# Extension of isometries in real Hilbert spaces

• Soon-Mo Jung
From the journal Open Mathematics

## Abstract

MSC 2010: 46B04; 46C99

## 1 Introduction

In the course of the development of mathematics in the last century, the problem of extending the domain of a function while keeping/preserving the characteristic properties of a function defined in a local domain has had a great influence on the development of functional analysis.

For example, in topology, the Tietze extension theorem states that all continuous functions defined on a closed subset of a normal topological space can be extended to the entire space.

## Theorem 1.1

(Tietze) Let X be a normal space, E be a nonempty closed subset of X, and let [ L , L ] be a closed real interval. If f : E [ L , L ] is a continuous function, then there exists a continuous extension of f to X , i.e., there exists a continuous function F : X [ L , L ] such that F ( x ) = f ( x ) for all x E .

The Tietze extension theorem has a wide range of applications and is an interesting theorem, so there are many variations in this theorem.

In 1972, Mankiewicz published his article [1] determining whether an isometry f : E Y from a subset E of a real normed space X into a real normed space Y admits an extension to an isometry from X onto Y . Indeed, he proved that every isometry f : E Y can be uniquely extended to an affine isometry between the whole spaces when either E and f ( E ) are both convex bodies or E is nonempty open connected and f ( E ) is open, where a convex body is a convex set with a nonempty interior.

## Theorem 1.2

(Mankiewicz) Let X and Y be real normed spaces, E be a nonempty subset of X , and let f : E f ( E ) be a surjective isometry, where f ( E ) is a subset of Y . If either both E and f ( E ) are convex bodies, or E is open and connected and f ( E ) is open, then f can be uniquely extended to an affine isometry F : X Y .

This conclusion particularly holds for the closed unit balls. Based on this fact, with the same research direction, Tingley [2] intuitively paid attention to the unit spheres and posed the following problem, which is now known as Tingley’s problem.

## Problem 1.1

(Tingley) Is every surjective isometry between the unit spheres of two Banach spaces a restriction to the unit sphere of a surjective real-linear isometry between the whole spaces?

Recently, many articles have been devoted to the study of the extension of isometries and Tingley’s problem. Among them, a result of Ding [3, Theorem 2.2], which is related to Problem 1.1, will be introduced.

## Theorem 1.3

(Ding) Let X and Y be real Hilbert spaces and let f : S 1 ( X ) S 1 ( Y ) be a function between unit spheres. If f ( S 1 ( X ) ) f ( S 1 ( X ) ) and f ( x 1 ) f ( x 2 ) x 1 x 2 for all x 1 , x 2 S 1 ( X ) , then f can be extended to a real-linear isometry from X into Y (see also [4,5, 6,7, 8,9, 10,11, 12,13]).

We observe that the domain of a local isometry is assumed to be bounded and contains at least two elements, but it need not be a convex body nor an open set. This indicates that the main results of this article are more general than those previously published.

## 2 Preliminaries

Throughout this article, the symbol R ω will denote the space of all real sequences. From now on, we denote by ( R ω , T ) the product space i = 1 R , where ( R , T R ) is the usual topological space. Then, since ( R , T R ) is a Hausdorff space, ( R ω , T ) is a Hausdorff space.

Let a = { a i } be a sequence of positive real numbers satisfying the following condition:

(2.1) i = 1 a i 2 < .

With this sequence a = { a i } , we define

M a = ( x 1 , x 2 , ) R ω : i = 1 a i 2 x i 2 < .

Then, M a is a vector space over R , and we can define an inner product , a on M a by

x , y a = i = 1 a i 2 x i y i

for all x = ( x 1 , x 2 , ) and y = ( y 1 , y 2 , ) of M a , with which ( M a , , a ) becomes a real inner product space. This inner product induces the norm in the natural way

x a = x , x a

for all x M a , so that ( M a , a ) becomes a real normed space.

## Remark 2.1

M a is the set of all elements x R ω satisfying x a 2 < , i.e.,

M a = { ( x 1 , x 2 , ) R ω : x a 2 < } .

We define the metric d a on M a by

d a ( x , y ) = x y a = x y , x y a

for all x , y M a . Thus, ( M a , d a ) is a real metric space. Let ( M a , T a ) be the topological space generated by the metric d a .

Similar to [18, Theorem 70.4], we can prove Remark 2.2 ( i ) .

## Remark 2.2

We note that

1. ( M a , , a ) is a Hilbert space over R ;

2. ( M a , T a ) is a Hausdorff space as a subspace of the Hausdorff space ( R ω , T ) .

