Extension of isometries in real Hilbert spaces

Soon-Mo Jung

doi:10.1515/math-2022-0518

Article Open Access

Extension of isometries in real Hilbert spaces

Soon-Mo Jung

Published/Copyright: October 28, 2022

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$Open Mathematics$

From the journal Open Mathematics Volume 20 Issue 1

Abstract

The main purpose of this article is to develop a theory that extends the domain of any local isometry to the whole space containing the domain, where a local isometry is an isometry between two proper subsets. In fact, the main purpose of this article consists of the following three detailed objectives: The first objective is to extend the bounded domain of any local isometry to the first-order generalized linear span. The second one is to extend the bounded domain of any local isometry to the second-order generalized linear span. The third objective of this article is to extend the bounded domain of any local isometry to the whole Hilbert space.

Keywords: isometry; extension of isometry; generalized linear span

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]).

The research in this article is strongly motivated by Theorems 1.2 and 1.3, and [14, Theorem 2.5], among others (refer to [15,16,17] also).

The main purpose of this article is to develop a theory that extends the (bounded) domain of any local isometry to the real Hilbert space M a containing the domain, where a local isometry is an isometry between two proper subsets of the Hilbert space M a , which is defined in Section 2 of this article. In Section 3, we introduce some concepts such as first-order generalized linear span and index set, which are essential to prove the final result of this article. Section 4 is devoted to the problem of extending the domain of a local isometry to the first-order generalized linear span. Solving this problem is the first objective of this article. We introduce the concept of a second-order generalized linear span in Section 5 and develop the theory of extension of the domain of a local isometry to the second-order generalized linear span in Section 7, which is the second objective of this article. Finally, we prove in Theorem 8.1 that the domain of a local isometry can be extended to the real Hilbert space M a including that domain, which is the third objective of this article.

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

( M a , ⟨ ⋅ , ⋅ ⟩ a ) is a Hilbert space over R ;
( 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 .

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.
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 4.1 is 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 ∑ ℓ = n 1 + 1 n 2 α k ℓ ( T − q ∘ f ∘ T p ) ( x k ℓ − p ) a = ∑ i = 1 m ∑ k = 1 m ∑ j = n 1 + 1 n 2 α i j ∑ ℓ = n 1 + 1 n 2 α k ℓ ⟨ ( T − q ∘ f ∘ T p ) ( x i j − p ) , ( T − q ∘ f ∘ T p ) ( x k ℓ − p ) ⟩ a = ∑ i = 1 m ∑ k = 1 m ∑ j = n 1 + 1 n 2 α i j ∑ ℓ = n 1 + 1 n 2 α k ℓ ⟨ x i j − p , x k ℓ − p ⟩ a = ∑ i = 1 m ∑ j = n 1 + 1 n 2 α i j ( x i j − p ) , ∑ k = 1 m ∑ ℓ = n 1 + 1 n 2 α k ℓ ( x k ℓ − 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 ∑ ℓ = 1 ∞ α k ℓ ( T − q ∘ f ∘ T p ) ( x k ℓ − p ) − ∑ k = 1 m 2 ∑ ℓ = 1 ∞ β k ℓ ( T − q ∘ f ∘ T p ) ( y k ℓ − 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 ∑ ℓ = 1 ∞ α k ℓ ( x k ℓ − p ) − ∑ k = 1 m 2 ∑ ℓ = 1 ∞ β k ℓ ( y k ℓ − 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 4.1 is 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 ∑ ℓ = 1 ∞ β k ℓ ( T − q ∘ f ∘ T p ) ( y k ℓ − p ) a = ∑ i = 1 m ∑ k = 1 n ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ β k ℓ ⟨ ( T − q ∘ f ∘ T p ) ( x i j − p ) , ( T − q ∘ f ∘ T p ) ( y k ℓ − p ) ⟩ a = ∑ i = 1 m ∑ k = 1 n ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ β k ℓ ⟨ x i j − p , y k ℓ − p ⟩ a = ∑ i = 1 m ∑ j = 1 ∞ α i j ( x i j − p ) , ∑ k = 1 n ∑ ℓ = 1 ∞ β k ℓ ( y k ℓ − 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 ) .

