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 1 Celal Bayar University Journal of Science A. Mutlu Volume 14, Issue 1, p 1-12 In 2006, Mustafa and Sims introduced the notion of gene- 1 . ralized metric space, as an alternative to the notions of dG xy,,,,, Gxyy Gxxy 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 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 (G 0) 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 G x,,,, x z G x y z , 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 Gyzx,,,,,, Gzxy Gzyx GXXX: satisfying the axioms (G1) , (G2) , (G3) and (G4) in DefinitionG 1.1 is called a ge- (G 4) Gxyz,,,,,, Gxaa Gayz 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 G x,,,, y y G y x x 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 . 2 Celal Bayar University Journal of Science A. Mutlu Volume 14, Issue 1, p 1-12 (N1) a a, b for all x,,,, y z a X (N2) a b, c a b , c G x,, y z G x,, y z , for all x, y , z X , . Then (N3) aa , :X X , a, b G a , b ,0 (N4) a, b b , a (N5) a, b a , a b defines a (semi)norm on X . In particular, to see (N2) and (N4) note that Then , is called a generalized seminorm, or se- Gabc, ,0 Gabbb , , Gbc , ,0 minorm on . If additionally G a,0,0 G b , c ,0 (N0) a, b 0 a b 0 and is satisfied, then , is called a norm. A Gab, ,0 G 0, baa , G 0, aba , . (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 . G Example 2.2: Let X , be a (semi)normed space, n then it is easy to show that the three mappings X Example 2.5: In the Euclidean space , consider two , , , , , : XX defined as arbitrary points a and b . If , and the origin define r t e 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 a,, b X are (semi)norms on X , such that usual-triangular norm on . More generally, if p . r t e 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 , a, b a b a b , 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 seminorm on is given 2.6 and 2.7 are always equi- (semi)norms. a, b max p a , p b , p a b . 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 a, b max a , b , a b . r (a) 0 0, (b) a, a a , Example 2.7: For any given seminorm , the -elliptic (c) a, b a , b , seminorm is defined as 1 (d) a b a, b a b . 1 2 a,[()()()] b p a2 p b 2 p a b 2 1/2 . G e 2 Proof: In particular, the usual-elliptic norm on n can be formulated as (a) 0 0aa 0 0. n 22 (b) a, a a , a a a ,0 a . a, b a b a b , e i1 i i i i (c) a, b a , a b b a, a where a a,, a , b b,, b n and the 1 n 1 n b a, b a a usual-elliptic norm on is defined as b,,,. b a b a a b a,. b a22 b ab e 11 (d) a b a,, b a b In follows is always assumed to have the usual-elliptic 22 norm on it. a, b a 0, b a 0, b a b . Definition 2.8: A (semi)norm , on X , with the X 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 , 4 Celal Bayar University Journal of Science A. Mutlu Volume 14, Issue 1, p 1-12 by Proposition 2.10 (c).