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We investigate the Möbius gyrovector spaces which are open balls centered at the origin in a real Hilbert space with the Möbius addition, the Möbius scalar multiplication, and the Poincaré metric introduced by Ungar. In particular, for an arbitrary point, we can easily obtain the unique closest point in any closed gyrovector subspace, by using the ordinary orthogonal decomposition. Further, we show that each element has the orthogonal gyroexpansion with respect to any orthogonal basis in a Möbius gyrovector space, which is similar to each element in a Hilbert space having the orthogonal expansion with respect to any orthonormal basis. Moreover, we present a concrete procedure to calculate the gyrocoefficients of the orthogonal gyroexpansion.

A. Ungar initiated study on gyrogroups and gyrovector spaces (cf. [

Gyrooperations are generally not commutative, associative, or distributive. Thus the theory of gyrovector spaces falls within the general area of nonlinear functional analysis. They are enjoying algebraic rules such as left and right gyroassociative, gyrocommutative, scalar distributive, and scalar associative laws, so there exist rich symmetrical structures which we should clarify precisely. Many elementary problems are still unsolved. We refer to [

In [

The importance of the orthogonal expansion of each vector with respect to an orthonormal basis in a Hilbert space cannot be overemphasized in both theory and application of functional analysis. In this paper we will introduce a concept of orthogonal gyroexpansion of each element with respect to an orthogonal basis in a Möbius gyrovector space and reveal analogies that it shares with its classical counterpart. Such problems seem to be quite fundamental and important for developing pure and applied mathematics, since one of the virtues of gyrovector spaces is that they have properties which are fully analogous to vector space properties. Moreover, the gyrocoefficients of the orthogonal gyroexpansion can be concretely calculated by a procedure that is given here.

The paper is organized as follows. Section

Let us briefly recall the definitions of two models of gyrovector spaces, that is, the Möbius and Einstein gyrovector spaces. For precise definitions and basic results of gyrocommutative gyrogroups and gyrovector spaces, see [

Let

The Möbius addition

We simply denote

In the limit of large

The Möbius addition (resp., Möbius scalar multiplication) reduces to the vector addition (resp., scalar multiplication) as

The Einstein addition

Note that each of the Einstein scalar multiplication and the operations on the set

An isomorphism from a gyrovector space

Let

Thus, most of results established for the Möbius gyrovector spaces in the sequel can be transformed to corresponding results for the Einstein gyrovector spaces by the isomorphism stated above.

We begin with consideration of a counterpart in a gyrovector space to the notion of linearly independent sets in a linear space.

A finite subset

Let

It is immediate to see the following lemma by the fact that

Let

each element is nonzero;

any subset is also gyrolinearly independent.

Suppose that

Without loss of generality, we may assume that

Let

Without loss of generality, we may assume that

Similarly, we may assume that

For any finite subset in

Thus we have

Although the contents in the rest of this section are actually known and used repeatedly in [

By using the definition of

Recall that the inverse element of

The following formulae hold:

for any

(i) It immediately follows from the definition of

(ii) By the Cauchy-Schwarz inequality, we have

Let

The previous lemma (i) shows that

Assume that the theorem is valid up to

In this section, we give a notion of orthogonal gyroexpansions with respect to a complete orthogonal sequence in the Möbius gyrovector space, which is fully analogous to the notion of the orthogonal expansions with respect to a complete orthonormal sequence in a Hilbert space. It is an application of the orthogonal gyrodecomposition which was established in [

The Möbius gyrodistance function

As pointed out in [

Let

At first, consider the case

Let one use the notations

(i) and (ii) immediately follow from the previous lemma. (iii) It is not difficult to see that we may assume

In the rest of the paper, we should concentrate to investigate the Möbius ball endowed with the Poincaré metric

For any sequence

It is obvious, because both

For any fixed

Suppose that

We should make sure of two definitions here. One of them is quite usual; another is very natural.

For any nonempty subset

A nonempty subset

From the definitions of

If

It suffices to show that

Let

Suppose that

Any finitely generated gyrovector subspace is

Let

From now on, we assume that the carrier

Let

Although this fact is well-known and it can be deduced by existing results and Theorem

Without loss of generality, we may assume that

Let

Denote by

Let

In a gyrocommutative gyrogroup, one has

This lemma can be obtained if we put

Let

Let

be the orthogonal gyrodecomposition of

Conversely, let

is the orthogonal gyrodecomposition. Thus

Note that

(i) Suppose that

(ii) Put

The following lemma plays a key role in our orthogonal gyroexpansion.

If

By [

(i) Let

Let

The series

converges to an element

The series

Note that parentheses are not necessary in the formula in (i) above by Lemma

Consider the sequence

Let

It is not difficult to see that we may assume

Next, we express

For

Suppose that we proceed up to the

Finally, from the uniqueness of the orthogonal gyrodecomposition with respect to the

Let

The

where

It is easy to deduce implications

The author declares that there are no conflicts of interest regarding the publication of this article.

The author would like to thank Professor Michio Seto for bringing information on literatures [