Celal Bayar University Journal of Science A. Mutlu Volume 14, Issue 1, p 1-12
A New Approach to G-Normed Spaces: Functionally Generalized Normed Spaces
Ali Mutlu1*, Utku Gürdal2, Kübra Özkan1
1 Manisa Celal Bayar University, Department of Mathematics, 45140, Manisa, Turkey, +90 236 201 3206 [email protected]
2 Mehmet Akif Ersoy University, Department of Mathematics, Burdur, Turkey
*Corresponding author
Received: 15 October 2017 Accepted: 15 February 2018
DOI: 10.18466/cbayarfbe.407993
Abstract In this paper, we introduce functionally generalized normed spaces as a generalization of G metric spaces and normed spaces. Some constructions are described within this structure and some related results are obtained. Keywords — completion, generalized Banach space, generalized metric space.
1 Introduction and Preliminaries linear space to a special type of trinary operation, to consi- der a norm-like function with two variables or to give an As a result of the strong link between the theories of met- additional algebraic operation on real numbers. ric spaces and normed spaces, there has been many foun- dations based on metric spaces and later adapted to nor- med structures or vice versa. Among many others, genera- lized metric spaces, widely known as G-metric spaces, In this study we introduce generalized pseudometric spa- which were first defined by Mustafa and Sims [1] as an al- ces, then we give a definition of generalized normed ternative to D metric spaces [2]; have been assumed to space, which is based on a type of norm functional with have connection to some normed-type structure, which two variables, without giving any additional algebraic would be called as generalized normed space. structure, such that the theory of normed spaces can be embedded into the theory of generalized normed spaces and every generalized normed space will have an underl- ying generalized metric space structure. We also define So far, various generalized normed space definitions were given [3,4]. There are some possible different approaches Banach spaces and construct product spaces, quoti- ent spaces and completions within the new structure. to normed spaces, change the binary operation of the
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In 2006, Mustafa and Sims introduced the notion of gene- 1 . ralized metric space, as an alternative to the notions of dxyGxyyGxxyG ,,,,, 2 metric space [5] and D metric space [2]. We give 2 an equivalent definition as follows: The family of all balls in a metric space XG, forms a base for a topology on X . This topo- Definition 1.1: [1] A function GXXX: , logy is the same as the topology corresponding to the met- where X is a nonempty set, is called a generalized met- ric dG [1]. ric, or G metric, on the set X , if the following are sa- tisfied for all a x, ,y , z X . Formerly pseudometric spaces were defined with the ( 0)G G x, y , z 0 implies x y z , name pseudo- metric space in [6]. While the axioms G and G of Definition 1.1 in [6] together force (G 1) G x x x, , 0 , 2 3 the structure to be a metric space, the purpose of the
(G 2) yz implies GxxzGxyz,,,, , definition is clear and since G2 was never used there, all results obtained were true. However for completeness, (G 3 ) Symmetry in all variables, that is here we redefine the same concept with alternative axioms and another name. Gxyz,,,,,, Gxzy Gyxz
Definition 1.3: Let be a nonempty set. A function Gy z,,,,,, xGz x yGz y x GXXX: satisfying the axioms (G 1), (G2) , (G3 ) and (G4) in Definition 1.1 is called a ge- ( 4)G Gx y,,,,,, zGx a aGa y z neralized pseudometric, or pseudometric, on the set
A generalized metric space, or metric space, is a pair X . A pseudometric space is a pair XG, , where XG,, where G is a metric on X. A met- G is a pseudometric on . ric space XG, with the property that GxyyGyxx,,,, for all x y, X is called 2 normed spaces symmetric. Let be any semigroup with the identity 0 , and ,: XX be any mapping. In this article, we Definition 1.2: [1] Let XG, be a metric space, consistently use the notation :,0, that is
xX0 and r 0 . Then the ball with center x0 aa ,0 for all aX . and radius r is defined as
B x,:{:,,} r y X G x y y r . 00 Definition 2.1: Let be a real vector space and ,: XX be a mapping satisfying the fol- Every metric G , generates a metric dG . For consis- tency, we consider it, among other equivalent forms, as lowing for all a,, b c X and .
