Orthogonal Gyroexpansion in Möbius Gyrovector Spaces

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.


Introduction
A. Ungar initiated study on gyrogroups and gyrovector spaces (cf.[1]).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.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]).He describes that gyrovector spaces provide the setting for hyperbolic geometry just as vector spaces provide the setting for Euclidean geometry.In particular, Möbius gyrovector spaces form the setting for the Poincaré ball model of hyperbolic geometry, and similarly, Einstein gyrovector spaces form the setting for the Beltrami-Klein ball model.Readers may consult [4,5] and the references therein for general information about gyrogroups and gyrovector spaces.
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 [6][7][8][9][10] as examples of recent papers for gyrovector spaces, their generalizations, and related matters.
In [8], Abe and the author of the present article showed that any finitely generated gyrovector subspace in the Möbius gyrovector space coincides with the intersection of the linear subspace generated by the same generators and the Möbius ball.As an application, they presented a notion of orthogonal gyrodecomposition and clarified the relation to the ordinary orthogonal decomposition.
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 2 is the preliminaries.In Section 3, we introduce a notion of gyrolinear

Preliminaries
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 [4].
Let V = (V, +, ⋅) be a real inner product space with a binary operation + and a positive definite inner product ⋅ and let V  be the ball for any fixed  > 0.
Definition 1 (see [4,Definitions 3.40 and 6.83]).The Möbius addition ⊕ M and the Möbius scalar multiplication ⊗ M are given by the equations for any a, b ∈ V  ,  ∈ R. The addition ⊕ M and scalar multiplication ⊗ M for the set ‖V  ‖ = {±‖a‖; a ∈ V  } in the axiom (VV) of gyrovector space are defined by the equations for any ,  ∈ ‖V  ‖,  ∈ R.
We simply denote ⊕ M , ⊗ M by ⊕, ⊗, 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 ⊗; (iii) gyroaddition ⊕; that is, and the parentheses are omitted in such cases.In general, we note that gyroaddition does not distribute with (both ordinary and gyro) scalar multiplications: In the limit of large ,  → ∞, the ball V  expands to the whole space V.The next proposition suggests that each result in linear analysis can be restored from the counterpart in gyrolinear analysis.Proposition 2 (see [4, p. 78]).The Möbius addition (resp., Möbius scalar multiplication) reduces to the vector addition (resp., scalar multiplication) as  → ∞; that is, Definition 3 (see [4,Definitions 3.45 and 6.86]).The Einstein addition ⊕ E and the Einstein scalar multiplication ⊗ E are given by the equations Note that each of the Einstein scalar multiplication and the operations on the set ‖V  ‖ is identical to the corresponding operation for the Möbius gyrovector spaces.Definition 4 (see [4,Definition 6.88]).An isomorphism from a gyrovector space ( 1 , ⊕ 1 , ⊗ 1 ) to a gyrovector space ( 2 , ⊕ 2 , ⊗ 2 ) is a bijective map  :  1 →  2 that preserves gyrooperations and keeps the inner product of normalized elements invariant; that is, for any a, b ∈  1 ,  ∈ R.
Theorem 5 (see [4, Table 6.1]).Let  EM : V  → V  be the map defined by the equation for any a ∈ V  .Then  EM is an isomorphism from the Möbius gyrovector space to the Einstein gyrovector space.
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.

Gyrolinear Independency
We begin with consideration of a counterpart in a gyrovector space to the notion of linearly independent sets in a linear space.Definition 6.A finite subset {a 1 , . . ., a  } ⊂ V  is gyrolinearly independent if, for any permutation ( 1 , . . .,   ) of {1, . . ., } and for any order of gyroaddition, the following implication holds: This means that {, , } is not gyrolinearly independent, and if we put then it is readily checked that ( It is immediate to see the following lemma by the fact that 1 ⊗ a = a and 0 ⊗ a = 0 and Definition 6.We omit the proof.Lemma 8. Let {a 1 , . . ., a  } ⊂ V  be gyrolinearly independent.Then (i) each element is nonzero; (ii) any subset is also gyrolinearly independent.Lemma 9. Suppose that {a 1 , a 2 } is linearly independent in V  and Then one has  1 = for  = 1, 2, then, from the definitions of ⊕, ⊗, it follows that and where we put This means that ( 1 ,  2 ) and ( 1 ,  2 ) are solutions to the system of equations where we put  =  1 / 2 and  = 1/ 2 .Then, we have  ̸ = 0 and 1 + (2 + ) <  2 by [8, Lemma 2.2].So we can apply [8,Theorem 2.4] to obtain that   =   , which yields that   =   for  = 1, 2. This completes the proof.Theorem 10.Let {a 1 , . . ., a  } be a linearly independent set in V  .Suppose that two gyrolinear combinations given the same order of gyroaddition and Then one has   =   ( = 1, . . ., ).
