Orthogonality and linear functionals in normed linear spaces

Convex sets and nearest points

“…There exists a point Z9*<p such that zi-/(y) and, since £ is strictly convex, w±.f(y) implies wERz [5,Theorem 4.3]. Let 7 be the open half-space determined by Rz which contains/(y) and let K be the open half-space determined by Rz+f(y) which contains <2>.…”

“…Let 7 be the open half-space determined by Rz which contains/(y) and let K be the open half-space determined by Rz+f(y) which contains <2>. If x£7 there exists a unique a£E such that x -af(y)±f(y) [5]. Now, a>0 since x -af(y) ERz and x is on the same side of Ezas is/(y).…”

Orthogonal Stability of an Additive-quartic Functional Equation in Non-Archimedean Spaces

“…It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [1]- [3], [7,14,24]). …”

Stability of a generalized quadratic functional equation in various spaces: a fixed point alternative approach

“…It is remarkable to note, however, that a real normed space of dimension greater than 2 is an inner product space if and only if the Birkhoff-James orthogonality is symmetric. There are several orthogonality notions on a real normed space such as Birkhoff-James, Boussouis, Singer, Carlsson, unitary-Boussouis, Roberts, Phythagorean, isosceles and Diminnie (see [11][12][13][14][15][16][17]). …”