# Continuous Quasi Gyrolinear Functionals on Möbius Gyrovector Spaces

Authors: Keiichi Watanabe
Journal: Journal of Function Spaces
 (2020)

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## Abstract

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


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## Body

## 1. 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 spaceℝ3 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. Section2 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.


## 2. 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].LetV=V,+,∙ be a real inner product space with a binary operation + and a positive definite inner product ∙. Let Vs be an open ball
(1)Vs=a∈V:a<s,for any fixed s>0, where a=a·a1/2.Definition 1 ([10], Definition 3.40, Definition 6.83).
The Mo¨bius addition ⊕M and the Mo¨bius scalar multiplication ⊗M are given by the equations
(2)a⊕Mb=1+2/s2a·b+1/s2b2a+1−1/s2a2b1+2/s2a·b+1/s4a2b2,r⊗Ma=stanhrtanh−1asaaifa≠0,r⊗M0=0,for any a,b∈Vs,r∈ℝ. The addition ⊕M and the scalar multiplication ⊗M for real numbers are defined by the equations
(3)a⊕Mb=a+b1+1/s2ab,r⊗Ma=stanhrtanh−1as,for any a,b∈−s,s,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,
(4)r1⊗sw1a1⊕sr2⊗sw2a2=r1⊗sw1a1⊕sr2⊗sw2a2,and the parentheses are omitted in such cases.In the limit of larges, s⟶∞, the ball Vs 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 Mo¨bius addition (resp., Möbius scalar multiplication) reduces to the ordinary addition (resp., scalar multiplication) as s⟶∞, that is,
(5)a⊕sb⟶a+bs⟶∞,r⊗sa⟶ras⟶∞,for any a,b∈V and r∈ℝ.Definition 3 ([10], Definition 2.7, (2.1), (6.286), (6.293)).
The inverse element ofa with respect to ⊕s obviously coincides with −a. We use the notation
(6)a⊝sb=a⊕s−bas in group theory. Moreover, the Möbius gyrodistance function d and Poincaré distance function h are defined by the equations
(7)da,b=b⊝sa,ha,b=tanh−1da,bs.
Ungar showed thath 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.
Lets>0. The following formulae hold:
(i)
a⊕sb2=a2+2a·b+b2/1+2/s2a·b+1/s4a2b2(ii)
a/s⊕1b/s=a⊕sb/s(iii)
r⊗1a/s=r⊗sa/sfor any a,b∈Vs 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−s,s together with the ordinary addition and multiplication have a familiar nature.Lemma 5.
The following identities hold:(8)a⊕sb=b⊕sa,a⊕sb⊕sc=a⊕sb⊕sc,0⊕sa=a⊕s0=a,−a⊕sa=a⊕s−a=0,1⊗sa=a,r1r2⊗sa=r1⊗sr2⊗sa,r1+r2⊗sa=r1⊗sa⊕sr2⊗sa,r⊗sa⊕sb=r⊗sa⊕sr⊗sb,for any a,b,c∈−s,s,r1,r2,r∈ℝ.It is known that the inequalitya−b≤2a⊖sb holds for any s>0 and any a,b∈Vs (for instance, see [11], Lemma 14(iii)). We have some reverse inequalities as follows.Lemma 6.
(i)
Ifa,b∈V and a,b≤1/2, then the inequality
(9)a⊖1b≤2a−bholds.(ii)
Leta,b∈V. If we take s>0 sufficiently large, then the inequality
(10)a⊖sb≤2a−bholds.Proof.
(i)
Ifa,b≤1/2, by the classical Schwarz inequality, we have
(11)1−2a·b+a2b2≥1−2ab+a2b2=1−ab2≥14,which yields
(12)a⊖1b2=a2−2a·b+b21−2a·b+a2b2≤4a−b2.(ii)
Fors>2maxa,b, it is easy to see
(13)as⊖1bs≤2as−bs,by (i) just established above, which implies a⊖sb≤2a−b. This completes the proof.Definition 7 ([11], Definition 32).
(i) Letajj be a sequence in Vs. We say that a series
(14)a1⊕sa2⊕sa3⊕s⋯⊕saj⊕s⋯converges if there exists an element x∈Vs such that hx,xj⟶0j⟶∞, where the sequence xjj is defined recursively by x1=a1 and xj=xj−1⊕saj. In this case, we say the series converges to x and denote
(15)x=a1⊕sa2⊕sa3⊕s⋯⊕saj⊕s⋯.
In addition, if the sequenceajj is orthogonal, then we shortly denote
(16)x=∑j=1∞⊕saj.