Proof.Without loss of generality, we may assume that  = 1.
Assume that the theorem is valid up to .Let {a 1 , . . ., a +1 } be a linearly independent set in V 1 and let the following formula show the latest gyroadditions.Put Then a, a . ( By the definition of ⊕, it follows that where we put Since {a 1 , . . ., a +1 } is linearly independent, we have  1 = ⋅ ⋅ ⋅ =   = 0, which implies that   1 = ⋅ ⋅ ⋅ =    = 0; that is, a  = 0.By the assumption of our induction, it follows that   =   for all .
Similarly, we may assume that a, a  , b, b  ̸ = 0, so {a, b} is linearly independent.By the definition of ⊕, we can rewrite the equation as so we obtain that Therefore, (27) can be changed to the following equation: where By the previous lemma, we can conclude that Then, the assumption of our induction shows that   =   ( = 1, . . .,  + 1).This completes the proof.
Theorem 11.For any finite subset in V  , two notions of linearly independent and gyrolinearly independent coincide.
Proof.(⇒) It immediately follows from the previous theorem.
(⇐) We may assume that  = 1.Assume that the theorem is valid up to , the number of elements of the finite set.Suppose that {a 1 , . . ., a +1 } ⊂ V 1 is gyrolinearly independent and By Lemma 8(ii) and the assumption of our induction, it suffices to show that  +1 = 0. On the contrary, assume that  +1 ̸ = 0.Then, it is obvious that Take a positive number  satisfying that Thus we have where we put From [8, Theorem 2.1], we can rewrite (35) in the form of We can also rewrite  ⊗ a in the form of by using [8, Theorem 3.3], so we obtain the following equation: Since {a 1 , . . ., a +1 } is assumed to be gyrolinearly independent, we can conclude that  1 = ⋅ ⋅ ⋅ =  +1 = 0, which implies that  +1 = 0.This is a contradiction and completes the proof.
Although the contents in the rest of this section are actually known and used repeatedly in [4], we give their proofs for the convenience of readers.
for any a, b ∈ V  .
Proof.By using the definition of ⊕, one can easily calculate the inner product of a ⊕ b with itself to obtain If we put u = a/, k = b/, then it is easy to factorize the second factor as hence we can conclude identity (40).
Definition 13 (see [4, Definition 2.7, (2.1)]).Recall that the inverse element of a is denoted by ⊖a in a gyrogroup, and one uses the notation as in group theory.
Lemma 14.The following formulae hold: Proof.(i) It immediately follows from the definition of ⊕.
(ii) By the Cauchy-Schwarz inequality, we have (iii) From (ii) just established, identity (40) in Lemma 12, and the fact that ⊖a = −a in V  , we have This completes the proof.
Proof.The previous lemma (i) shows that for any a, b in V  .On the other hand, if {a, b} is orthogonal, then it follows from identity (40) in Lemma 12 that Thus the theorem holds for  = 2.
Assume that the theorem is valid up to .Let {c  } +1 =1 be an orthogonal set in V  and let the following equation show the latest gyroaddition ⊕.If we put then {a, b} is orthogonal.From the case of  = 2, it follows that Due to the assumption of our induction, we can conclude that This completes the proof.

The Poincaré Metric and Orthogonal Gyroexpansion in the Möbius Gyrovector Space
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 [8, Theorem 4.2], and we present an explicit procedure to obtain the orthogonal gyroexpansions in the Möbius gyrovector space.
Definition 16 (see [4, Definition 6.8, 6.17There are a number of literatures dealing with relationship between hyperbolic geometry and Hilbert spaces.In particular, Goebel and Reich [11] introduced the hyperbolic metric  on the open unit ball of a complex Hilbert space, from a viewpoint of holomorphic function theory.They developed the study of the Hilbert ball, which leads to research on convexity, nonexpansive mappings, fixed point theorems, and so forth, and [11] is cited in many bibliography such as [12].
