Continuous Quasi Gyrolinear Functionals on Möbius Gyrovector Spaces

Abstract

We investigate a class of functionals on Möbius gyrovector spaces, which consists of a counterpart to bounded linear functionals on Hilbert spaces.

On Quasi Gyrolinear Maps between Möbius Gyrovector Spaces Induced from Finite Matrices

“…Theorem 7. [19] (Theorem 27). Let V be a real Hilbert space, let {e j } ∞ j=1 be a complete orthonormal sequence in V, and let {c j } ∞ j=1 be a square summable sequence of real numbers.…”

“…Now we define the notion of quasi-gyrolinearity for maps between two Möbius gyrovector spaces. It seems that [19] (Theorem 15) provides sufficiently reasonable motivation for making the following definitions.…”

“…(cf. [19] (Definition 17)) Let U and V be two real inner product spaces, and let T : U → V be a bounded linear operator. A map f : U 1 → V 1 is said to be quasi-gyrolinear with respect to T if the family { f s } defined by Formula (2) satisfies the following conditions:…”

“…It follows from sc j → x j + y j and c j → 0 as s → ∞ that s tanh −1 c j → x j + y j . By applying [19] (Lemma 21), we can obtain…”

“…In recent years, we have clarified the structure of Möbius gyrovector spaces to some extent, such as the structure of finitely generated gyrovector subspaces, orthogonal gyrodecomposition of any element with respect to any closed gyrovector subspace, orthogonal gyroexpansion of any element with respect to an arbitrary orthogonal basis with weight sequence, Cauchy-Schwarz-type inequalities, and continuous quasi gyrolinear functionals induced by any square summable sequence of real numbers (cf. [14][15][16][17][18][19]). The purpose of this article is to present a class of continuous maps between Möbius gyrovector spaces induced by finite matrices, which can be regarded as a certain counterpart to bounded linear operators on real Hilbert spaces.…”