Note that parentheses are not necessary in the formula above by [11], Lemma 31.
(ii) Letajj be a sequence in ℝ with ∣aj∣<s for all j. We say that a series
(17)∑j=1∞⊕saj=a1⊕sa2⊕s⋯⊕saj⊕s⋯converges if there exists x∈ℝ with ∣x∣<s such that xj⟶x, where the sequence xjj is defined recursively by x1=a1 and xj=xj−1⊕saj. In this case, we say the series converges to x and denote
(18)x=∑j=1∞⊕saj.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).
LetV be a real inner product space. For any a,b∈V, s>maxa,b and c∈V with c≤1, the following inequality holds:
(19)a·c⊖sb·c≤c⊗sa⊖sbor
(20)ha·c,b·c≤cha,b,that is,
(21)a·c−b·c1−1/s2a·cb·c≤stanhctanh−11sa2−2a·b+b21−2/s2a·b+1/s4a2b2.
The equality holds if and only if one of the following conditions is satisfied:(i)
a=b(ii)
c=0(iii)
c=1 and a=λc,b=μc for some real numbers λ,μRemark 9.
Note thata·c⊖sb·c≠a⊖sb·c in general. Moreover, the following inequality does not hold:
(22)a·c⊖sb·c≤a⊖sb·c.
Indeed, takeV=ℝ2, s=1 and
(23)a=120,b=12515,c=01.
Then, it is immediate to check(24)a·c⊖sb·c=15>12175−8=a⊖sb·c.


## 3. 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.Lemma 10.
Assume thata1,a2∈−1,1,r1,r2,t1,t2∈ℝ and
(25)r1⊗a1⊕r2⊗a2=t1⊗a1⊕t2⊗a2,r1⊗a1⊕−r2⊗a2=t1⊗a1⊕−t2⊗a2.
Then, at least one of the following (i)~(iv) holds.(i)
r1=t1 and r2=t2(ii)
r1=t1 and a2=0(iii)
r2=t2 and a1=0(iv)
a1=a2=0Proof.
Note that the Mo¨bius addition ⊕ is commutative and associative on the open interval −1,1. Thus, we have
(26)r1⊗a1⊕r1⊗a1=r1⊗a1⊕r2⊗a2⊕r1⊗a1⊕−r2⊗a2=t1⊗a1⊕t2⊗a2⊕t1⊗a1⊕−t2⊗a2=t1⊗a1⊕t1⊗a1,which implies that
(27)2⊗r1⊗a1=2⊗t1⊗a1,tanh2r1tanh−1a1=tanh2t1tanh−1a1,(28)r1=t1or
(29)a1=0.
Moreover, together with the fact that−b1⊕b2=−b1⊕−b2, we also obtain
(30)r2⊗a2⊕r2⊗a2=r1⊗a1⊕r2⊗a2⊕−r1⊗a1⊕−r2⊗a2=t1⊗a1⊕t2⊗a2⊕−t1⊗a1⊕−t2⊗a2=t2⊗a2⊕t2⊗a2,and hence, r2=t2 or a2=0. Obviously, it yields the conclusion of the lemma. This completes the proof.Theorem 11.
LetV be a separable real Hilbert space with dimV≥2. Consider the Poincaré metric h on the ball V1 and the interval −1,1, respectively. If a continuous map f:V1⟶−1,1 satisfies
(31)fx⊕y=fx⊕fy,for any x,y∈V1, then fx=0 for all x∈V1.Proof.
At first, it follows from a standard argument using condition (31) and the continuity of f that
(32)fr⊗x=r⊗fx,for any x∈V1,r∈ℝ.
We may assumeV is countably infinite dimensional. Take any complete orthonormal sequense ejj=1∞ of V. Put
(33)aj=fej2j=1,2.
We use conditions (31) and (32), by the following two ways (I) and (II)
(34)Ix=y=e12⊕e22,IIx=y=e12⊕−e22.
In the case (I), it is easy to see that(35)x=y=e12⊕e22=10e1+6e217,x·y=817,x=y=817,1+2x·y+x2y2=54172,x⊕y=45e1+1225e2.
We can expressx⊕y∈V1 as a gyrolinear combination of e1/2,e2/2 to obtain
(36)e12⊕e22⊕e12⊕e22=r1⊗e12⊕r2⊗e22for a unique pair of real numbers r1,r2. Put x1=4/5e1,x2=12/25e2. Applying [7], Theorem 4.2, we can rewrite
(37)45e1+1225e2=λ1x1⊕λ2x2,where λj are given by the formulae
(38)λ1=4/52+12/252+1−4/52+12/252+12−44/5224/52=1169−3924123·53·54,λ2=4/52+12/252−1+4/52+12/252+12−44/52212/252=−81+3924123·3·52·2512.
