Abstract
We investigate a class of functionals on Möbius gyrovector spaces, which consists of a counterpart to bounded linear functionals on Hilbert spaces.
Highlights
Ungar initiated a study on gyrogroups and gyrovector spaces
We investigate a class of functionals on Möbius gyrovector spaces, which consists of a counterpart to bounded linear functionals on Hilbert spaces
The first known gyrogroup was the ball of Euclidean space R3 endowed with Einstein’s velocity addition associated with the special theory of relativity. Another example of a gyrogroup is the open unit disc in the complex plain endowed with the Möbius addition. Ungar extended these gyroadditions to the ball of an arbitrary real inner product space, introduced a common gyroscalar multiplication, and observed that the ball endowed with gyrooperations are gyrovector spaces
Summary
Ungar initiated a study on gyrogroups and gyrovector spaces. Gyrovector spaces are generalized vector spaces, with which they share important analogies, just as gyrogroups are analogous to groups. Another example of a gyrogroup is the open unit disc in the complex plain endowed with the Möbius addition Ungar extended these gyroadditions to the ball of an arbitrary real inner product space, introduced a common gyroscalar multiplication, and observed that the ball endowed with gyrooperations are gyrovector spaces (cf [2, 3]). The celebrated Riesz-Fréchet theorem is one of the most fundamental theorems in both theory and application of functional analysis It states that every bounded linear functional on a Hilbert space can be represented as a map taking the value of the inner product of each variable vector and a fixed vector. We investigate a certain class of continuous functionals on the Möbius gyrovector spaces corresponding to linear functionals induced by the inner product and reveal analogies that it shares with the Riesz representation theorem. It can be regarded as a counterpart to continuous linear functionals on real Hilbert spaces, and we might call it quasi gyrolinear functionals
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