weak limit of the sequence 1{ }n nx 
 and is denoted as
w
nx x , n  .
Theorem 3. Let ( ,( , | ))L    be a 2-pre-Hilbert space. If
w
nx x ,
w
ny y , n  and n  , for
n  , then
w
n n n nx y x y      , for n  .
Proof. The validity of this theorem is as a direct implication of Definition 1, the fact that every
convergent real sequence is bounded and the following equalities:
lim ( , | ) lim ( , | ) lim ( , | )
lim ( , | ) lim ( )( , | )
n n n n n n n n
n n n
n n n
n n
x y x y u v x x u v y y u v
x x u v x u v
       
  
  
 
      
    
lim ( , | ) lim ( )( , | )n n n
n n
y y u v y u v  
 
    . ■
Corollary 1 ([5]). Let ( ,( , | ))L    be a 2-pre-Hilbert space. If
w
nx x ,
w
ny y , for n   and
,  R , then
w
n nx y x y      , for n  .
Proof. The validity of this Corollary is as a direct implication of Theorem 3 for , , 1n n n      .
■
Lemma 1 ([5]). Let ( ,( , | ))L    be a 2-pre-Hilbert space.
а) If nx x , n  , then
w
nx x , n  .
б) If
w
nx x and '
w
nx x , n  , then 'x x . ■
Theorem 4. Let ( ,( , | ))L    be a real 2-pre-Hilbert space. If nx x , ny y , n  and
n  , for n  , then
lim ( , | ) ( , | )n n n n
n
x y z x y z   

 , for every z L . (2)
Proof. Let nx x , ny y , n  and n  , for n  . Then, Theorem 1 а) implies
lim || , || 0n n
n
x x z 

  , lim || , || 0n n
n
y y z 

  , for every z L . (3)
Properties of 2-inner product and the Cauchy-Buniakowsky-Schwarz inequality imply
0 |( , | ) ( , | ) |
|( , | ) ( , | ) ( , | ) ( , | ) |
|( , | ) ( , | ) | |( , | ) ( , | ) |
|( , | ) | |( , | ) |
|| , || || , || ||
n n n n
n n n n n n n n
n n n n n n n n
n n n n n n
n n n n n n
x y z x y z
x y z x y z x y z x y z
x y z x y z x y z x y z
x y y z x x y z
x z y y z x x
   
       
       
     
    
 
   
   
   
     , || || , ||.z y z

3. About the 2-Banach Spaces
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 5| May. 2014 | 30|
Finally, letting n  , in the last inequality, by Theorem 2 b) and equality (3) follows the equality (2). ■
Corollary 2 ([5]). Let ( ,( , | ))L    be a real 2-pre-Hilbert space. If nx x and ny y , for n  ,
then lim ( , | ) ( , | )n n
n
x y z x y z

 , for every z L .
Proof. The validity of this Corollary is as a direct implication of Theorem 4, 1, 1n n n    . ■
Theorem 5. Let 1{ }n nx 
 be a sequence in 2-pre-Hilbert space L . If
lim || , || || , ||n
n
x y x y

 and lim ( , | ) ( , | )n
n
x x y x x y

 , for every y L ,
then lim n
n
x x

 .
Proof. Let y L . Then
2
2 2
|| , || ( , | )
( , | ) 2( , | ) ( , | )
|| , || 2( , | ) || , || ,
n n n
n n n
n n
x x y x x x x y
x x y x x y x x y
x y x x y x y
   
  
  
So,
2 2 2
2
lim || , || lim || , || 2 lim ( , | ) || , ||
2 || , || 2( , | ) 0.
n n n
n n n
x x y x y x x y x y
x y x x y
  
   
  
Finally, arbitrariness of y implies nx x , n  . ■
A. White in [4] gave the term Cauchy sequence in a vector 2-normed space L of linearly independent
vectors ,y z L , i.e. the sequence 1{ }n nx 
 in a vector 2-normed space L is Cauchy sequence, if there exist
,y z L such that y and z are linearly independent and
,
lim || , || 0n m
m n
x x y

  and
,
lim || , || 0n m
m n
x x z

  . (4)
Further, White defines 2-Banach space, as a vector 2-normed space in which every Cauchy sequence is
convergent and have proved that each 2-normed space with dimension 2 is 2-Banach space, if it is defined
over complete field. The main point in this state is linearly independent of the vectors y and z , and by
equalities (4) are sufficient for getting the proof. Later, this definition about Cauchy sequence into 2-normed
space is reviewed, and the new definition is not contradictory with the state that 2-normed space with
dimension 2 is 2-Banach space.
Definition 2 ([6]). The sequence 1{ }n nx 
 in a vector 2-normed space L is Cauchy sequence if
,
lim || , || 0n m
m n
x x y

  , for every y L . (4)
Theorem 6. Let L be a vector 2-normed space.
i) If 1{ }n nx 
 is a Cauchy sequence in L , then for every y L the real sequence 1{|| , ||}n nx y 
 is
convergent.
ii) If 1{ }n nx 
 and 1{ }n ny 
 are Cauchy sequences in L and 1{ }n n 
 is Cauchy sequence in R , then
1{ }n n nx y 
 and 1{ }n n nx 
 are Cauchy sequences in L .
Proof. The proof is analogous to the proof of Theorem 1.1, [4]. ■
Theorem 7. Let L be a vector 2-normed space, 1{ }n nx 
 be a Cauchy sequence in L and 1{ }km kx 
 be
a subsequence of 1{ }n nx 
 . If lim km
k
x x