The definition of  is equivalent to where for any elements ,  in the Hilbert ball.If we identify R 2 with C, then it is easy to see that ℎ and  coincide with the Poincaré metric on D. In general, however, ℎ and  do not coincide for higher dimensional spaces.We clarify the relationship between ℎ and  below.
Lemma 18.Let V be a real inner product space.Then the norm of the Einstein addition of two elements is given by the equation for any a, b ∈ V  .
Proof.At first, consider the case  = 1.From the definition of ⊕ E , it is easy to calculate the inner product of a⊕ E b with itself as follows: Thus the lemma holds for  = 1.For general  > 0, let a, b ∈ V  .If we put u = a/ and k = b/, then it is immediate to see that and we can easily deduce identity (57) by applying the case  = 1 to u, k.This completes the proof.
Theorem 19.Let one use the notations  and  in V  by where for a, b ∈ V  .Then the following identities hold: Proof.(i) and (ii) immediately follow from the previous lemma.(iii) It is not difficult to see that we may assume  = 1.By (ii) just established, it suffices to show that for real number 0 ≤  < 1.For any a, b ∈ V 1 , if we put  = ‖a‖,  = ‖b‖, and  = (a/‖a‖) ⋅ (b/‖b‖), then, by the definition or the axioms of gyrovector spaces, we have By identity (40) in Lemma 12, On the other hand, identity (57) in the previous lemma shows that This completes the proof.
In the rest of the paper, we should concentrate to investhe Möbius ball endowed with the Poincaré metric ℎ introduced by Ungar.We can perform gyrolinear algebraic operations which behave quite well for orthogonal sequences in the Möbius gyrovector spaces, like as linear algebraic ones in Hilbert spaces.
Lemma 21.For any fixed a ∈ V  , the map is continuous, where one considers the metric ℎ on both sets.
We should make sure of two definitions here.One of them is quite usual; another is very natural.
For any nonempty  of V  , we denote  ⊥ as the orthogonal complement of  in V; that is, A nonempty subset  of V  is a gyrovector subspace if  is closed under gyroaddition and gyroscalar multiplication; that is, a, b ∈  and  ∈ R imply that a⊕b ∈  and ⊗a ∈ .
Proof.From the definitions of ⊕ and ⊗, it is immediate to see that  ⊥ ∩ V  forms a gyrovector subspace.Moreover,  ⊥ ∩ V  is obviously ℎ-closed by the previous lemma.
Proof.It suffices to show that (a  , a) → 0. By the assumption ‖a‖ < , we can obtain where we used identity (40) in Lemma 12.
Proposition 24.Let  be a gyrovector subspace of V  .Then the closure  ℎ with respect to the metric ℎ is also a gyrovector subspace.
Proof.Suppose that a, b ∈  ℎ .There exist sequences This completes the proof.
From now on, we assume that the carrier V of the Möbius gyrovector space V  is complete as a metric space with respect to the norm induced by the inner product.Thus, V is a real Hilbert space.
Theorem 26.Let V be a real Hilbert space.Then (V  , ℎ) is a complete metric space.
Although this fact is well-known and it can be deduced by existing results and Theorem 19, we give a direct proof here in order to show how gyrovector space approach is fully analogous to vector space approach.
Proof.Without loss of generality, we may assume that  = 1.Suppose that {a  } ∞ =1 is a Cauchy sequence in (V 1 , ℎ).From Lemmas 14(iii) and 20(ii), it follows that which implies that {a  } is a Cauchy sequence with respect to the norm of V. Hence there exists a unique element a ∈ V such that ‖a‖ ≤ 1, ‖a  − a‖ → 0. In order to show that ‖a‖ < 1, on the contrary, we assume that ‖a‖ = 1.By the assumption that {a  } is a Cauchy sequence in (V 1 , ℎ), there exists a natural number  0 such that for any  ≥  0 and any .On the other hand, from identity (40) Lemma 12, we have Now we fix  ≥  0 and let  → ∞.Then, from the fact that a + → a and the assumption ‖a‖ = 1, we can obtain which is a contradiction.This implies that ‖a‖ < 1.By Lemma 23, the proof is complete.