## Definition 2.1

Given c in M a , the translation by c is the mapping T c : M a M a defined by T c ( x ) = x + c for all x M a .

## 3 First-order generalized linear span

In [14, Theorem 2.5], we were able to extend the domain of a d a -isometry f to the entire space when the domain of f is a nondegenerate basic cylinder (see Definition 6.1 for the exact definition of nondegenerate basic cylinders). However, we shall see in Definition 4.1 and Theorem 4.2 that the domain of a d a -isometry f can be extended to its first-order generalized linear span whenever f is defined on a bounded set that contains more than one element.

From now on, it is assumed that E , E 1 , and E 2 are subsets of M a , and each of them contains more than one element, unless specifically stated for their cardinalities. If the set has only one element or no element, this case will not be covered here because the results derived from this case are trivial and uninteresting.

## Definition 3.1

Assume that E is a nonempty bounded subset of M a and p is a fixed element of E . We define the first-order generalized linear span of E with respect to p as

GS ( E , p ) = p + i = 1 m j = 1 α i j ( x i j p ) M a : m N ; x i j E and α i j R for all  i  and  j .

We remark that if a bounded subset E of M a contains more than one element, then E is a proper subset of its first-order generalized linear span GS ( E , p ) , because x = p + ( x p ) GS ( E , p ) for any x E and p + α ( x p ) GS ( E , p ) for any α R , which implies that GS ( E , p ) is unbounded. Moreover, we note that α x + β y M a for all x , y M a and α , β R , because α x + β y a α x a + β y a < . Therefore, GS ( E , p ) p is a real vector space, because the double sum in the definition of GS ( E , p ) guarantees α x + β y GS ( E , p ) p for all x , y GS ( E , p ) p and α , β R and because GS ( E , p ) p is a subspace of a real vector space M a (cf. Lemma 5.3 ( i ) ).

For each i N , we set e i = ( 0 , , 0 , 1 , 0 , ) , where 1 is in the i th position. Then, 1 a i e i is a complete orthonormal sequence in M a .

## Definition 3.2

Let E be a nonempty subset of M a .

1. We define the index set of E by

Λ ( E ) = { i N : there are an x E and an α R { 0 } satisfying x + α e i E } .

Each i Λ ( E ) is called an index of E . If Λ ( E ) N , then the set E is called degenerate. Otherwise, E is called nondegenerate.

2. Let β = { β i } i N be another complete orthonormal sequence in M a . We define the β -index set of E by

Λ β ( E ) = { i N : there are an x E and an α R { 0 } satisfying x + α β i E } .

Each i Λ β ( E ) is called a β -index of E .

We will find that the concept of an index set in Hilbert space sometimes takes over the role that the concept of dimension plays in vector space. According to the definition above, if i is a β -index of E , i.e., i Λ β ( E ) , then there are x E and x + α β i E for some α 0 . Since x x + α β i , we remark that if Λ β ( E ) , then the set E contains at least two elements.

In the following lemma, we prove that if i is an index of E and p E , then the first-order generalized linear span GS ( E , p ) contains the line through p in the direction e i .

## Lemma 3.1

Assume that E is a bounded subset of M a , and GS ( E , p ) is the first-order generalized linear span of E with respect to a fixed element p E . If i Λ ( E ) , then p + α e i GS ( E , p ) for all α R .

## Proof

By Definition 3.2, if i Λ ( E ) , then there exist an x E and an α 0 0 , which satisfy x + α 0 e i E . Since x E and x + α 0 e i E , by Definition 3.1, we obtain

p + α 0 β e i = p + β ( x + α 0 e i p ) β ( x p ) GS ( E , p )

for all β R . Setting α = α 0 β in the above relation, we obtain p + α e i GS ( E , p ) for any α R .□

We now introduce a lemma, which is a generalized version of [14, Lemma 2.3] and whose proof runs in the same way. We prove that the function T q f T p : E 1 p E 2 q preserves the inner product. This property is important in proving the following theorems as a necessary condition for f to be a d a -isometry.

## Lemma 3.2

Assume that E 1 and E 2 are bounded subsets of M a that are d a -isometric to each other via a surjective d a -isometry f : E 1 E 2 . Assume that p is an element of E 1 and q is an element of E 2 with q = f ( p ) . Then, the function T q f T p : E 1 p E 2 q preserves the inner product, i.e.,

( T q f T p ) ( x p ) , ( T q f T p ) ( y p ) a = x p , y p a

for all x , y E 1 .

