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International OPEN

Journal

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Of Modern Engineering Research (IJMER)

About the 2-Banach Spaces Risto Malčeski1, Katerina Anevska2 1, 2

(Faculty of informatics/ FON University, Skopje, Macedonia)

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.

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) || x, y || 0 , for each x, y  L и || x, y || 0 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 n  1 be a natural number, L be a real vector space, dim L  n and (,  | ) be real function on L  L  L such that: i) ( x, x | y )  0 , for each x, y  L и ( x, x | y)  0 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) ( x  x1, y | z)  ( x, y | z)  ( x1, y | z) , for each x1, 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

|| x, y || ( x, x | y )1/2 , for each x, y  L defines 2-norm. So, we get 2-normed vector space ( L,|| ,  ||) .

(1)

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 {xn }n1 into vector 2-normed space is called as convergent if there exists x  L such that lim || xn  x, y || 0 , for every y  L . Vector x  L is called as bound of the n

sequence

{xn }n1

and we note lim xn  x or xn  x , n   , ([4]). n

Theorem 1 ([4]). Let ( L,|| ,  ||) be a 2-normed space over L . а) If lim xn  x , lim yn  y , lim n   and lim n   , then n

n

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n

n

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About the 2-Banach Spaces lim (n xn  n yn )   x   y .

n

б) If lim xn  x and lim xn  y , then x  y . ■ n

n

Theorem 2 ([4]). а) Let L be a vector 2-normed space, {xn }n1 be the sequence in L and y  L is such that lim || xn  xm , y || 0 . Then the real sequence {|| xn  x, y ||} n 1 is convergent for every x  L . n

b) Let L be a vector 2-normed space, {xn }n1 be a sequence in L and y  L is such that lim || xn  x, y || 0 . Then lim || xn , y |||| x, y || . ■

n

n

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 {xn }n1 in the 2-pre-Hilbert space ( L,(,  | )) is called as weakly convergent if there exists x  L such that lim ( xn  x, y | z )  0 , for every y, z  L . The vector x  L is n

w

called as weak limit of the sequence {xn }n1 and is denoted as xn  x , n   . w

w

Theorem 3. Let ( L,(,  | )) be a 2-pre-Hilbert space. If xn  x , yn  y , n   and n   , for w

n   , then n xn  n yn   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 (n xn  n yn   x   y, u | v )  lim (n xn   x, u | v )  lim (  n yn   y, u | v ) n

n

n

 lim n ( xn  x, u | v )  lim (n   )( x, u | v)  n

n

 lim n ( yn  y, u | v)  lim ( n   )( y, u | v) . ■ n

n

w

w

Corollary 1 ([5]). Let ( L,(,  | )) be a 2-pre-Hilbert space. If xn  x , yn  y , for n   and w

 ,  R , then  xn   yn  x   y , for n   . Proof. The validity of this Corollary is as a direct implication of Theorem 3 for n   , n   , n  1 . ■ Lemma 1 ([5]). Let ( L,(,  | )) be a 2-pre-Hilbert space. w

а) If xn  x , n   , then xn  x , n   . w

w

б) If xn  x and xn  x ' , n   , then x  x ' . ■ Theorem 4. Let ( L,(,  | )) be a real 2-pre-Hilbert space. If xn  x , yn  y , n   and n   , for n   , then

lim (n xn , n yn | z )  ( x,  y | z ) , for every z  L .

n

Proof. Let xn  x , yn  y , n   and n   , for n   . Then, Theorem 1 а) implies lim || n xn   x, z || 0 , lim || n yn   y , z || 0 , for every z  L . n

n

(2)

(3)

Properties of 2-inner product and the Cauchy-Buniakowsky-Schwarz inequality imply 0 | ( n xn ,  n yn | z )  ( x,  y | z ) | | ( n xn ,  n yn | z )  ( n xn ,  y | z )  (n xn ,  y | z )  ( x,  y | z ) | | ( n xn ,  n yn | z )  ( n xn ,  y | z ) |  | ( n xn ,  y | z )  ( x,  y | z ) | | ( n xn ,  n yn   y | z ) |  | (n xn   x,  y | z ) | ||  n xn , z ||  ||  n yn   y, z ||  ||  n xn   x, z ||  ||  y, z || .