Therefore, we have(39)45e1+1225e2=λ1x1⊕λ2x2=1169−3924123·53e1⊕−81+3924123·3·52e2=r1⊗e12⊕r2⊗e22,where rj are given by the formulae
(40)r1=tanh−11169−39241/23·53tanh−11/2(41)r2=tanh−1−81+39241/23·3·52tanh−11/2.
On the other hand, it is easy to check(42)tanh2tanh−112=2⊗12=2·1/21+1/22=45,which implies that rj≠2.
Sincef satisfies formulae (31) and (32), it follows from taking the value of f in (36) that
(43)2⊗a1⊕2⊗a2=a1⊕a2⊕a1⊕a2=fe12⊕fe22⊕fe12⊕fe22=fe12⊕e22⊕e12⊕e22=fr1⊗e12⊕r2⊗e22=r1⊗a1⊕r2⊗a2.
In the case (II), a similar calculation shows that
(44)x=y=e12⊕−e22=10e1−6e217,x⊕y=45e1−1225e2,e12⊕−e22⊕e12⊕−e22=r1⊗e12⊕−r2⊗e22,where r1 and r2 are identically given by the formulae (40) and (41), respectively, and we obtain
(45)2⊗a1⊕−2⊗a2=a1⊕−a2⊕a1⊕−a2=fe12⊕−1⊗fe22⊕fe12⊕−1⊗fe22=fe12⊕−e22⊕e12⊕−e22=fr1⊗e12⊕−r2⊗e22=r1⊗a1⊕−r2⊗a2.
Therefore, iff satisfies formulae (31) and (32), then we must have the system of equations
(46)r1⊗a1⊕r2⊗a2=2⊗a1⊕2⊗a2,r1⊗a1⊕−r2⊗a2=2⊗a1⊕−2⊗a2.
From the fact thatr1≠2,r2≠2 and Lemma 10, we have a1=a2=0. Since the argument above is valid for any pair of distinct members ej1,ej2, we have fej/2=0 for all j.
It follows from [11], Theorem 35, that an arbitrary element x in V1 has an orthogonal gyroexpansion as
(47)x=∑j=1∞⊕rj⊗ej2.
Thus, we obtain(48)fr1⊗e12⊕⋯⊕rj⊗ej2=r1⊗fe12⊕⋯⊕rj⊗fej2=0,for j=1,2,⋯, which implies that fx=0 for all x∈V1 by the continuity of f. This completes the proof.The case of 1-dimensional real inner product space is exceptional.Theorem 12.
LetV be a real inner product space with dimV=1 and let e be an element in V with e=1.
For an arbitrary real numberc, the formula
(49)fte=c⊗t,for −1<t<1 defines a map f:V1⟶−1,1 which satisfies (31) and (32) for any x,y∈V1,r∈ℝ.
Conversely, if a mapf:V1⟶−1,1 satisfies (31) and (32) for any x,y∈V1,r∈ℝ, then c=tanh−1fwe/tanh−1w does not depend on 0<w<1 and f is given by formula (49) for −1<t<1.Proof.
Letc be an arbitrary real number. Suppose that the map f:V1⟶−1,1 is defined by the formula (49). A straightforward calculation shows that f satisfies (31) and (32) for any x,y∈V1,r∈ℝ.
Conversely, suppose that a mapf:V1⟶−1,1 satisfies (31) and (32) for any x,y∈V1,r∈ℝ. Let 0<w<1 be a fixed number. Any element te−1<t<1 in V1 can be expressed as
(50)te=r⊗we=tanhrtanh−1wewewe=tanhrtanh−1weby a unique real number r=tanh−1t/tanh−1w. Then, we have
(51)fte=fr⊗we=r⊗fwe=tanhrtanh−1fwe=tanhtanh−1ttanh−1wtanh−1fwe=tanhtanh−1fwetanh−1wtanh−1t=c⊗t,for any −1<t<1. The argument above includes that the value of c does not depend on w. This completes the proof.


## 4. 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.
LetX1,d1 and X2,d2 be two metric spaces. A map f:X1⟶X2 is said to be Lipschitz continuous if
(52)supx,y∈X1,x≠yd2fx,fyd1x,y<∞holds. Then, the left-hand side of (52) is called the Lipschitz constant of f and denoted by Lf.Definition 14.
