A study of symmetric points in Banach spaces

Abstract

We completely characterize the left-symmetric points, the rightsymmetric points, and, the symmetric points in the sense of Birkhoff-James, in a Banach space. We obtain a complete characterization of the left-symmetric (right-symmetric) points in the infinity sum of two Banach spaces, in terms of the left-symmetric (right-symmetric) points of the constituent spaces. As an application of this characterization, we explicitly identify the left-symmetric (right-symmetric) points of some well-known three-dimensional polyhedral Banach spaces.2010 Mathematics Subject Classification. Primary 46B20, Secondary 52A21.

Norm derivatives and geometry of bilinear operators

“…In [21], Sain et al obtained a complete characterization of the left-symmetric points and the right-symmetric points in a normed linear space in terms of the positive part and the negative part of the elements of the space. As a consequence of Theorem 2.1, Theorem 2.2 and results obtained in [21], we obtain the following complete characterizations of the left-symmetric points and the right-symmetric points of a normed linear space in terms of the norm derivatives ρ ± .…”

Study on Orthogonal Sets for Birkhoff Orthogonality

“…Similarly, x ∈ X is called a right symmetric point if y ⊥ B x implies that x ⊥ B y for all y ∈ X. We note that in the study of symmetry in Birkhoff orthogonality, many scholars have focused on characterizing left symmetric points and right symmetric points in different types of normed spaces [7][8][9][10][11][12][13][14]. However, there exist x, y ∈ X, which are neither left symmetric points nor right symmetric points but satisfy x ⊥ B y and y ⊥ B x.…”

Birkhoff-James Orthogonality and Its Local Symmetry in Some Sequence Spaces

“…Characterizations of the smooth points, the left-symmetric points and the right-symmetric points of a given Banach space are of paramount importance in understanding the geometry of the Banach space. We refer the readers to [1], [6], [7], [10], [11], [12], [17], [18], [19], [20], [21] for some prominent work in this direction.…”