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(N1) a a b , for all x y, ,z , ,a X
(N2) ,,abcabc GxyzGxyz,, ,, ,
for all xyzX,,, . Then (N3) aa , :X X , a , b G a , b ,0 (N4) ,,abba (N5) ,,abaab defines a (semi)norm on X . In particular, to see (N2) and (N4) note that Then , is called a generalized seminorm, or G se- GabcGabbbGbc,,0,,,,0 minorm on X . If additionally GaGbc,0,0,,0 (N0) abab,00 and is satisfied, then , is called a norm. A G a, bGbaaGab ,00,,0,, a . (semi)normed space is a pair X ,,, where , is a (semi)norm on .
To avoid confusion with older definitions, we also use the Example 2.4: If p and q are G (semi)norms on X , name "functionally generalized normed space". and , then pq and p are (semi)norms on X .
Example 2.2: Let X , be a (semi)normed space, then it is easy to show that the three mappings Example 2.5: In the Euclidean space n , consider two ,, ,, ,: XX defined as arbitrary points a and b . If , and the origin define rte a triangle, then set ab, equal to half of the perimeter t a, b max{a , b , a b }, r of the triangle. If , and the origin together define a 1 line, then let ab, denote the maximum length of the a,( b a b a b ) t t 2 line segments defined by these points. And if and 1 2 2 2 a, b ( a b a b )1/2 are equal to the origin, then suppose that ab,0 . e t 2 n Then ‖‖, t is a norm on . We call this the for all abX,, are (semi)norms on X , such that usual-triangular norm on . More generally, if p . rte is any seminorm on a vector space then the -trian- gular seminorm on X is formulated as 1 Example 2.3: Let G be a symmetric (pseudo)met- a, b ( p ( a ) p ( b ) p ( a b )) . ric space on a vector space X , with the properties t 2
Gxayaza ,, Gxyz ,, , In particular on , the usual-triangular norm is defi- ned as 3 Celal Bayar University Journal of Science A. Mutlu Volume 14, Issue 1, p 1-12
1 for all a b, , c, d X , ababab, , t 2 (N3 ) a , b a , b for all a b, X , for all . ab, .
Example 2.6 : Let be a seminorm on a vector space p Example 2.9: (semi)normed in Examples 2.2, 2.5, X. The -rectangular G seminorm on X is given 2.6 and 2.7 are always equi- (semi)norms.
abpapbpab,max,,. r Proposition 2.10: If X ,, is a seminormed space, the following are true for all a b, X In particular, the usual-rectangular norm on is gi- ven by ababab,max,,. r (a) 00, (b) ,,a aa
Example 2.7: For any given seminorm , the -elliptic (c) ,,,a bab seminorm is defined as 1 (d) aba bab , . 1 2 a,[( bp )( ap )() bp ] ab 22 2 1/2 . e 2 Proof: In particular, the usual-elliptic norm on n can be formulated as (a) 0000.aa
n 22 (b) a ,,,0. aa a aaa a, baba b , e i1 iiii (c) ,,a ba abb a a , where aaa ,, , bbb ,, n and the 1 n 1 n b a, b aa usual-elliptic norm on is defined as b,,, b ab. aa b a,. babab 22 e 11 (d) aba ba b ,, In follows is always assumed to have the usual-elliptic 22 norm on it. a,0, bab abab0, .