Letgs be a real valued function of a real variable s. For any real constant α, gs=osαs⟶∞ means that gs/sα⟶0 as s⟶∞. In particular, gs=o1s⟶∞ means that gs⟶0 as s⟶∞.Theorem 15.
LetV be a real inner product space, c∈V with c≤1, and consider the functional f defined by the formula
(53)fx=x·c,for any x∈V. Then,
(i)
The restriction off to the Möbius gyrovector space V1 is Lipschitz continuous with the Lipschitz constant c, if we consider the Poincaré metric on both the ball V1 and the interval −1,1(ii)
For anyε>0, the functional f satisfies the following conditions:
(54)−fx+y⊕sfx⊕sy=os−2+εs⟶∞,−fx+fy⊕sfx⊕sfy=os−2+εs⟶∞,−frx⊕sfr⊗sx=os−2+εs⟶∞,−rfx⊕sr⊗sfx=os−2+εs⟶∞,for any x,y∈V and any r∈ℝ.Proof.
(i) For a while, let us denote the restriction off to V1 by f simply. The Lipschitz continuity of f is an immediate consequence of the Schwarz type inequality related to the Möbius operations. Actually, it follows from Theorem 8 that
(55)x·c⊖1y·c≤c⊗1x⊖1yor
(56)hfx,fy≤chx,yfor any x,y∈V1, which shows
(57)Lf=supx,y∈V1,x≠yhfx,fyhx,y≤c.
On the other hand, putc=Lf. Then, for any x,y∈V1, we have
(58)tanh−1fx⊖1fy=hfx,fy≤chx,y=ctanh−1x⊖1y.
By takingx=c/2s and y=0, we have
(59)c22s=fc2s=fx⊖1f0≤tanhctanh−1x⊖10=tanhctanh−1c2s,which yields
(60)c22≤stanhctanh−1c/2s⟶cc2s⟶∞.
Thus, we can obtainc≤c.
(ii) For anyx,y∈V and sufficiently large real number s, it is immediate to see that
(61)−fx+y⊕sfx⊕sy=−x·c+y·c⊕s1+2/s2x·y+1/s2y2x·c+1−1/s2x2y·c1+2/s2x·y+1/s4x2y2=−x·c+y·c1+2/s2x·y+1/s4x2y2+1+2/s2x·y+1/s2y2x·c+1−1/s2x2y·c1+o1,and it is also straightforward to calculate the numerator of this formula as
(62)−x·c+2s2x·cx·y+1s4x2y2x·c+y·c+2s2y·cx·y+1s4x2y2y·c+x·c+2s2x·yx·c+1s2y2x·c+y·c−1s2x2y·c=1s2−2y·cx·y+y2x·c−x2y·c+1s4−x2y2x·c−x2y2y·c,which yields the first formula of (ii). Next,
(63)−fx+fy⊕sfx⊕sfy=−x·c+y·c⊕sx·c+y·c1+1/s2x·cy·c=−x·c+y·c1+1/s2x·cy·c+x·c+y·c1+o1=−1/s2x·c+y·cx·cy·c1+o1,which yields the second formula of (ii). For any ε>0, real numbers r and x, it is elementary to check
(64)stanhrtanh−1xs=rx+r1−r2x33s2+os−4+εs⟶∞.
We can obtain(65)−frx⊕sfr⊗sx=−rx·c+stanhrtanh−1x/sx/x·c1+o1.
By formula (64), the numerator of this formula can be calculated as
(66)−rx·c+rx+13s2r1−r2x3+os−4+εxx·c=13s2r1−r2x·cx2+os−4+ε,which shows the third formula of (ii). We also obtain
(67)−rfx⊕sr⊗sfx=−rx·c+stanhrtanh−1x·c/s1+o1.
The numerator of this formula can be calculated as(68)−rx·c+rx·c+13s2r1−r2x·c3+os−4+ε=13s2r1−r2x·c3+os−4+ε,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. Theorem11 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 Mo¨bius gyrovector spaces which is corresponding to the linearity of functionals on inner product spaces.For any general (not necessarily linear) mappingf from the Möbius gyrovector space V1 to the interval −1,1, we associate a family fs of mappings defined as follows.Definition 16.
LetV be a real inner product space. For any mapping f:V1⟶−1,1 and any positive real number s, we define a map fs:Vs⟶−s,s by
(69)fsx=sfxs,for any element x∈Vs.It seems that Theorem15 provides sufficiently reasonable motivation for making the following definitions.Definition 17.
