Continuous Quasi Gyrolinear Functionals on Möbius Gyrovector Spaces

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


Introduction
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 ℝ 3 endowed with Einstein's velocity addition associated with the special theory of relativity (cf. [1]). 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]). Although gyrooperations are generally not commutative, associative, or distributive, 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.
Abe and Hatori [4] introduced the notion of generalized gyrovector spaces (GGVs), which is a generalization of the notion of real inner product gyrovector spaces by Ungar. Hatori [5] showed several substructures of positive invertible elements of a unital C * -algebra are actually GGVs. Abe [6] introduced the notion of normed gyrolinear spaces, which is a further generalization of the notion of GGVs. Although they are complicated objects from the viewpoint of the present article and we do not deal with them here, they will provide advanced research subjects.
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. [1-3, 5, 7-14]).
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 2 is the preliminaries. In Section 3, we show a triviality of continuous gyrolinear functionals on the Möbius gyrovector spaces. In Section 4, we investigate the relationship between the Möbius operations and the linear functionals induced by the inner product and consider a representation theorem of Riesz type. In Section 5, we present a class of continuous functionals that are induced by square summable sequences of real numbers. It can be regarded as a counterpart to continuous linear functionals on real Hilbert spaces, and we might call it quasi gyrolinear functionals.

Preliminaries
Let us briefly recall the definition of the Möbius gyrovector spaces. For precise definitions, basic results of gyrocommutative gyrogroups and gyrovector spaces, see monograph [9] or [10] by Ungar. For elementary facts on inner product spaces, for instance, one can refer to [8].
Let V = ðV, +, •Þ be a real inner product space with a binary operation + and a positive definite inner product •.
for any a, b ∈ V s , r ∈ ℝ. The addition ⊕ M and the scalar multiplication ⊗ M for real numbers are defined by the equations for any a, b ∈ ð−s, sÞ, r ∈ ℝ.
We simply denote ⊕ M , ⊗ M by ⊕ s , ⊗ s , respectively. If several kinds of operations appear in a formula simultaneously, we always give priority by the following order: (i) ordinary scalar multiplication, (ii) gyroscalar multiplication ⊗ s , and (iii) gyroaddition ⊕ s , that is, and the parentheses are omitted in such cases.
In the limit of large s, s ⟶ ∞, the ball V s expands to the whole space V. The next proposition suggests that each result in linear analysis can be recaptured from the counterpart in gyrolinear analysis. Proposition 2 ([10], after Remark 3.41, [3], p.1054). The M€ o bius addition (resp., Möbius scalar multiplication) reduces to the ordinary addition (resp., scalar multiplication) as s ⟶ ∞, that is, for any a, b ∈ V and r ∈ ℝ.
Definition 3 ([10], Definition 2.7, (2.1), (6.286), (6.293)). The inverse element of a with respect to ⊕ s obviously coincides with −a. We use the notation as in group theory. Moreover, the Möbius gyrodistance function d and Poincaré distance function h are defined by the equations Ungar showed that h satisfies the triangle inequality ([10], (6.294)).
The following identities are easy consequence of the definition. One can refer to [11], Lemma 12, Lemma 14 (i).

Lemma 4.
Let s > 0. The following formulae hold: for any a, b ∈ V s and r ∈ ℝ.
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 ð−s, sÞ together with the ordinary addition and multiplication have a familiar nature.
It is known that the inequality ka − bk ≤ 2ka ⊖ s bk holds for any s > 0 and any a, b ∈ V s (for instance, see [11], Lemma 14(iii)). We have some reverse inequalities as follows.
(ii) Let a, b ∈ V. If we take s > 0 sufficiently large, then the inequality holds. Proof.
(i) If kak, kbk ≤ 1/ ffiffi ffi 2 p , by the classical Schwarz inequality, we have which yields (ii) For s > ffiffi ffi 2 p max fkak, kbkg, it is easy to see by (i) just established above, which implies ka ⊖ s bk ≤ 2ka − bk. This completes the proof.