 , then lim n
n
x x

 .
Proof. Let y L and 0  is given. The sequence 1{ }n nx 
 is Cauchy, and thus exist 1n N such
that 2
|| , ||n px x y   , for 1,n p n . Further, lim km
k
x x

 , so exist 2n N such that 2
|| , ||kmx x y   , for
2km n . Letting 0 1 2max{ , }n n n we get that for 0, kn m n is true that
2 2
|| , || || , || || , ||k kn n m mx x y x x y x x y           ,

4. About the 2-Banach Spaces
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The arbitrariness of y L implies lim n
n
x x

 . ■
Theorem 8. Let L be a 2-pre-Hilbert space. If 1{ }n nx 
 and 1{ }n ny 
 are Cauchy sequences in L ,
then for every z L the sequence 1{( , | )}n n nx y z 
 is Cauchy in R .
Proof. Let z L . The properties of inner product and Cauchy-Buniakovski-Swartz inequality imply
|( , | ) ( , | ) | |( , | ) ( , | ) | |( , | ) ( , | ) |
|( , | ) | |( , | ) |
|| , || || , || || , || || , ||.
n n m m n n m n m n m m
n m n m n m
n m n m n m
x y z x y z x y z x y z x y z x y z
x x y z x y y z
x x z y z x z y y z
    
   
     
Finally, by the last inequality, equality (4) and Theorem 6 i) is true that for given z L the sequence
1{( , | )}n n nx y z 
 is Cauchy sequence. ■
III. CHARACTERIZATION OF 2-BANACH SPACES
Theorem 9. Each 2-normed space L with finite dimension is 2-Banach space.
Proof. Let L be a 2-normed space with finite dimension, 1{ }n nx 
 be a Cauchy sequence in L and
a L . By dim 1L  , exist b L such that the set { , }a b is linearly independent. By Theorem 1, [7],
2 2 1/2
, ,2|| || (|| , || || , || )a bx x a x b  , x L (5)
defines norm in L , and by Theorem 6, [7] the sequence 1{ }n nx 
 is Cauchy into the space , ,2( ,|| || )a bL  . But
normed space L with finite dimension is Banach, so exist x L such that , ,2lim || || 0n a b
n
x x

  . So, by (5)
we get
lim || , || lim || , || 0n n
n n
x x a x x b
 
    . (6)
We’ll prove that for every y L ,
lim || , || 0n
n
x x y

  . (7)
Clearly, if y a , then (6) and the properties of 2-norm imply (7). Let the set { , }y a be linearly independent.
Then , ,2( ,|| || )y aL  is a Banach space, and moreover because of the sequence 1{ }n nx 
 is Cauchy sequence in
, ,2( ,|| || )y aL  , exist 'x L such that
, ,2lim || ' || 0n y a
n
x x

  . (8)
But in space with finite dimension each two norms are equivalent (Theorem 2, [8], pp. 29), and thus exist
, 0m M  such that
, ,2 , ,2 , ,2|| ' || || ' || || ' ||n y a n a b n y am x x x x M x x     ,
This implies , ,2lim || ' || 0n a b
n
x x

  and moreover , ,2lim || || 0n a b
n
x x

  . So, we may conclude 'x x . By (8),
, ,2lim || || 0n y a
n
x x

  , and thus (7) holds. Finally the arbitrariness of y implies the validity of the theorem i.e.
the proof of the theorem is completed. ■
Definition 3. Let L be a 2-normed space and 1{ }n nx 
 be a sequence in L . The series
1
n
n
x


 is
called as summable in L if the sequence of its partial sums 1{ }m ms 
 ,
1
m
m n
n
s x

  converges in L . If 1{ }m ms 

converges to s , then s is called as a sum of the series
1
n
n
x


 , and is noted as
1
n
n
s x


 
The series
1
n
n
x


 is called as absolutely summable in L if for each y L holds
1
|| , ||n
n
x y


  .
Theorem 10. The 2-normed space L is 2-Banach if and only if each absolutely summable series of
elements in L is summable in L .

5. About the 2-Banach Spaces
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Proof. Let L be a 2-Banach space, 1{ }n nx 
 be a sequence in L and for each y L holds
1
|| , ||n
n
x y


  . Then the sequence 1{ }m ms 
 of partial sums of the series
1
n
n
x


 is Cauchy in L , because of
1 2 1 2|| , || || ... , || || , || || , || ... || , ||m k m m m m k m m m ks s y x x x y x y x y x y               ,
for every y L and for every , 1m k  . But, L is a 2-Banach space, and thus the sequence 1{ }m ms 
 converge
in L . That means, the series
1
n
n
x


 is summable in L .
Conversely, let each absolutely summable series be summable in L . Let 1{ }n nz 
 is a Cauchy
sequence in L and y L . Let 1m N is such that 1
|| , || 1m mz z y  , for 1m m . By induction we determine
2 3, ,...m m such that 1k km m  and for each km m holds 2
1|| , ||km m
k
z z y  . Letting 1k kk m mx z z
  ,
1,2,...k  , we get 2
1
1 1
|| , ||k
kk k
x y
 
 
    . The arbitrarily of y L implies the series
1
k
k
x


 is absolutely
summable. Thus, by assumption, the series
1
k
k
x


 is summable in L . But, 1
1
1
k
k
m m i
i
z z x


   . Hence, the
subsequence 1{ }km kz 
 of the Cauchy sequence 1{ }n nz 
 converge in L . Finally, by Theorem 7, the sequence
1{ }n nz 
 converge in L . It means L is 2-Banach space. ■
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