(i) Letc∈V and c≤1. A mapping f:V1⟶−1,1 is said to be quasi gyrolinear with respect to c if the family fs defined by formula (69) satisfies the following conditions:
(70)fsx⊕sy⟶x+y·cs⟶∞,(71)fsx⊕sfsy⟶x·c+y·cs⟶∞,(72)fsr⊗sx⟶rx·cs⟶∞,(73)r⊗sfsx⟶rx·cs⟶∞,for any element x,y∈V and any real number r∈ℝ.
(ii) A mappingf:V1⟶−1,1 is said to be asymptotically gyrolinear if the family fs defined by formula (69) satisfies the following conditions:
(74)−fsx⊕sfsy⊕sfsx⊕sy⟶0s⟶∞,(75)−r⊗sfsx⊕sfsr⊗sx⟶0s⟶∞,for any element x,y∈V and any real number r∈ℝ.Remark 18.
Letc∈V and c≤1. (i) Every quasi gyrolinear map from V1 to −1,1 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 V1 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 thatus,vs are elements in Vs defined for sufficiently large real number s such that us,vs converge to constant vectors a,b as s⟶∞, respectively. Then, us⊕svs⟶a+b as s⟶∞.Although we already know thatr⊗sa⟶ra as s⟶∞ for any constants r,a∈ℝ, we need a lemma for the case where a is replaced by a function gs which converges to a constant as s⟶∞.Lemma 20.
There exists a functionCr which depends on r and the formula
(76)r⊗1x−rx≤Crx3−12≤x≤12.holds.Proof.
For a while, assumer>0. Put fx=r⊗1x=tanhrtanh−1x. For any x∈−1,1, it follows from Maclaurin’s theorem that there exists a number 0<θ<1 satisfying the equation
(77)fx=f0+f′0x+f″θx2!x2.
It is elementary to see(78)f′x=1−tanhrtanh−1x2r1−x2f′0=r,f″x=1−tanhrtanh−1x22r1−x22−rtanhrtanh−1x+x,so that we obtain
(79)r⊗1x−rx=f″θx2!x2=1−tanhrtanh−1θx2r1−θx22−rtanhrtanh−1θx+θxx2.
Now we restrictx≤1/2. If 0<r<1, then we have r⊗1θx=r⊗1θx<θx<x, which yields
(80)r⊗1x−rx≤169rr+1x3.
Ifr>1, then we have r⊗1θx=r⊗1θx<rθx<rx, which yields
(81)r⊗1x−rx≤169rr2+1x3.
Forr<0, since we have the relation r⊗1x−rx=−r⊗1x−−rx, a similar argument shows the conclusion. This completes the proof.Lemma 21.
Letr,a∈ℝ. Suppose that gs is a real valued function defined for sufficiently large real number s which satisfies the condition gs⟶a as s⟶∞. Then, r⊗sgs⟶ra as s⟶∞.Proof.
For anyx∈ℝ, if we take s≥2x, then obviously −1/2≤x/s≤1/2. It follows from Lemma 20 that
(82)r⊗1xs−rxs≤Crxs3,r⊗sx−rx≤Crx3s2.
Therefore, for sufficiently larges>0, we obtain
(83)r⊗sgs−rgs≤Crgs3s2,and hence,
(84)r⊗sgs−ra≤r⊗sgs−rgs+rgs−ra≤Crgs3s2+rgs−a⟶0as s⟶∞. This completes the proof.The following result can be considered as a representation theorem of Riesz type in the Möbius gyrovector space.Theorem 22.
LetV be a real Hilbert space. Suppose that f:V1⟶−1,1 is asymptotically gyrolinear and Lipschitz continuous with Lf≤1 considering the Poincaré metrics on both the Möbius gyrovector space V1 and the interval −1,1. Additionally, assume that f satisfies the following condition:
(85)Foranyelementx∈V,lims⟶∞fsxexistsasarealnumber.
Then, there exists a unique elementc∈V satisfying c≤Lf and f is quasi gyrolinear with respect to c.Proof.
Putc=Lf. Note that it implies
(86)fx⊖1fy≤c⊗1x⊖1y,for any x,y∈V1. For any x,y∈V and any sufficiently large s>0, we have
(87)fsx⊖sfsys=fsxs⊖1fsys=fxs⊖1fys≤c⊗1xs⊖1ys=c⊗1x⊖sys=c⊗sx⊖sys,which yields
(88)fsx⊖sfsy≤c⊗sx⊖sy.
Now, it follows from Lemma6(ii) that
(89)x⊕sy⊖sx+y≤2x⊕sy−x+y⟶0s⟶∞,r⊗sx⊖srx≤2r⊗sx−rx⟶0s⟶∞.
