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

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. The first known gyrogroup was the ball of Euclidean space

Abe and Hatori [

In this article, we concentrate on the Möbius gyrovector spaces, because they are most fundamental among real inner product gyrovector spaces. There are notions of the Einstein gyrovector spaces and the PV gyrovector spaces by Ungar, and they are isomorphic to the Möbius gyrovector spaces, so most results on each space can be directly translated to other two spaces. In the Möbius gyrovector spaces, one can consider counterparts to various notions of Hilbert spaces such as the orthogonal decomposition and the closest point property with respect to any closed linear subspace, orthogonal expansion with respect to any orthonormal basis, and the Cauchy-Bunyakovsky-Schwarz inequality (cf. [

We study some aspects of the Möbius gyrovector spaces from some viewpoints of basic theory of functional analysis. 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. This fact makes duality in Hilbert spaces much closer to finite-dimensional duality, and for that reason, it is a particularly useful tool. 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.

The paper is organized as follows. Section

Let us briefly recall the definition of the Möbius gyrovector spaces. For precise definitions, basic results of gyrocommutative gyrogroups and gyrovector spaces, see monograph [

Let

The M

We simply denote

In the limit of large

The M

The inverse element of

Ungar showed that

The following identities are easy consequence of the definition. One can refer to [

Let

Note that the Möbius operations generally are not commutative, associative, or distributive. Furthermore, the ordinary scalar multiplication does not distribute the Möbius addition.

However, the restricted Möbius operations to the interval

The following identities hold:

It is known that the inequality

If

Let

If

For

(i) Let

In addition, if the sequence

Note that parentheses are not necessary in the formula above by [

(ii) Let

Recently, the following Schwarz type inequality related to the Möbius operations in real inner product spaces was obtained, which is an extension of a similar type inequality obtained in a preceding paper [

Let

The equality holds if and only if one of the following conditions is satisfied:

Note that

Indeed, take

Then, it is immediate to check

In this section, we denote

Assume that

Then, at least one of the following (i)~(iv) holds.

Note that the M

Moreover, together with the fact that

Let

At first, it follows from a standard argument using condition (

We may assume

We use conditions (

In the case (I), it is easy to see that

We can express

Therefore, we have

On the other hand, it is easy to check

Since

In the case (II), a similar calculation shows that

Therefore, if

From the fact that

It follows from [

Thus, we obtain

The case of 1-dimensional real inner product space is exceptional.

Let

For an arbitrary real number

Conversely, if a map

Let

Conversely, suppose that a map

In this section, we investigate the relationship between the Möbius operations and the linear functional which takes the value of the inner product of each vector and a fixed vector. Then, a representation theorem of Riesz type is considered.

We need a well-known notion of continuity of mappings between metric spaces and a notation for asymptotic behavior of functions in elementary calculus.

Let

Let

Let

The restriction of

For any

(i) For a while, let us denote the restriction of

On the other hand, put

By taking

Thus, we can obtain

(ii) For any

We can obtain

By formula (

The numerator of this formula can be calculated as

In the rest of this section, we consider a representation theorem of Riesz type. Theorem

For any general (not necessarily linear) mapping

Let

It seems that Theorem

(i) Let

(ii) A mapping

Let

(ii) The restriction of the mapping defined by formula (

The following lemma can be verified immediately by the definition of the Möbius addition

Suppose that

Although we already know that

There exists a function

For a while, assume

It is elementary to see

Now we restrict

If

For

Let

For any

Therefore, for sufficiently large

The following result can be considered as a representation theorem of Riesz type in the Möbius gyrovector space.

Let

Then, there exists a unique element

Put

Now, it follows from Lemma

Therefore, by inequality (

By Lemma

We can estimate

Similarly, by Lemma

Instead of condition (

In this section, we present a class of continuous functionals on Möbius gyrovector spaces, which consists of a counterpart to continuous linear functionals on real Hilbert spaces.

Let

Suppose that

For any

From the proof of Lemma

Moreover, we can take a positive number

Let

Then, the following inequality holds:

Note that

Thus, we obtain

We show that, if a sequence

Let

Note that

We present a class of continuous functionals on the Möbius gyrovector spaces, which seems to be a counterpart of bounded linear functionals on real Hilbert spaces.

Let

Since the sequence

Thus, if

Take arbitrary elements

It follows from

By letting

Assume that we have shown

Now, it is straightforward to see

By multiplying

We can also express as

Then, a similar argument for the addition shows that

No data were used to support this study.

The author declares that there are no conflicts of interest.

The author typed, read, and approved the final manuscript.

This work was supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.