## Proof

Since T q f T p : E 1 p E 2 q is a d a -isometry, we have

( T q f T p ) ( x p ) ( T q f T p ) ( y p ) a = ( x p ) ( y p ) a

for any x , y E 1 . If we put y = p in the previous equality, then we obtain

( T q f T p ) ( x p ) a = x p a

for each x E 1 . Moreover, it follows from the previous equality that

( T q f T p ) ( x p ) ( T q f T p ) ( y p ) a 2 = ( T q f T p ) ( x p ) ( T q f T p ) ( y p ) , ( T q f T p ) ( x p ) ( T q f T p ) ( y p ) a = x p a 2 2 ( T q f T p ) ( x p ) , ( T q f T p ) ( y p ) a + y p a 2

and

( x p ) ( y p ) a 2 = ( x p ) ( y p ) , ( x p ) ( y p ) a = x p a 2 2 x p , y p a + y p a 2 .

Finally, comparing the last two equalities yields the validity of our assertion.□

## 4 First-order extension of isometries

In the previous section, we made all the necessary preparations to extend the domain E 1 of the surjective d a -isometry f : E 1 E 2 to its first-order generalized linear span GS ( E 1 , p ) .

Although E 1 is a bounded set, GS ( E 1 , p ) p is a real vector space. Now we will extend the d a -isometry T q f T p defined on the bounded set E 1 p to the d a -isometry T q F T p defined on the vector space GS ( E 1 , p ) p .

## Definition 4.1

Assume that E 1 and E 2 are nonempty bounded subsets of M a that are d a -isometric to each other via a surjective d a -isometry f : E 1 E 2 . Let p be a fixed element of E 1 and let q be an element of E 2 that satisfies q = f ( p ) . We define a function F : GS ( E 1 , p ) M a as

( T q F T p ) i = 1 m j = 1 α i j ( x i j p ) = i = 1 m j = 1 α i j ( T q f T p ) ( x i j p )

for any m N , x i j E 1 , and for all α i j R satisfying i = 1 m j = 1 α i j ( x i j p ) M a .

We note that in the definition above, it is important for the argument of T q F T p to belong to M a . Now, we show that the function F : GS ( E 1 , p ) M a is well defined.

## Lemma 4.1

Assume that E 1 and E 2 are bounded subsets of M a that are d a -isometric to each other via a surjective d a -isometry f : E 1 E 2 . Let p be an element of E 1 and let q be an element of E 2 that satisfy q = f ( p ) . The function F : GS ( E 1 , p ) M a given in Definition 4.1is well defined.

## Proof

First, we will check that the range of F is a subset of M a . For any m , n 1 , n 2 N with n 2 > n 1 , x i j E 1 , and for all α i j R , it follows from Lemma 3.2 that

(4.1) i = 1 m j = 1 n 2 α i j ( T q f T p ) ( x i j p ) i = 1 m j = 1 n 1 α i j ( T q f T p ) ( x i j p ) a 2 = i = 1 m j = n 1 + 1 n 2 α i j ( T q f T p ) ( x i j p ) , k = 1 m = n 1 + 1 n 2 α k ( T q f T p ) ( x k p ) a = i = 1 m k = 1 m j = n 1 + 1 n 2 α i j = n 1 + 1 n 2 α k ( T q f T p ) ( x i j p ) , ( T q f T p ) ( x k p ) a = i = 1 m k = 1 m j = n 1 + 1 n 2 α i j = n 1 + 1 n 2 α k x i j p , x k p a = i = 1 m j = n 1 + 1 n 2 α i j ( x i j p ) , k = 1 m = n 1 + 1 n 2 α k ( x k p ) a = i = 1 m j = n 1 + 1 n 2 α i j ( x i j p ) a 2 = i = 1 m j = 1 n 2 α i j ( x i j p ) i = 1 m j = 1 n 1 α i j ( x i j p ) a 2 .

Indeed, equality (4.1) holds for all m , n 1 , n 2 N .

We now assume that i = 1 m j = 1 α i j ( x i j p ) M a for some x i j E 1 and α i j R , where m is a fixed positive integer. Then, since ( M a , T a ) is a Hausdorff space on account of Remark 2.2 ( i i ) and the topology T a is consistent with the metric d a and with the norm a , the sequence i = 1 m j = 1 n α i j ( x i j p ) n converges to i = 1 m j = 1 α i j ( x i j p ) (in M a ), and hence, the sequence i = 1 m j = 1 n α i j ( x i j p ) n is a Cauchy sequence in M a .