Definition 7 ([11]
, Definition 32). (i) Let fa j g j be a sequence in V s . We say that a series converges if there exists an element x ∈ V s such that hðx, recursively by x 1 = a 1 and x j = x j−1 ⊕ s a j . In this case, we say the series converges to x and denote In addition, if the sequence fa j g j is orthogonal, then we shortly denote Note that parentheses are not necessary in the formula above by [11], Lemma 31.
(ii) Let fa j g j be a sequence in ℝ with |a j | <s for all j. We say that a series converges if there exists x ∈ ℝ with |x | <s such that x j ⟶ x, where the sequence fx j g j is defined recursively by x 1 = a 1 and x j = x j−1 ⊕ s a j . In this case, we say the series converges to x and denote 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 [13]. See also [12] for a discrete Cauchy type inequality restricted to real numbers. Theorem 8 ([14], Theorem 3.8). Let V be a real inner product space. For any a, b ∈ V, s > max fkak, kbkg and c ∈ V with kck ≤ 1, the following inequality holds: 3 Journal of Function Spaces that is, The equality holds if and only if one of the following conditions is satisfied: Moreover, the following inequality does not hold: Indeed, take V = ℝ 2 , s = 1 and Then, it is immediate to check

Continuous Gyrolinear Functionals
In this section, we denote ⊕ 1 , ⊗ 1 by ⊕ , ⊗ for simplicity, respectively, and we show a triviality of continuous gyrolinear functionals on the Möbius gyrovector spaces. It is an application of the orthogonal gyroexpansion with respect to an orthonormal basis in a Hilbert space, which was established in [11]. At first, we consider an elementary system of equations with the Möbius operations restricted to real numbers.
(i) r 1 = t 1 and r 2 = t 2 (ii) r 1 = t 1 and a 2 = 0 (iii) r 2 = t 2 and a 1 = 0 Proof. Note that the M€ obius addition ⊕ is commutative and associative on the open interval ð−1, 1Þ. Thus, we have which implies that or Moreover, together with the fact that −ðb 1 ⊕ b 2 Þ = −b 1 ⊕ −b 2 , we also obtain and hence, r 2 = t 2 or a 2 = 0. Obviously, it yields the conclusion of the lemma. This completes the proof.
Theorem 11. Let V be a separable real Hilbert space with dim V ≥ 2. Consider the Poincaré metric h on the ball V 1 and the interval ð−1, 1Þ, respectively. If a continuous map f : for any x, y ∈ V 1 , then f ðxÞ = 0 for all x ∈ V 1 .

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Proof. At first, it follows from a standard argument using condition (31) and the continuity of f that for any x ∈ V 1 , r ∈ ℝ.
We may assume V is countably infinite dimensional. Take any complete orthonormal sequense fe j g ∞ j=1 of V. Put We use conditions (31) and (32), by the following two ways (I) and (II) In the case (I), it is easy to see that r , 1 + 2x · y + x k k 2 y k k 2 = 5 4 17 2 , x ⊕ y = 4 5 e 1 + 12 25 We can express x ⊕ y ∈ V 1 as a gyrolinear combination of e 1 /2, e 2 /2 to obtain for a unique pair of real numbers r 1 , r 2 . Put x 1 = ð4/5Þe 1 , x 2 = ð12/25Þe 2 . Applying [7], Theorem 4.2, we can rewrite where λ j are given by the formulae where r j are given by the formulae On the other hand, it is easy to check which implies that r j ≠ 2.
Since f satisfies formulae (31) and (32), it follows from taking the value of f in (36) that In the case (II), a similar calculation shows that where r 1 and r 2 are identically given by the formulae (40) and (41), respectively, and we obtain 5 Journal of Function Spaces Therefore, if f satisfies formulae (31) and (32), then we must have the system of equations From the fact that r 1 ≠ 2, r 2 ≠ 2 and Lemma 10, we have a 1 = a 2 = 0. Since the argument above is valid for any pair of distinct members e j 1 , e j 2 , we have f ðe j /2Þ = 0 for all j.
It follows from [11], Theorem 35, that an arbitrary element x in V 1 has an orthogonal gyroexpansion as Thus, we obtain for j = 1, 2, ⋯, which implies that f ðxÞ = 0 for all x ∈ V 1 by the continuity of f . This completes the proof.
The case of 1-dimensional real inner product space is exceptional.

Theorem 12.
Let V be a real inner product space with dim V = 1 and let e be an element in V with kek = 1.
Proof. Let c be an arbitrary real number. Suppose that the map f : 1Þ is defined by the formula (49). A straightforward calculation shows that f satisfies (31) and (32) for any x, y ∈ V 1 , r ∈ ℝ.
Conversely, suppose that a map f : V 1 ⟶ ð−1, 1Þ satisfies (31) and (32) for any x, y ∈ V 1 , r ∈ ℝ. Let 0 < w < 1 be a fixed number. Any element teð−1 < t < 1Þ in V 1 can be expressed as by a unique real number r = tanh −1 t/tanh −1 w. Then, we have for any −1 < t < 1. The argument above includes that the value of c does not depend on w. This completes the proof.

