International OPEN

Journal

ACCESS

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 }n1 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 }n1

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

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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 }n1 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 }n1 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 }n1 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 }n1 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 }n1 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 }n1 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 }n1 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 }n1 is a Cauchy sequence in L , then for every y L the real sequence {|| xn , y ||} n 1 is convergent. ii) If {xn }n1 and { yn } n 1 are Cauchy sequences in L and {n }n 1 is Cauchy sequence in R , then

{xn yn }n1 and {n xn }n1 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 }n1 be a Cauchy sequence in L and {xmk }k1 be a subsequence of {xn }n1 . If lim xmk x , then lim xn x . k

n

Proof. Let y L and 0 is given. The sequence {xn }n1 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 }n1 and { yn } n 1 are Cauchy sequences in L , then for every z L the sequence {( xn , yn | z )}n1 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 )}n1 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 }n1 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 }n1 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 }n1 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 }n1 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 }m1

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 }n1 be a sequence in L and for each y L holds

n 1

n 1

|| xn , y || . Then the sequence {sm }m1 of partial sums of the series xn is Cauchy in

L , because of

|| smk sm , y |||| xm1 xm2 ... xmk , y |||| xm1, y || || xm2 , y || ... || xmk , 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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