Therefore, by inequality (88) and Lemma 21, we can obtain
(90)fsx⊕sy⊖sfsx+y≤c⊗sx⊕sy⊖sx+y⟶0s⟶∞,(91)fsr⊗sx⊖sfsrx≤c⊗sr⊗sx⊖srx⟶0s⟶∞,for any x,y∈V and r∈ℝ. Define a functional F:V⟶ℝ by the formula
(92)Fx=lims→∞fsx,for any x∈V.
By Lemma19, formulae (74) and (90), for any ε>0, there exists a real number s0>0 such that, if s≥s0 then
(93)−Fx+Fy+fsx⊕sfsy<ε,−fsx⊕sfsy⊕sfsx⊕sy<ε,fsx⊕sy⊖sfsx+y<ε,fsx+y−Fx+y<ε.
We can estimate(94)−Fx+Fy+Fx+y≤−Fx+Fy+fsx⊕sfsy+−fsx⊕sfsy+fsx⊕sy+−fsx⊕sy+fsx+y+−fsx+y+Fx+y≤−Fx+Fy+fsx⊕sfsy+2−fsx⊕sfsy⊕sfsx⊕sy+2fsx⊕sy⊖sfsx+y+∣−fsx+y+Fx+y∣<6ε,for any s≥s0, which implies Fx+y=Fx+Fy.
Similarly, by Lemma21, formulae (75), (91), and (88), we can obtain the formulae
(95)Frx=rFx,Fx−Fy≤cx−y,for any x,y∈V and r∈ℝ. Thus, F:V⟶ℝ is a bounded linear functional which is defined on the whole space V and F≤c. It follows from the classical Riesz theorem that there exists a unique element c∈V satisfying Fx=x·c for any x∈V and c=F. Moreover, it is easy to verify formulae (70)–(73) by using formula (90), formula (91), Lemma 19, and Lemma 20. Thus, f is quasi gyrolinear with respect to c. This completes the proof.Remark 23.
Instead of condition (85), if we assume that there exists a positive homogeneous map f~:V⟶ℝ which is an extension of f, then there exists a unique element c∈V satisfying
(96)f~x=x·c,for any x∈V.


## 5. Continuous Quasi Gyrolinear Functionals

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.Lemma 24.
Letcjj=1∞ and ajj=1∞ be square summable sequences of real numbers. Note that
(97)wedonotassume∑j=1∞cj2≤1or∑j=1∞aj2≤1.
Suppose thats0 is a positive real number and ujs is a real valued function defined for s≥s0 and j=1,2,⋯, which satisfies the following conditions:
(98)−s<ujs<s,fors≥s0andj=1,2,⋯,ujs⟶aj,ass⟶∞foranyj=1,2,⋯,∑j=1∞⊕s2ujs2≤K,fors≥s0,where K is a constant which does not depend on s. Then, the series ∑j=1∞⊕scj⊗sujs converges for any s≥s0, and
(99)∑j=1∞⊕scj⊗sujs⟶∑j=1∞cjajas s⟶∞.Proof.
For anys≥s0, any natural number n and p, we can apply the gyrotriangle inequality and a Cauchy type inequality ([12], Remark 2.6) to obtain
(100)−∑j=1n⊕scj⊗sujs⊕s∑j=1n+p⊕scj⊗sujs=∑j=n+1n+p⊕scj⊗sujs≤∑j=n+1n+p⊕scj⊗sujs=∑j=n+1n+p⊕scj⊗sujs≤∑j=n+1∞cj21/2⊗s∑j=n+1∞⊕s2ujs21/2,which yields that the sequence ∑j=1n⊕scj⊗sujsn=1∞ is a Cauchy sequence in the interval −s,s,h (cf. [11], Lemma 20(ii)), and hence, it converges.
From the proof of Lemma20, if 0<r<1, then the inequality
(101)r⊗1x−rx≤4rx3−12≤x≤12holds. By putting s1=2K1/2+1, a similar argument to the proof of Lemma 21 shows that
(102)r⊗sK1/2−rK1/2≤4rK3/2s2,0≤r⊗sK1/2≤rK1/2+4rK3/2,for any s≥s1. Therefore, for any positive number ε, there exists a natural number j0 such that
(103)∑j=j0+1∞cjaj<ε,sups≥s1∑j=j0+1∞cj21/2⊗sK1/2<ε.