Mappings That Take Values of Inner Product
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. Definition 13. Let ðX 1 , d 1 Þ and ðX 2 , d 2 Þ be two metric spaces. A map f : X 1 ⟶ X 2 is said to be Lipschitz continuous if sup x,y∈X 1 ,x≠y holds. Then, the left-hand side of (52) is called the Lipschitz constant of f and denoted by Lð f Þ.
Theorem 15. Let V be a real inner product space, c ∈ V with kck ≤ 1, and consider the functional f defined by the formula for any x ∈ V. Then, (ii) For any ε > 0, the functional f satisfies the following conditions: for any x, y ∈ V and any r ∈ ℝ.
Proof. (i) For a while, let us denote the restriction of f to V 1 by f simply. The Lipschitz continuity of f is an immediate consequence of the Schwarz type inequality related to the Mobius operations. Actually, it follows from Theorem 8 that for any x, y ∈ V 1 , which shows On the other hand, put c = Lð f Þ. Then, for any x, y ∈ V 1 , we have By taking x = c/2s and y = 0, we have which yields Thus, we can obtain kck ≤ c: (ii) For any x, y ∈ V and sufficiently large real number s, it is immediate to see that and it is also straightforward to calculate the numerator of this formula as which yields the first formula of (ii). Next, which yields the second formula of (ii). For any ε > 0, real numbers r and x, it is elementary to check

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We can obtain By formula (64), the numerator of this formula can be calculated as which shows the third formula of (ii). We also obtain The numerator of this formula can be calculated as which shows the fourth formula of (ii). This completes the proof.
In the rest of this section, we consider a representation theorem of Riesz type. Theorem 11 shows that, in a certain sense, the gyroadditivity (31) is too much strong for continuous functionals. Therefore, it is natural to introduce a suitable notion for functionals on the M€ obius gyrovector spaces which is corresponding to the linearity of functionals on inner product spaces.
For any general (not necessarily linear) mapping f from the Möbius gyrovector space V 1 to the interval ð−1, 1Þ, we associate a family f f s g of mappings defined as follows.
Definition 16. Let V be a real inner product space. For any mapping f : V 1 ⟶ ð−1, 1Þ and any positive real number s, we define a map f s : V s ⟶ ð−s, sÞ by for any element x ∈ V s .
It seems that Theorem 15 provides sufficiently reasonable motivation for making the following definitions.
Definition 17. (i) Let c ∈ V and kck ≤ 1. A mapping f : V 1 ⟶ ð−1, 1Þ is said to be quasi gyrolinear with respect to c if the family f f s g defined by formula (69) satisfies the following conditions: r for any element x, y ∈ V and any real number r ∈ ℝ.
(ii) A mapping f : V 1 ⟶ ð−1, 1Þ is said to be asymptotically gyrolinear if the family f f s g defined by formula (69) satisfies the following conditions: for any element x, y ∈ V and any real number r ∈ ℝ.
Remark 18. Let c ∈ V and kck ≤ 1. (i) Every quasi gyrolinear map from V 1 to ð−1, 1Þ with respect to c is asymptotically gyrolinear. It is easy to check by using Lemma 5.
(ii) The restriction of the mapping defined by formula (53) to the Möbius gyrovector space V 1 is quasi gyrolinear by Theorem 15(ii).
The following lemma can be verified immediately by the definition of the Möbius addition ⊕ s , so we omit the proof.
Lemma 19. Suppose that uðsÞ, vðsÞ are elements in V s defined for sufficiently large real number s such that uðsÞ, vðsÞ converge to constant vectors a, b as s ⟶ ∞, respectively. Then, uðsÞ ⊕ s vðsÞ ⟶ a + b as s ⟶ ∞.
Although we already know that r ⊗ s a ⟶ ra as s ⟶ ∞ for any constants r, a ∈ ℝ, we need a lemma for the case where a is replaced by a function gðsÞ which converges to a constant as s ⟶ ∞.
Lemma 20. There exists a function CðrÞ which depends on r and the formula holds.
Lemma 21. Let r, a ∈ ℝ. Suppose that gðsÞ is a real valued function defined for sufficiently large real number s which satisfies the condition gðsÞ ⟶ a as s ⟶ ∞. Then, r ⊗ s gðsÞ ⟶ ra as s ⟶ ∞.