Definition 2.8: A (semi)norm , on X , with the following properties (N2 ) and (N3 ), which are Proposition 2.11: Let X ,, be a (semi)nor- stronger than (N2) and (N3), respectively, is called an med space. Then XG, is a (pseudo)metric equi- (semi)norm on X. ,
space on X , where GXXX, : is given by (N2 ) a b , c d a , c b , d ,
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by Proposition 2.10 (c). Gxyzxyxz, ,,, for all x y,, z X . In (G4) Observe that addition, G , is symmetric and Gabab, ,,0, for all a b,. X Gxyzxyxyxz, ,,, Proof: Let x y, ,z , a X xaayzy , xaayzy , (G1) ,,0,00.Gxxx,
GxaaGayz,, ,,,,. (G2) From (N1) and (N4) we get If, in addition , is a G norm, then (G0) is obtai- Gxxzxzxz,,0, , ned by
xyxzGxyz, , ,,). Gx, y zxy,,0,0 xz (G3 ) By (N4) we have xyxzxyz 0.
Since GxyzGxzy,, ,,,,, Gxyyxyxy ,,, GyxzGyzx,, ,,,, , yxyx, and
Gyxx, ,,, GzxyGzyx,, ,,,, . G is symmetric. Also, , By Proposition 2.10 (c) and (G3 ) , G, x,,, y z x y x z
x y, x y x z Gabab, ,,0,
=x y , z y for all a b, X . y x, y z
G, y,, x z Proposition 2.12 : Let X ,, be a (semi)nor- and med space. Then the induced functional : X , de- fined as G, x,,, y z x y x z x z, x y aa ,0 x y, z y for all aX , is a (semi)norm on X. z x, z y Proof: Follows directly from the definitions.
G, z,, x y
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Proposition 2.12 and Example 2.2 together shows that, a , in the corresponding topological space. In that case we G (semi)normed space properly generalize (semi) nor- write aaaann, lim or simply l imaan . med spaces. n
Every G seminormed space X ,, will be assumed (ii) Let X ,, be a G seminormed space. A to have the topology of the corresponding seminormed sequence an on X is called G Cauchy if and only space X , . Also, since if it is a Cauchy sequence on the pseudometric space
( ,Xd ) G . 1 , dxyGxyyGxxy,[,,,,] G , ,, 2 (iii) A G seminormed space X ,, is called 1 [,0,]xyxyxy G complete if and only if every G Cauchy sequence 2 on X is convergent. xy by Proposition 2.10 (b) and (N4) , the pseudometrics, obtained by the seminorm and the G pseudometric, 3 Banach Spaces both corresponding to a fixed seminorm, are same. Now we are in a position to define generalized Banach spaces.
Definition 3.1 : A complete normed space is called a Since every seminorm , defines a G pseudo- Banach space. metric G which also defines a pseudometric d that , G , has an underlying topology , the theory of G se- dG , Proposition 3.2 : A normed space X ,, is a minormed spaces is only isometrically different from the theory of seminormed spaces, a similar case with the case Banach space if and only if X , is a Banach of G metric spaces and metric spaces. So, many con- space. cepts defined on topological, pseudometric, and pse- udometric spaces, are readily available on normed spaces. Proof: The topology induced by , is equal to the topo- logy of the norm and the definitions of G conver- Definition 2.13: Let X ,, and Y,, be two 1 2 gence, G Cauchy sequence and G completeness on G seminormed spaces. A function f: X Y is cal- X ,, are equivalent to the definitions of conver- led G continuous if and only if it is continuous with gence, Cauchy sequence and completeness on X , . respect to the corresponding topologies.