Moreover, we can take a positive numbers2 such that
(104)∑j=1j0cjaj−∑j=1j0⊕scj⊗sujs<ε,for any s>s2. Then, by [11], Lemma 14(iii), and [12], Remark 2.6, we can obtain
(105)∑j=1∞cjaj−∑j=1∞⊕scj⊗sujs≤∑j=1∞cjaj−∑j=1j0cjaj+∑j=1j0cjaj−∑j=1j0⊕scj⊗sujs+∑j=1j0⊕scj⊗sujs−∑j=1∞⊕scj⊗sujs≤∑j=j0+1∞cjaj+∑j=1j0cjaj−∑j=1j0⊕scj⊗sujs+2−∑j=1j0⊕scj⊗sujs⊕s∑j=1∞⊕scj⊗sujs<2ε+2∑j=j0+1∞⊕scj⊗sujs≤2ε+2∑j=j0+1∞cj21/2⊗s∑j=j0+1∞⊕s2ujs21/2<4εfor sufficiently large s. This completes the proof.Theorem 25.
Letejj=1∞ be a complete orthonormal sequence in a real Hilbert space V and let x be an arbitrary element in V1. We can apply the orthogonal gyroexpansion ([11], Theorem 35) to get a unique sequence r1,r2,⋯ of real numbers such that
(106)x=∑j=1∞⊕1rj⊗1ej2.
Then, the following inequality holds:(107)∑j=1∞rj21/2≤tanh−1xtanh−11/2.Proof.
Note that(108)x2=∑j=1∞⊕1rj⊗1122by [11], Theorem 36(iii). Let cjj=1∞ be an arbitrary square summable sequence of real numbers. Then, for any natural number n, by identities of Lemma 5 and a Cauchy type inequality ([12], Theorem 2.5), we obtain the following estimation:
(109)∑j=1ncjrj⊗112=∑j=1n⊕1cjrj⊗112=∑j=1n⊕1cj⊗1rj⊗112≤∑j=1ncj21/2⊗1∑j=1n⊕1rj⊗11221/2.
Thus, we obtain(110)∑j=1∞cjrj⊗112≤c⊗1xor
(111)∑j=1∞cjrj≤ctanh−1xtanh−11/2,where c is a unique element in V defined by the formula c=∑j=1∞cjej. Because cj is an arbitrary square summable sequence, we can conclude inequality (107). This completes the proof.We show that, if a sequencexkk in a Möbius gyrovector space converges to an element x, then the jth gyrocoefficient of xk converges to that of x.Lemma 26.
Letejj=1∞ be a complete orthonormal sequence in a real Hilbert space V. Suppose that xkk is a sequence and x is an element in V1 such that hxk,x⟶0 as k⟶∞. Let
(112)xk=∑j=1∞⊕1rk,j⊗1ej2,x=∑j=1∞⊕1rj⊗1ej2be the orthogonal gyroexpansions. Then,
(113)rk,j⟶rjk⟶∞,for any j=1,2,⋯.Proof.
Note thatxk−x⟶0 as k⟶∞ (cf. [11], Lemma 14(iii), Lemma 20). A similar argument on formula (118) in the proof of [11], Theorem 35, shows that
(114)rk,1=tanh−1λk,11xk·e1tanh−11/2,r1=tanh−1λ11x·e1tanh−11/2,where
(115)λk,11=xk2+1−xk2+12−4xk·e122xk·e12,λ11=x2+1−x2+12−4x·e122x·e12,which obviously yields rk,1⟶r1 as k⟶∞. Applying a similar argument to
(116)−rk,1⊗1e12⊕1xk=∑j=2∞⊕1rk,j⊗1ej2,−r1⊗1e12⊕1x=∑j=2∞⊕1rj⊗1ej2,we obtain rk,2⟶r2 as k⟶∞, and so on. This completes the proof.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.Theorem 27.
LetV be a real Hilbert space, let ejj=1∞ be a complete orthonormal sequence in V and let cjj=1∞ be a square summable sequence of real numbers. Consider the Poincaré metrics on both the Möbius gyrovector space V1 and the interval −1,1. Then, we can define a map f:V1⟶−1,1 by the equation
(117)fx=∑j=1∞cjrj⊗112,for any x∈V1 whose orthogonal gyroexpansion is given by formula (106). Moreover, f is continuous and quasi gyrolinear with respect to c=∑j=1∞cjej.Proof.
