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DEMONSTRATIO MATHEMATICAVol. XXXIINo 31999Seong Sik KimFARTHEST POINTS A N D B O U N D E D 2-FUNCTIONALSA b s t r a c t . In this paper, we give some theorems on further characterizations andexistences of e-farthest points in linear 2-normed spaces in terms of bounded linear 2functionals.I. IntroductionLet X be a linear space of dimension greater than 1 and ||-,-|| be areal-valued function on X x X which satisfies the following conditions:(Ni) ||a, 6|| = 0 if and only if a and b are linearly dependent,(N 2 ) i m i t i m i ,(N3) ||aa,6|| = |a|||a,6||, where a is real,( N O ||a + M | <I M I +i m i .Il», -|| is called a 2-norm on X and (X, ||-, -||) is called a linear 2-normed space([3]). Note that the 2-norm is non-negative and ||a, 6|| = ||a + b,b\\.The following definitions and Theorem 1.1 are given in [5] and [9]:DEFINITION 1.1. A 2-functional f is a real-valued mapping with domainA X C, where A and C are linear manifolds of a linear 2-normed space(XJvll).DEFINITION1.2. A 2-functional / is said to be linear if(1) / ( a + c, b
Demonstratio Mathematica – de Gruyter
Published: Jul 1, 1999
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