We know by (4.1) and the definition of Cauchy sequences that for each ε > 0 , there exists an integer N ε > 0 such that

i = 1 m j = 1 n 2 α i j ( T q f T p ) ( x i j p ) i = 1 m j = 1 n 1 α i j ( T q f T p ) ( x i j p ) a = i = 1 m j = 1 n 2 α i j ( x i j p ) i = 1 m j = 1 n 1 α i j ( x i j p ) a < ε

for all integers n 1 , n 2 > N ε , which implies that i = 1 m j = 1 n α i j ( T q f T p ) ( x i j p ) n is also a Cauchy sequence in M a . By Remark 2.2 ( i ) , we observe that ( M a , , a ) is a real Hilbert space. Thus, M a is not only complete, but also a Hausdorff space, so the Cauchy sequence i = 1 m j = 1 n α i j ( T q f T p ) ( x i j p ) n converges in M a , i.e., by Definition 4.1, we have

( T q F T p ) i = 1 m j = 1 α i j ( x i j p ) = i = 1 m j = 1 α i j ( T q f T p ) ( x i j p ) = lim n i = 1 m j = 1 n α i j ( T q f T p ) ( x i j p ) M a ,

which implies

F p + i = 1 m j = 1 α i j ( x i j p ) M a + q = M a

for all x i j E 1 and α i j R with i = 1 m j = 1 α i j ( x i j p ) M a , i.e., the image of each element of GS ( E 1 , p ) under F belongs to M a .

We now assume that i = 1 m 1 j = 1 α i j ( x i j p ) = i = 1 m 2 j = 1 β i j ( y i j p ) M a for some m 1 , m 2 N , x i j , y i j E 1 , and α i j , β i j R . It then follows from Definition 4.1 and Lemma 3.2 that

( T q F T p ) i = 1 m 1 j = 1 α i j ( x i j p ) ( T q F T p ) i = 1 m 2 j = 1 β i j ( y i j p ) a 2 = i = 1 m 1 j = 1 α i j ( T q f T p ) ( x i j p ) i = 1 m 2 j = 1 β i j ( T q f T p ) ( y i j p ) a 2 = i = 1 m 1 j = 1 α i j ( T q f T p ) ( x i j p ) i = 1 m 2 j = 1 β i j ( T q f T p ) ( y i j p ) , k = 1 m 1 = 1 α k ( T q f T p ) ( x k p ) k = 1 m 2 = 1 β k ( T q f T p ) ( y k p ) a = = i = 1 m 1 j = 1 α i j ( x i j p ) i = 1 m 2 j = 1 β i j ( y i j p ) , k = 1 m 1 = 1 α k ( x k p ) k = 1 m 2 = 1 β k ( y k p ) a = i = 1 m 1 j = 1 α i j ( x i j p ) i = 1 m 2 j = 1 β i j ( y i j p ) a 2 = 0 ,

which implies that

( T q F T p ) i = 1 m 1 j = 1 α i j ( x i j p ) = ( T q F T p ) i = 1 m 2 j = 1 β i j ( y i j p )

for all m 1 , m 2 N , x i j , y i j E 1 , and α i j , β i j R , satisfying i = 1 m 1 j = 1 α i j ( x i j p ) = i = 1 m 2 j = 1 β i j ( y i j p ) M a .□

We prove in the following theorem that the domain of a d a -isometry f : E 1 E 2 can be extended to the first-order generalized linear span GS ( E 1 , p ) whenever E 1 is a nonempty bounded subset of M a . Therefore, Theorem 4.2 is a generalization of [19, Theorem 2.2] for M a .

## Theorem 4.2

Assume that E 1 and E 2 are bounded subsets of M a that are d a -isometric to each other via a surjective d a -isometry f : E 1 E 2 . Assume that p is an element of E 1 and q is an element of E 2 with q = f ( p ) . The function F : GS ( E 1 , p ) M a defined in Definition 4.1is a d a -isometry and the function T q F T p : GS ( E 1 , p ) p M a is a linear d a -isometry. In particular, F is an extension of f.