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 .
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

⟨ ( 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 ℓ ∈ GS n ( E 1 , p ) ∩ B r ( p ) , and α i j , β k ℓ ∈ 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 ∑ ℓ = 1 ∞ β k ℓ ( y k ℓ − 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 ∑ ℓ = 1 ∞ β k ℓ ( T − q ∘ F n ∘ T p ) ( y k ℓ − p ) a

= ∑ i = 1 m 1 ∑ k = 1 m 2 ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ β k ℓ ⟨ ( T − q ∘ F n ∘ T p ) ( x i j − p ) , ( T − q ∘ F n ∘ T p ) ( y k ℓ − p ) ⟩ a = ∑ i = 1 m 1 ∑ k = 1 m 2 ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ β k ℓ ⟨ x i j − p , y k ℓ − p ⟩ a = ∑ i = 1 m 1 ∑ j = 1 ∞ α i j ( x i j − p ) , ∑ k = 1 m 2 ∑ ℓ = 1 ∞ β k ℓ ( y k ℓ − p ) a = ⟨ u − p , v − p ⟩ a

for all u , v ∈ GS n + 1 ( E 1 , p ) . By mathematical induction, we may then conclude that our assertion is true for all n ∈ N .□

When n = 1 and p = p ′ , the first assertion in ( i ) of the following lemma is obvious, so we have used that fact several times before, omitting the proof. The assertion ( i v ) in the following lemma seems to be related in some way to Proposition 5.1.

Lemma 5.3

Let E be a bounded subset of M a and p , p ′ ∈ E . Assume that r is a positive real number satisfying E ⊂ B r ( p ) .

GS n ( E , p ) − p ′ is a vector space over R for each n ∈ N .
GS n ( E , p ) ⊂ GS n + 1 ( E , p ) for each n ∈ N .
GS 2 ( E , p ) = GS ( E , p ) ¯ , where GS ( E , p ) ¯ is the closure of GS ( E , p ) in M a .
Λ ( GS n ( E , p ) ) = Λ ( GS n ( E , p ) ∩ B r ( p ) ) for all n ∈ N .

Proof

( i ) By using Definitions 3.1 and 5.1, we prove that GS ( E , p ) − p ′ is a real vector space. (We can prove similarly for the case of n > 1 .) Given x , y ∈ GS ( E , p ) − p ′ , we may choose some m 1 , m 2 ∈ N , u i j , v i j ∈ E , and α i j , β i j ∈ R such that x = ( p − p ′ ) + ∑ i = 1 m 1 ∑ j = 1 ∞ α i j ( u i j − p ) ∈ M a and y = ( p − p ′ ) + ∑ i = 1 m 2 ∑ j = 1 ∞ β i j ( v i j − p ) ∈ M a . Since M a is a real vector space, α ∑ i = 1 m 1 ∑ j = 1 ∞ α i j ( u i j − p ) + β ∑ i = 1 m 2 ∑ j = 1 ∞ β i j ( v i j − p ) ∈ M a for all α , β ∈ R .

Moreover, we see that

α x + β y = p + ( 1 − α − β ) ( p ′ − p ) + ∑ i = 1 m 1 ∑ j = 1 ∞ α α i j ( u i j − p ) + ∑ i = 1 m 2 ∑ j = 1 ∞ β β i j ( v i j − p ) − p ′ ∈ GS ( E , p ) − p ′

for all α , β ∈ R . Hence, GS ( E , p ) − p ′ is a real vector space as a subspace of real vector space M a .

( i i ) Let r be a positive real number with E ⊂ B r ( p ) . If x ∈ GS n ( E , p ) for some n ∈ N , then x − p ∈ GS n ( E , p ) − p . Since GS n ( E , p ) − p is a real vector space by ( i ) and B r ( p ) − p = B r ( 0 ) , we can choose a (positive or negative but sufficiently small) real number μ ≠ 0 such that μ ( x − p ) ∈ ( GS n ( E , p ) − p ) ∩ ( B r ( p ) − p ) . We note that

(5.1) ( GS n ( E , p ) − p ) ∩ ( B r ( p ) − p ) = { v − p ∈ M a : v ∈ GS n ( E , p ) ∩ B r ( p ) } .

Thus, we see that μ ( x − p ) = v − p for some v ∈ GS n ( E , p ) ∩ B r ( p ) . Since x = p + 1 μ ( v − p ) , it holds that x ∈ GS n + 1 ( E , p ) . Therefore, we conclude that GS n ( E , p ) ⊂ GS n + 1 ( E , p ) for every n ∈ N .