Definition 2.14: (i) Let X ,, be a G seminor- Proposition 3.3: Let X ,, be a G seminormed space. Then , is jointly G continuous in both vari- med space. A sequence an on X is G convergent ables. to a point aX if and only if it converges to the point
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Proof: Since abGab,,,0 , for all a b, X , Corollary 3.6 : Let X ,, be a G normed space. the Then a linear functional f on X (that is a function proof follows from Proposition 8 in [1]. fX: which is linear with respect to vector space Definition 3.4 : Let ( ,X , ) and (Y , , ) be G structures) is G bounded if and only if it is G conti- 1 2 nuous. seminormed spaces, A function f X: Y is called bounded, if there exists a constant such that Proposition 3.7 : Let ( ,X , ) and (Y , , ) be G 1 2 Banach spaces, and B X Y , denote the set of all fxfyxy,, G 2 1 G bounded linear mappings from X to Y. Then the for all x y, X . functions
0 # ,: XX and ,:XX Proposition 3.5 : A function defined as follows are G norms on BXY , such that fXY: (,,)(,,) 0 # 12 (,,,)BXY and (,,,)BXY be G Ba- between G seminormed spaces is G bounded if and nach spaces. only if it is bounded with respect to the induced semi- (a) norms and . 1 2 0 fgfxg,sup,:1, xxxX . 2 1 Proof: If there exists constant such that (b) f x ,, f y x y # 2 1 fg, sup fxgy , :, xy 1, xyX, . 2 1 for all xyX,, then
0 # f x f x, f x Proof: It is straightforward to show that , and , 22 x, x x . are both G norms on BXY , . Moreover noting that, 11 by N1 , x 1 if and only if there exists a yX Conversely, assume that fXY: (,)(,) is bo- 0 # 12 such that xy,1, it is seen that and are both unded by a constant . Then f is G bounded, since for every x, y X , equal to the operator norm on BXY, defined as
f x, f y f x f y ffxxsup{:1, xX }, 2 2 2 2 1 (x y ) 2 x , y 1 1 1 which makes BXY , complete [7] and thus G 0 # by Proposition 2.10 (d). complete with respect to , and ,.
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Definition 3.8 : Let X ,, be a G Banach space. GxyzGxyzGxyzt ,,,, ,, 111 222,, 12 Then X denotes the set of all G continuous (or equivalently, G bounded) linear functionals on X and 22 GxyzGxyzGxyz,,(,, ,,), 1/2 e ,, 111 222 X ,, is called the dual space of X if 12 0 ,, and the equi-dual space of X if where # xxxyyyzzz, , , , , XX , , , . 1 2 1 2 1 2 1 2 define G pseudometrics on X. Let G denote
GGrt, or Ge. Note that G satisfies 4 Product of Seminormed Spaces GxayazaGxyz ,, ,, Proposition 4.1: Let Xin,,, 1,2,,, be i i and GxyzGxyz,, ,,, for all G (semi)normed spaces, where n 2. Consider set xyzaX,,,. So ,:, XX n XX i i1 abGab,,,0 with the structure of product of real vector spaces X i . defines a G seminorm on X. Also, for i r t e ,, we Then the functions ,, ,, ,:, XX have Gabab,,0,, which proves that , are 012 i i i defined as G seminorms on X. Moreover, these G seminorms have the topology of the product space, since they induce (a) ,max,abab ii the three known product space seminorms. ri1in
n (b) a , b a, b tii1 ii 5 Quotient Space of a Seminormed Space 1/2 n 2 (c) a , b a, b for all Let (,,)X be a G (semi)normed space, and let U eii1 ii be a linear subspace of X. Define a relation on X by a a11,,,,, ann b b b in X are G ab abU, for all a,. b X Consider the (semi)norms on X. quotient set
XUaaX/:. Proof: Note that, in each case it is enough to prove for n 2. Other results follow from induction. Consider the XU/ is also a vector space with the operations defined induced G pseudometrics G , and G , . Then by as abab : and aa: for all 1 2 Theorem 2.3 of [6], the rules a ,/ b X U and .
Gxyzr , , max G,, xyzG1 , 1 , 1 , xyz2 , 2 , 2 12 Now we give a G (semi)norm on XU/.