Since the sequencerjj is square summable by Theorem 25, the ordinary infinite series ∑j=1∞cjrj converges absolutely; hence, the map f is well defined. Suppose that xkk is a sequence and x is an element in V1 such that hxk,x⟶0 as k⟶∞. Let
(118)xk=∑j=1∞⊕1rk,j⊗1ej2,x=∑j=1∞⊕1rj⊗1ej2be the orthogonal gyroexpansions. By Theorem 25,
(119)∑j=1∞rk,j21/2≤tanh−1xktanh−11/2,which is bounded with respect to k, because xk⟶x<1. Therefore, for any positive number ε, there exists a natural number j0 such that
(120)∑j=j0+1∞∣cjrk,j∣≤∑j=j0+1∞cj21/2∑j=j0+1∞rk,j21/2<εfor all k, and ∑j=j0+1∞cjrj<ε. By Lemma 26, we can take sufficiently large k0 such that, if k≥k0, then
(121)∑j=1j0cjrk,j−∑j=1j0cjrj<ε.
Thus, ifk≥k0, then we have
(122)∑j=1∞rk,jcj−∑j=1∞rjcj≤∑j=1j0rk,jcj−∑j=1j0rjcj+∑j=j0+1∞rk,jcj+∑j=j0+1∞rjcj<3ε,which implies that hfxk,fx⟶0. This copmpletes the proof of the continuity of the functional f.
Take arbitrary elementsx,y∈V, and put xj=x·ej,yj=y·ej for j=1,2,⋯. Then, it follows from the definition of ⊕1 that
(123)sxs⊕1ys·ej=1+2/s2x·y+1/s2y2xj+1−1/s2x2yj1+2/s2x·y+1/s4x2y2⟶xj+yjas s⟶∞. For sufficiently large s>0, applying the orthogonal gyroexpansion, there exists a unique sequence r1s,r2s,⋯ of real numbers such that
(124)xs⊕1ys=∑j=1∞⊕1rjs⊗1ej2=∑j=1∞⊕1tjsej,where we put tjs=tanhrjstanh−11/2. Then, we have
(125)x⊕sys2=xs⊕1ys2=∑j=1∞⊕1tjsej2=∑j=1∞⊕1tjs2.
It follows fromx⊕sy⟶x+y that x⊕sy/s⟶0 and hence tjs⟶0 as s⟶∞. Put z1s=∑j=2∞⊕1tjsej. Note that z1s≤x⊕sy/s⟶0 as s⟶∞.
(126)xs⊕1ys·e1=t1se1⊕1z1s·e1=1+z1s2t1se1+1−t1s2z1s1+t1s2z1s2·e1=1+z1s2t1s1+t1s2z1s2x⊕sy·e1=1+z1s2st1s1+t1s2z1s2.
By lettings⟶∞ in the formula above, we have st1s⟶x+y·e1=x1+y1.
Assume that we have shownstjs⟶xj+yjj=1,⋯,n.
(127)−st1se1⊕s⋯⊕sstnsen⊕sx⊕sy·en+1=1+zn+1s2stn+1s1+tn+1s2zn+1s2,where we put zn+1s=∑j=n+2∞⊕1tjsej. From the assumption of induction and Lemma 19, it follows that
(128)stn+1s⟶−x1+y1e1+⋯+xn+ynen+x+y·en+1=xn+1+yn+1.
Now, it is straightforward to see(129)fsx⊕sy=sfx⊕sys=sfxs⊕1ys=s∑j=1∞cjrjs⊗112=∑j=1∞cjrjs⊗ss2=∑j=1∞⊗scjrjs⊗ss2=∑j=1∞⊗scj⊗sstjs.
By multiplyings2 to both sides of formula (125), we have
(130)∑j=1∞⊕s2stjs2=x⊕sy2,which is bounded with respect to s, because x⊕sy⟶x+y as s⟶∞. Thus, we can apply Lemma 24 by putting aj=xj+yj,ujs=stjs to obtain
(131)fsx⊕sy⟶x+y·cas s⟶∞, where c is a unique element in V defined by the formula c=∑j=1∞cjej. By taking y=0 in the formula just established, it is obvious that fsx⟶x·c, so we have
(132)fsx⊕sfsy⟶x·c+y·c,r⊗sfsx⟶rx·cas s⟶∞. Finally, it is easy to see
(133)sr⊗1xs·ej=stanhrtanh−1xsxjx⟶rxj.
We can also express as(134)r⊗1xs=∑j=1∞⊕1rjs⊗1ej2=∑j=1∞⊕1tjsej.
Then, a similar argument for the addition shows thatstjs⟶rxj, and hence, we can apply Lemma 24 to obtain
(135)fsr⊗sx=∑j=1∞⊕scj⊗sstjs⟶∑j=1∞cjrxj=rx⋅cas s⟶∞.



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