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About the 2-Banach Spaces 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 xn  x and yn  y , for n   , then lim ( xn , yn | z )  ( x, y | z) , for every z  L . n

Proof. The validity of this Corollary is as a direct implication of Theorem 4, n  n  1, n  1 . ■ Theorem 5. Let {xn }n1 be a sequence in 2-pre-Hilbert space L . If lim || xn , y |||| x, y || and lim ( xn , x | y )  ( x, x | y ) , for every y  L , n

n

then lim xn  x . n

Proof. Let y  L . Then || xn  x, y ||2  ( xn  x, xn  x | y )  ( xn , xn | y )  2( xn , x | y )  ( x, x | y ) || xn , y ||2 2( xn , x | y )  || x, y ||2 ,

So, lim || xn  x, y ||2  lim || xn , y ||2 2 lim ( xn , x | y )  || x, y ||2

n

n

n

 2 || x, y || 2( x, x | y )  0. Finally, arbitrariness of y implies xn  x , n   . ■ 2

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 {xn }n1 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 || xn  xm , y || 0 and

m,n

lim || xn  xm , z || 0 .

m,n

(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 {xn }n1 in a vector 2-normed space L is Cauchy sequence if

lim || xn  xm , y || 0 , for every y  L .

(4)

m,n

Theorem 6. Let L be a vector 2-normed space. i) If {xn }n1 is a Cauchy sequence in L , then for every y  L the real sequence {|| xn , y ||} n 1 is convergent.  ii) If {xn }n1 and { yn } n 1 are Cauchy sequences in L and {n }n 1 is Cauchy sequence in R , then

{xn  yn }n1 and {n xn }n1 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, {xn }n1 be a Cauchy sequence in L and {xmk }k1 be a subsequence of {xn }n1 . If lim xmk  x , then lim xn  x . k 

n

Proof. Let y  L and   0 is given. The sequence {xn }n1 is Cauchy, and thus exist n1  N such that || xn  x p , y || 2 , for n, p  n1 . Further, lim xmk  x , so exist n2  N such that || xmk  x, y || 2 , for k 

mk  n2 . Letting n0  max{n1, n2} we get that for n, mk  n0 is true that || xn  x, y |||| xn  xmk , y ||  || xmk  x, y || 2  2   ,

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About the 2-Banach Spaces The arbitrariness of y  L implies lim xn  x . ■ n

Theorem 8. Let L be a 2-pre-Hilbert space. If {xn }n1 and { yn } n 1 are Cauchy sequences in L , then for every z  L the sequence {( xn , yn | z )}n1 is Cauchy in R . Proof. Let z  L . The properties of inner product and Cauchy-Buniakovski-Swartz inequality imply | ( xn , yn | z )  ( xm , ym | z ) || ( xn , yn | z )  ( xm , yn | z ) |  | ( xm , yn | z)  ( xm , ym | z ) | | ( xn  xm , yn | z ) |  | ( xm , yn  ym | z ) | || xn  xm , z ||  || yn , z ||  || xm , z ||  || yn  ym , z || . Finally, by the last inequality, equality (4) and Theorem 6 i) is true that for given z  L the sequence

{( xn , yn | z )}n1 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, {xn }n1 be a Cauchy sequence in L and a  L . By dim L  1 , exist b  L such that the set {a, b} is linearly independent. By Theorem 1, [7],

|| x ||a,b,2  (|| x, a ||2  || x, b ||2 )1/2 , x  L

(5)

defines norm in L , and by Theorem 6, [7] the sequence {xn }n1 is Cauchy into the space ( L,||  ||a,b,2 ) . But normed space L with finite dimension is Banach, so exist x  L such that lim || xn  x ||a,b,2  0 . So, by (5) n

we get

lim || xn  x, a || lim || xn  x, b || 0 .

(6)

lim || xn  x, y || 0 .

(7)

n

n

We’ll prove that for every y  L , n

Clearly, if y   a , then (6) and the properties of 2-norm imply (7). Let the set { y, a} be linearly independent. Then ( L,||  || y,a,2 ) is a Banach space, and moreover because of the sequence {xn }n1 is Cauchy sequence in ( L,||  || y,a,2 ) , exist x '  L such that

lim || xn  x ' || y,a,2  0 .