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Proposition 5.1: Let ( ,X , ) be a G seminormed a u b v ,. space, and let U be a linear subspace of X. Then the 2 function ,://, XUXU defined as / This implies au. Then ()au is a sequ- 2 1/n abaubv,inf, / uvU, ence in convergent to 0. Since ua1/n 0,
is a G seminorm on XU/. Moreover if , is a we have (ua ) .1/n U is a closed subset of X , G norm and the subspace U is closed in X , then also hence 0, a a U which gives a 0. , is a norm. / G
Proof: (N1) Since auvau,,0 and 0,U Corollary 5.2: Let ( ,X , ) be a G Banach space, and let U be a closed linear subspace of X. Then aa , 0inf,inf au vau (/,,)XU is a Banach space. //u, v U u U / G inf,,.au bvab u, v U / Proof: It follows from the fact that quotient space of a Banach space to a closed subspace is also a Banach space (N2) We have uuuuUU:,, thus and the induced norm of is equal to the quoti- / XU/ inf,infabu cvabuu cv , ent norm corresponding to [7]. u, v U u,, u v U infinf,aubu cv u U u, v U abc ,. 6 G Completion of a G Normed Space // Here we construct a G completion of a G normed space, which will be unique up to G norm-isomorp- (N3 ) For 0, 00inf0.au hism and defined as follows. //uU For 0, observe that UuuUU :, so
Definition 6.1: A function f:(,,)(,,) X Y a inf a u inf a v 12 / u U v U between G seminormed spaces is called a G norm- inf a u a . isomorphism if and only if it is a linear bijection preser- uU ving the values of the G seminorm, that is, a one-to- (N4) and (N5) are immediately seen. one and surjective function such that fxyfxfy Now we suppose that , is a norm and is clo- / G U sed. Let ab,0. Then and / f x ,, f y x y infa u , b v 0, 2 1 u, v U for all , x , y X . so for every 0 there exist u, v U such that 9 Celal Bayar University Journal of Science A. Mutlu Volume 14, Issue 1, p 1-12
Two G seminormed spaces are said to be G isomo- ˆ for all , aabbX, . nn rophic, if there exists a G norm-isomorphism from one to the other.
Lemma 6.4: Let X ,, be a G normed space. De- Definition 6.2: Let ,: XX be a G fine ,: XX with (semi)normed space and A be a linear subspace of X. abab,lim,, Then also the restriction function nn nn
,|: AA AA for all a b X,. ˆ Then , is a norm on nn X . is a G (semi)norm on A. Together with this Proof: First, to show that , is well-defined, we note (semi)norm, A is called a G subspace of X. We that simply write , instead of , | . AA abaaabnnnmmn,,
aabbabnmnmmm ,, Definition 6.3 : A G completion of a G normed hence the inequality space X ,, is a G Banach space (Y , , ) such ababaabb,, that X ,, is G isomorphic to a G subspace nnmmnmnm (,,)X of (Y , , ) , such that X is a dense sub- 0 0 hold. On the other hand, since ()an is a Cauchy sequence set of Y with respect to the topology of ,. on X ,,, ˆ 1 Let X ,, be a G normed space. We denote by X , daaaaaaaaGnm(,)(, nmnmnm ) , 22 the set of all G Cauchy sequences on X. Define a re- for every given 0 and sufficiently large numbers m lation ~ on Xˆ with and n.
ab~ :lim0 ab nn nn In particular aa and similarly bb nm2 nm2 on X. Definitions of G Cauchy sequence and G which imply that (,)ab is a Cauchy sequence on . convergence are essentially equivalent to definitions of nn
Cauchy sequence and convergence definitions for induced Thus lim, abnn exists. normed space X , and it is known from the classical ˆ theory of normed spaces, that ~ is an equivalence rela- Let aabbXnnnn ,,, such that tion on Xˆ [7]. and aann bbnn . ˆ Let XX : / ~ be the quotient vector space with opera- tions
a b : an b n , a : an
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Then limlim0,aabbnnnn and we have Lemma 6.6: Let X 0 be the subset of X, which con- sists of equivalence classes of constant sequences on X , lim,limabaaabnn nnnn , that is Then is a XxxX0 :. X 0 ,, lim aaabnnnn , dense subspace of ( ,X , ) and G lim,limabbbbann nnnn , fXX:,,(,,), f a a lim,.abnn 0
Similarly lim,lim,,ababnn nn and this yields is a G norm-isomorphism.