## Proof

( a ) Let u and v be arbitrary elements of the first-order generalized linear span GS ( E 1 , p ) of E 1 with respect to p . Then,

(4.2) u p = i = 1 m j = 1 α i j ( x i j p ) M a and v p = i = 1 n j = 1 β i j ( y i j p ) M a

for some m , n N , x i j , y i j E 1 , and α i j , β i j R . Then, according to Definition 4.1, we have

(4.3) ( T q F T p ) ( u p ) = i = 1 m j = 1 α i j ( T q f T p ) ( x i j p ) , ( T q F T p ) ( v p ) = i = 1 n j = 1 β i j ( T q f T p ) ( y i j p ) .

( b ) By Lemma 3.2, (4.2), and (4.3), we obtain

(4.4) ( T q F T p ) ( u p ) , ( T q F T p ) ( v p ) a = i = 1 m j = 1 α i j ( T q f T p ) ( x i j p ) , k = 1 n = 1 β k ( T q f T p ) ( y k p ) a = i = 1 m k = 1 n j = 1 α i j = 1 β k ( T q f T p ) ( x i j p ) , ( T q f T p ) ( y k p ) a = i = 1 m k = 1 n j = 1 α i j = 1 β k x i j p , y k p a = i = 1 m j = 1 α i j ( x i j p ) , k = 1 n = 1 β k ( y k p ) a = u p , v p a

for all u , v GS ( E 1 , p ) . That is, T q F T p preserves the inner product. Indeed, equality (4.4) is an extended version of Lemma 3.2.

( c ) By using equality (4.4), we further obtain

d a ( F ( u ) , F ( v ) ) 2 = F ( u ) F ( v ) a 2 = ( T q F T p ) ( u p ) ( T q F T p ) ( v p ) a 2 = ( T q F T p ) ( u p ) ( T q F T p ) ( v p ) ,

( T q F T p ) ( u p ) ( T q F T p ) ( v p ) a = u p , u p a u p , v p a v p , u p a + v p , v p a = ( u p ) ( v p ) , ( u p ) ( v p ) a = ( u p ) ( v p ) a 2 = u v a 2 = d a ( u , v ) 2

for all u , v GS ( E 1 , p ) , i.e., F is a d a -isometry.

( d ) Now, let u and v be arbitrary elements of GS ( E 1 , p ) . Then, it holds that u p GS ( E 1 , p ) p , v p GS ( E 1 , p ) p , and α ( u p ) + β ( v p ) GS ( E 1 , p ) p for any α , β R , because GS ( E 1 , p ) p is a real vector space.

We obtain

( T q F T p ) ( α ( u p ) + β ( v p ) ) α ( T q F T p ) ( u p ) β ( T q F T p ) ( v p ) a 2 = ( T q F T p ) ( α ( u p ) + β ( v p ) ) α ( T q F T p ) ( u p ) β ( T q F T p ) ( v p ) , ( T q F T p ) ( α ( u p ) + β ( v p ) ) α ( T q F T p ) ( u p ) β ( T q F T p ) ( v p ) a .

Since α ( u p ) + β ( v p ) = w p for some w GS ( E 1 , p ) , we further use (4.4) to obtain

( T q F T p ) ( α ( u p ) + β ( v p ) ) α ( T q F T p ) ( u p ) β ( T q F T p ) ( v p ) a 2 = w p , w p a α w p , u p a β w p , v p a α u p , w p a + α 2 u p , u p a + α β u p , v p a β v p , w p a + α β v p , u p a + β 2 v p , v p a = 0 ,

which implies that the function T q F T p : GS ( E 1 , p ) p M a is linear.

( e ) Finally, we set α 11 = 1 and α i j = 0 for any ( i , j ) ( 1 , 1 ) , and x 11 = x in (4.2) and (4.3) to see

( T q F T p ) ( x p ) = ( T q f T p ) ( x p )

for every x E 1 , which implies that F ( x ) = f ( x ) for every x E 1 , i.e., F is an extension of f .□

## 5 Second-order generalized linear span

For any element x of M a and r > 0 , we denote by B r ( x ) the open ball defined by B r ( x ) = { y M a : y x a < r } .

Definitions 3.1 and 4.1 will be generalized to the cases of n 2 in the following definition. We introduce the concept of n th-order generalized linear span GS n ( E 1 , p ) , which generalizes the concept of first-order generalized linear span GS ( E , p ) . Moreover, we define the d a -isometry F n , which extends the domain of a d a -isometry f to GS n ( E 1 , p ) .

It is surprising, however, that this process of generalization does not go far. Indeed, we will find in Proposition 5.4 and Theorem 7.2 that GS 2 ( E 1 , p ) and F 2 are their limits.

