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# About the 2-Banach Spaces

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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

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### About the 2-Banach Spaces

1. 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. 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. 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. 4. About the 2-Banach Spaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 5| May. 2014 | 31| 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. 5. About the 2-Banach Spaces | IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss. 5| May. 2014 | 32| 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. ■ REFERENCES [1] S. Gähler, Lineare 2-normierte Räume, Math. Nachr. 28, 1965, 1-42 [2] C. Diminnie, S. Gähler, and A. White, 2-Inner Product Spaces, Demonstratio Mathematica, Vol. VI, 1973, 169-188 [3] R. Ehret, Linear 2-Normed Spaces, doctoral diss., Saint Louis Univ., 1969 [4] A. White, 2-Banach Spaces, Math. Nachr. 1969, 42, 43-60 [5] H. Gunawan, On Convergence in n-Inner Product Spaces, Bull. Malaysian Math. Sc. Soc. (Second Series) 25, 2002, 11-16 [6] H. Dutta, On some 2-Banach spaces, General Mathematics, Vol. 18, No. 4, 2010, 71-84 [7] 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 [8] S. Kurepa, Functional analysis, (Školska knjiga, Zagreb, 1981) [9] A. Malčeski, R. Malčeski, Convergent sequences in the n-normed spaces, Matematički bilten, 24 (L), 2000, 47-56 (in macedonian) [10] A. Malčeski, R. Malčeski, n-Banach Spaces, Zbornik trudovi od vtor kongres na SMIM, 2000, 77-82, (in macedonian) [11] C. Diminnie, S. Gähler, and A. White, 2-Inner Product Spaces II, Demonstratio Mathematica, Vol. X, No 1, 1977, 525-536 [12] R. Malčeski, Remarks on n-normed spaces, Matematički bilten, Tome 20 (XLVI), 1996, 33-50, (in macedonian) [13] R. Malčeski, A. Malčeski, A. n-seminormed space, Godišen zbornik na PMF, Tome 38, 1997, 31-40, (in macedonian) [14] S. Gähler, and Misiak, A. Remarks on 2-Inner Product, Demonstratio Mathematica, Vol. XVII, No 3, 1984, 655- 670 [15] W. Rudin, Functional Analysis, (2nd ed. McGraw-Hill, Inc. New York, 1991)