On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

On Orthogonally Additive Functions With Big Graphs

Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

On Closed Subsets of Root Systems

Let R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

Thick sets and quasisymmetric maps

1.1. Thickness. Let E be a real inner product space. For a finite sequence of points a0, . . . ,ak in E we let a0. . . ,ak denote the convex hull of the set {a0, . . . , ak}. If these points are affinely independent, the set Δ = a0. . .ak is a k-simplex with vertices a0. . . ,ak. It has a well-defined k-volume written as mk(Δ) or briefly as m(Δ). We are interested in sets A ⊂ E which are “nowhere too flat in dimension k”. More precisely, suppose that A ⊂ E, q > 0 and that k: is a positive integer. We let denote the closed ball with center x and radius r. We say that A is (q, k)-thick if for each x ∈ A and r> 0 such that A\ ≠ there is a k-simplex Δ with vertices in A ∩ such that mk(Δ) ≥ qr.

Inner Bogoliubov automorphisms of the minimal C* Weyl algebra

Within the context of orthogonal geometry, isometries of a real inner product space induce Bogoliubov automorphisms of its associated Clifford algebras. The question whether or not such automorphisms are inner is of considerable interest and importance. Inner Bogoliubov automorphisms were fully characterized for the C* Clifford algebra by Shale and Stinespring [14] and for the W* Clifford algebra by Blattner [2]: each case engenders a corresponding notion of spin group, constructed as a group of units inside the Clifford algebra [4].

Spinors and canonical hermitian forms

The spaceSof spinors associated to a2m-dimensional real inner product space (V, B) carries a canonical Hermitian form 〈 〉 determined uniquely up to real multiples. This form arises as follows: the complex Clifford algebraC(V) of (V, B) is naturally provided with an antilinear involution; this induces an involution on EndSvia the spin representation; this is the adjoint operation corresponding to 〈 〉.