## Definition 5.1

Let E 1 be a nonempty bounded subset of M a that is d a -isometric to a subset E 2 of M a via a surjective d a -isometry f : E 1 E 2 . Let p be an element of E 1 and q an element of E 2 with q = f ( p ) . Assume that r is a positive real number satisfying E 1 B r ( p ) .

1. We define GS 0 ( E 1 , p ) = E 1 and GS 1 ( E 1 , p ) = GS ( E 1 , p ) . In general, we define the nth-order generalized linear span of E 1 with respect to p as GS n ( E 1 , p ) = GS ( GS n 1 ( E 1 , p ) B r ( p ) , p ) for all n N .

2. We define F 0 = f and F 1 = F , where F is defined in Definition 4.1. Moreover, for any n N , we define the function F n : GS n ( E 1 , p ) M a by

( T q F n T p ) i = 1 m j = 1 α i j ( x i j p ) = i = 1 m j = 1 α i j ( T q F n 1 T p ) ( x i j p )

for all m N , x i j GS n 1 ( E 1 , p ) B r ( p ) , and α i j R with i = 1 m j = 1 α i j ( x i j p ) M a .

## Proposition 5.1

Let E be a nonempty bounded subset of M a . If s and t are positive real numbers that satisfy E B s ( p ) B t ( p ) , then

GS ( GS ( E , p ) B s ( p ) , p ) = GS ( GS ( E , p ) B t ( p ) , p ) .

## Proof

Assume that 0 < s < t . Then, there exists a real number c > 1 with s > t c , and it is obvious that B t c ( p ) B s ( p ) . Assume that x is an arbitrary element of GS ( GS ( E , p ) B t ( p ) , p ) . Then, there exist some m N , u i j GS ( E , p ) B t ( p ) , and α i j R such that x = p + i = 1 m j = 1 α i j ( u i j p ) M a . We note that

( GS ( E , p ) p ) ( B t ( p ) p ) = { u p M a : u GS ( E , p ) B t ( p ) } .

Since GS ( E , p ) p is a real vector space, t c < s , and since u i j p ( GS ( E , p ) p ) ( B t ( p ) p ) for any i and j , we have

1 c ( u i j p ) ( GS ( E , p ) p ) ( B s ( p ) p ) .

Hence, we can choose a v i j GS ( E , p ) B s ( p ) such that 1 c ( u i j p ) = v i j p . Thus, we obtain

x = p + i = 1 m j = 1 α i j ( u i j p ) = p + i = 1 m j = 1 c α i j ( v i j p ) GS ( GS ( E , p ) B s ( p ) , p ) ,

which implies that GS ( GS ( E , p ) B t ( p ) , p ) GS ( GS ( E , p ) B s ( p ) , p ) .

The reverse inclusion is obvious, since B s ( p ) B t ( p ) .□

We generalize Lemma 3.2 and formula (4.4) in the following lemma. Indeed, we prove that the function T q F n T p : GS n ( E 1 , p ) p M a preserves the inner product. This property is important in proving the following theorems as a necessary condition for F n to be a d a -isometry.

## Lemma 5.2

Let E 1 be a bounded subset of M a that is d a -isometric to a subset E 2 of M a via a surjective d a -isometry f : E 1 E 2 . Assume that p and q are elements of E 1 and E 2 , which satisfy q = f ( p ) . If n N , then

( T q F n T p ) ( u p ) , ( T q F n T p ) ( v p ) a = u p , v p a

for all u , v GS n ( E 1 , p ) .

## Proof

Our assertion for n = 1 was already proved in (4.4). Considering Proposition 5.1, assume that r is a positive real number satisfying E 1 B r ( p ) . Now we assume that the assertion is true for some n N . Let u , v be arbitrary elements of GS n + 1 ( E 1 , p ) . Then, there exist some m 1 , m 2 N , x i j , y k GS n ( E 1 , p ) B r ( p ) , and α i j , β k R such that

u p = i = 1 m 1 j = 1 α i j ( x i j p ) M a and v p = k = 1 m 2 = 1 β k ( y k p ) M a .

Using Definition 5.1 ( i i ) and our assumption, we obtain

( T q F n + 1 T p ) ( u p ) , ( T q F n + 1 T p ) ( v p ) a = i = 1 m 1 j = 1 α i j ( T q F n T p ) ( x i j p ) , k = 1 m 2 = 1 β k ( T q F n T p ) ( y k p ) a

= i = 1 m 1 k = 1 m 2 j = 1 α i j = 1 β k ( T q F n T p ) ( x i j p ) , ( T q F n T p ) ( y k p )