Proof: Note that X is a linear subspace of X , thus abab, ,. 0 nn nn X ,, is a G subspace of ( ,X , ). On the 0
(N0) We show that, if ab nn ,0, other hand, in the Banach space ( ,X ) , the subspace
X 0 is isometric to X. So it is dense in X . Also we then abnn0, where 0 is the see that f is linear. And the observation that sequence with the constant value 0 . fafbab,, ab,0 nn lim,,a ba b
implies limabnn , 0. Since aab ,, completes the proof.
lim0lim0,aann so that and Similarly Corollary 6.7: Every G normed space ( ,X , ) has a n ~0 an 0. a G completion.
bn 0. Next, we prove the uniqueness of G completions up (N1), (N2), (N3), (N4) and (N5) follow easily. to G norm-isomorphism.
Lemma 6.5: The space X,, is complete. Theorem 6.8: If (,,)X is a G normed space, then Proof: The induced norm corresponding to its G completions are G isomorphic. G norm ,, is equal to the completion norm of the X , Proof: Let (,,)Y1 and (,,)Y2 be two G 1 2 which makes X complete, or equivalently G comp- completions a given G normed space (X , , ). Let lete. X i be dense subsets of Yi such that there exist G Now we show that the G normed space (,,)X can norm-isomorphisms fii:,,(,,) X X for i be densely embedded into a subspace of the G Banach i 1,2. space X ,,.
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Since is dense in if then there exists a X1 Y1, aY 1, limlim,lim,fafbgafb22nm 2 m mn 22m sequence in X f X , which G converges to a. 11 gagb,. 2 We denote this sequence as fa1 n , where aXn 1. Hence we conclude that abgagb,,. 1 2 Since it is G convergent, fa1 n is a G Cauchy sequence, and it is a Cauchy sequence in the induced nor- med space. Note that f 1 exists and it is a G norm- 1 References 1 isomorphism, and so is ff. Every G norm- 21 1. Mustafa, Z. and Sims, B.; A new approach to generali- isomorphism is an isometric isomorphism on induced nor- zed metric spaces, Journal of Nonlinear and Convex med spaces, thus it preserves Cauchy sequences. Thence Analysis. 2006, 7(2), 289-297.
fa2 n is a Cauchy sequence on the induced normed space, and G Cauchy on the G normed space 2. Dhage, B. C.; Generalized metric space and mapping f2 X X2 Y 2 and since Y2 is complete, with fixed point. Bulletin of the Calcutta Mathematical Society. 1992, 84, 329-336. fa2 n G converges to a point in Y2. We set this point as the value ga of g at a. 3. Mohammedpour, A. and Shobe, N.; Stability of addi- It is easy to verify that g is a well-defined linear bijec- tive mappings in generalized normed space, Tam- suiOxf. Jornal of Informatics and Mathematical Scien- tion. Also a b, Y and fa , fb are 1 n 1 m ces. 2013, 29(2), 201-217. sequences G convergent respectively to a and b on
X1. So 4. Shrivastava, R., Animesh, G. and Yadava, R. N.; Some mapping on G Banach space. International faga2 n and fbgb2 m . Journal of Mathematical Science and Engineering Applied. 2011, 5 (IV), 245-260. Since f1 and f2 are G norm-isomorphism
ann,, bfafb 11nm 1 5. Gahler, S.; 2-Metrische Raume und ihre topologische struktur. (German) Mathematische Nachrichten. 1963, fafb22nm,. 26, 115-148. 2
Since G seminorms are continuous in both variables, we have 6. Fernandez, J. and Malviya, N.; Pseudo- G metric spaces and pseudo- metric product spaces. South limlimf1 an , f 1 b m lim, a f1 bm m n 11m Asian Journal of Mathematics. 2013, 3 (1), 339-343. ab, 1 and 7. Maddox, I. J.; Elements of functional analysis. Second edition. Cambridge University Press, Cambridge, 1988; pp. 242
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