Therefore, the orthogonal gyrodecomposition is applicable to ℎclosed gyrovector subspaces in the sense of [8,Theorem 4.2].
Proof.Denote by  the closure of  with respect to the norm topology.It suffices to show that  =  ∩ V  .One of the inclusions (⊂) is trivial.If x ∈  ∩ V  , then there exists a sequence {x  } ⊂  such that ‖x  − x‖ → 0. So we can apply Lemma 23 to obtain that ℎ(x  , x) → 0. Hence x ∈  and this completes the proof.
Theorem 28.Let  be a gyrovector subspace of V  and x ∈ V  .Then one has Proof.(⇒) Suppose that x ∈  ⊥ ∩ V  , m ∈ .By identity (40) in Lemma 12, we obtain (⇐) Suppose that m ∈ .For an arbitrary positive real number , take Since  ⊗ m ∈ , it follows from identity (40) in Lemma 12 that (78) By the axiom (V7), we have ‖ ⊗ m‖ = || ⊗ ‖m‖ =  ⊗ ‖m‖ and the inequality which yields the following inequality: Note that x ⋅ m and  have the same signature, so we have Therefore, we can obtain the inequality Since  > 0 is arbitrary, we can let  → +0 and conclude that x ⋅ m = 0.This completes the proof.
Lemma 29.In a gyrocommutative gyrogroup, one has This lemma can be obtained if we put ,  and  as ⊖b, a and ⊖c, respectively, in [4, Theorem 3.9].However, we give a proof for the convenience of readers by using gyroautomorphic inverse property ⊖(a ⊕ b) = ⊖a ⊖ b, left gyroassociative law (G3), gyrocommutativity (G6), and gyroautomorphism (G4).
Theorem 30.Let  be an ℎ-closed gyrovector subspace of V  and x ∈ V  .
because each gyroautomorphism preserves the norm.Thus we can apply Theorem 28 and obtain that z ∈  ⊥ .This completes the proof.
The following lemma plays a key role in our orthogonal gyroexpansion.
Lemma 31.If {u, k, w} is an orthogonal set in V  , then the associative law holds; that is, where for all u, k, w ∈ V  .See also [10,Proposition 2.14] for a proof by hand calculation.If {u, k, w} is orthogonal, then we have A =  = 0, so that gyr[u, k]w = w.This completes the proof.
Definition 32.(i) Let {a  }  be a sequence in V  .One says that a series converges if there exists an element x ∈ V  such that ℎ(x, x  ) → 0 ( → ∞), where the sequence {x  }  is defined recursively by x 1 = a 1 and x  = x −1 ⊕ a  .In this case, we say the series converges to x and denote (ii) Let {  }  be a sequence in R with |  | <  for all .We say that a series converges if there exists  ∈ R with || <  such that   → , where the sequence {  }  is defined recursively by  1 =  1 and   =  −1 ⊕   .In this case, we say the series converges to  and denote Theorem 33.Let {e  } ∞ =1 be an orthonormal sequence in a real Hilbert space V. Let {  } ∞ =1 be a sequence in R such that 0 <   <  for all .For any sequence {  } ∞ =1 in R, the following are equivalent: converges to an element x ∈ V  .
Note that parentheses are not necessary in the formula in (i) above by Lemma 31.
Proof.(i) ⇒ (ii).Put By the assumption, we have ℎ(x, x  ) → 0. It follows from Lemma 20(i) and Lemma 14(iii) that ‖x  ‖ → ‖x‖.Thus we have lim By the assumption, for any  > 0, there exists a number  0 such that The last inequality implies that ( −   )/(1 − (1/ Note that the strict inequality  <  is crucial in the argument above.This implies that {x  }  is a Cauchy sequence with respect to the metric ℎ by Lemma 20(ii).Since (V  , ℎ) is complete by Theorem 26, there exists a unique element x ∈ V  such that ℎ(x, x  ) → 0. This completes the proof.
Example 34.Consider the sequence {  } ∞ =1 in R defined by   = 1/2.For  = 1, it is easy to see that which does not converge to an element  ∈ R with || < 1.This example can be considered as a counterpart in the Möbius gyrovector space to the series ∑ ∞ =1 (1/2).
Next, we express y  in the form of a gyrolinear combination and present a concrete procedure to seek the gyrocoefficients   .