(8)

n

But in space with finite dimension each two norms are equivalent (Theorem 2, [8], pp. 29), and thus exist m, M  0 such that m || xn  x ' || y,a,2 || xn  x ' ||a,b,2  M || xn  x ' || y ,a,2 , This implies lim || xn  x ' ||a,b,2  0 and moreover lim || xn  x ||a,b,2  0 . So, we may conclude x '  x . By (8), n

n

lim || xn  x || y,a,2  0 , and thus (7) holds. Finally the arbitrariness of y implies the validity of the theorem i.e.

n

the proof of the theorem is completed. ■ 

Definition 3. Let L be a 2-normed space and {xn }n1 be a sequence in L . The series  xn is n 1

m

  called as summable in L if the sequence of its partial sums {sm }m 1 , sm   xn converges in L . If {sm }m1

n 1

n 1

n 1

converges to s , then s is called as a sum of the series  xn , and is noted as s   xn 

n 1

n 1

The series  xn is called as absolutely summable in L if for each y  L holds  || xn , 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 . | IJMER | ISSN: 2249–6645 |

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About the 2-Banach Spaces Proof. Let L be a 2-Banach space, {xn }n1 be a sequence in L and for each y  L holds 

n 1

n 1

 || xn , y ||   . Then the sequence {sm }m1 of partial sums of the series  xn is Cauchy in

L , because of

|| smk  sm , y |||| xm1  xm2  ...  xmk , y |||| xm1, y ||  || xm2 , y || ... || xmk , y || ,  for every y  L and for every m, k  1 . But, L is a 2-Banach space, and thus the sequence {sm }m 1 converge

in L . That means, the series  xn is summable in L . n 1

Conversely, let each absolutely summable series be summable in L . Let {zn } n 1 is a Cauchy sequence in L and y  L . Let m1  N is such that || zm  zm1 , y || 1 , for m  m1 . By induction we determine m2 , m3 ,... such that mk  mk 1 and for each m  mk holds || zm  zmk , y || 12 . Letting xk  zmk 1  zmk , k 

k  1, 2,... , we get  || xk , y ||  

1 2 k k 1

k 1

  . The arbitrarily of y  L implies the series  xk is absolutely k 1 k 1

summable. Thus, by assumption, the series  xk is summable in L . But, zmk  zm1   xi . Hence, the k 1

subsequence {zn } n 1

{zmk } k 1

of the Cauchy sequence

{zn } n 1

i 1

converge in L . Finally, by Theorem 7, the sequence

converge in L . It means L is 2-Banach space. ■

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

S. Gähler, Lineare 2-normierte Räume, Math. Nachr. 28, 1965, 1-42 C. Diminnie, S. Gähler, and A. White, 2-Inner Product Spaces, Demonstratio Mathematica, Vol. VI, 1973, 169-188 R. Ehret, Linear 2-Normed Spaces, doctoral diss., Saint Louis Univ., 1969 A. White, 2-Banach Spaces, Math. Nachr. 1969, 42, 43-60 H. Gunawan, On Convergence in n-Inner Product Spaces, Bull. Malaysian Math. Sc. Soc. (Second Series) 25, 2002, 11-16 H. Dutta, On some 2-Banach spaces, General Mathematics, Vol. 18, No. 4, 2010, 71-84 R. Malčeski, K. Anevska, Families of norms generated by 2-norm, American Journal of Engineering Research (AJER), e-ISSN 2320-0847 p-ISSN 2320-0936, Volume-03, Issue-05, pp-315-320 S. Kurepa, Functional analysis, (Školska knjiga, Zagreb, 1981) A. Malčeski, R. Malčeski, Convergent sequences in the n-normed spaces, Matematički bilten, 24 (L), 2000, 47-56 (in macedonian) A. Malčeski, R. Malčeski, n-Banach Spaces, Zbornik trudovi od vtor kongres na SMIM, 2000, 77-82, (in macedonian) C. Diminnie, S. Gähler, and A. White, 2-Inner Product Spaces II, Demonstratio Mathematica, Vol. X, No 1, 1977, 525-536 R. Malčeski, Remarks on n-normed spaces, Matematički bilten, Tome 20 (XLVI), 1996, 33-50, (in macedonian) R. Malčeski, A. Malčeski, A. n-seminormed space, Godišen zbornik na PMF, Tome 38, 1997, 31-40, (in macedonian) S. Gähler, and Misiak, A. Remarks on 2-Inner Product, Demonstratio Mathematica, Vol. XVII, No 3, 1984, 655670 W. Rudin, Functional Analysis, (2nd ed. McGraw-Hill, Inc. New York, 1991)

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