( i i i ) Let x be an arbitrary element of GS ( E , p ) ¯ . Then, there exists some sequence { x n } in GS ( E , p ) that converges to x , where x n ≠ x for all n ∈ N . We now set y 1 = x 1 and y i = x i − x i − 1 for each integer i ≥ 2 . Then, we have

x n = ∑ i = 1 n y i ,

where y i = ( x i − p ) − ( x i − 1 − p ) ∈ GS ( E , p ) − p for i ≥ 2 . Since GS ( E , p ) − p is a real vector space and B r ( p ) − p = B r ( 0 ) , we can select a real number μ i ≠ 0 such that

μ i y i ∈ GS ( E , p ) − p and μ i y i ∈ B r ( p ) − p

for every integer i ≥ 2 . Thus, it follows from (5.1) that

x n = ∑ i = 1 n y i = y 1 + ∑ i = 2 n 1 μ i ( μ i y i ) = x 1 + ∑ i = 2 n 1 μ i ( v i − p ) ,

where v i ∈ GS ( E , p ) ∩ B r ( p ) for i ≥ 2 . Since the sequence { x n } is assumed to converge to x , the sequence x 1 + ∑ i = 2 n 1 μ i ( v i − p ) n converges to x . Hence, we have

(5.2) x 1 + ∑ i = 2 ∞ 1 μ i ( v i − p ) = lim n → ∞ x n = x ∈ M a .

(Since M a is a Hausdorff space, x is the unique limit point of the sequence { x n } .)

Furthermore, there exists a real number μ 1 ≠ 0 that satisfies μ 1 ( x 1 − p ) ∈ GS ( E , p ) − p and μ 1 ( x 1 − p ) ∈ B r ( p ) − p , i.e., μ 1 ( x 1 − p ) ∈ ( GS ( E , p ) − p ) ∩ ( B r ( p ) − p ) . Thus, there exists a v 1 ∈ GS ( E , p ) ∩ B r ( p ) such that μ 1 ( x 1 − p ) = v 1 − p or x 1 − p = 1 μ 1 ( v 1 − p ) . Therefore,

(5.3) x = p + ( x 1 − p ) + ∑ i = 2 ∞ 1 μ i ( v i − p ) = p + ∑ i = 1 ∞ 1 μ i ( v i − p ) ,

where v i ∈ GS ( E , p ) ∩ B r ( p ) for each i ∈ N . It follows from (5.2) that ∑ i = 1 ∞ 1 μ i ( v i − p ) ∈ M a . Thus, by (5.3), we see that x ∈ GS 2 ( E , p ) , which implies that GS ( E , p ) ¯ ⊂ GS 2 ( E , p ) .

On the other hand, let y ∈ GS 2 ( E , p ) . Then, there are some m ∈ N , v i j ∈ GS ( E , p ) ∩ B r ( p ) , and α i j ∈ R such that y = p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( v i j − p ) ∈ M a . Let us define y n = p + ∑ i = 1 m ∑ j = 1 n α i j ( v i j − p ) for every n ∈ N . Since v i j − p ∈ GS ( E , p ) − p for all i and j and GS ( E , p ) − p is a real vector space, we know that y n − p = ∑ i = 1 m ∑ j = 1 n α i j ( v i j − p ) ∈ GS ( E , p ) − p , and hence, y n ∈ GS ( E , p ) for all n ∈ N . Since GS ( E , p ) is a Hausdorff space, y is the unique limit point of the sequence { y n } . Thus, we see that

y = p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( v i j − p ) = lim n → ∞ y n ∈ GS ( E , p ) ¯ ,

which implies that GS 2 ( E , p ) ⊂ GS ( E , p ) ¯ .

( i v ) Let i ∈ Λ ( GS n ( E , p ) ) . In view of Definition 3.2, there exist x ∈ GS n ( E , p ) and α ≠ 0 with x + α e i ∈ GS n ( E , p ) . Furthermore, x = p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) for some m ∈ N , u i j ∈ GS n − 1 ( E , p ) ∩ B r ( p ) , and α i j ∈ R . Since ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) + α e i = x − p + α e i ∈ GS n ( E , p ) − p , it holds that μ ( ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) + α e i ) ∈ ( GS n ( E , p ) − p ) ∩ ( B r ( p ) − p ) for any sufficiently small μ ≠ 0 , or equivalently, it follows from (5.1) that

(5.4) p + ∑ i = 1 m ∑ j = 1 ∞ μ α i j ( u i j − p ) + μ α e i ∈ GS n ( E , p ) ∩ B r ( p ) .

On the other hand, since ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) = x − p ∈ GS n ( E , p ) − p , it holds that ∑ i = 1 m ∑ j = 1 ∞ μ α i j ( u i j − p ) ∈ ( GS n ( E , p ) − p ) ∩ ( B r ( p ) − p ) for any sufficiently small μ ≠ 0 . Hence, it follows from (5.1) that p + ∑ i = 1 m ∑ j = 1 ∞ μ α i j ( u i j − p ) ∈ GS n ( E , p ) ∩ B r ( p ) for any sufficiently small μ ≠ 0 . Thus, by Definition 3.2 and (5.4), it holds that i ∈ Λ ( GS n ( E , p ) ∩ B r ( p ) ) , which implies that Λ ( GS n ( E , p ) ) ⊂ Λ ( GS n ( E , p ) ∩ B r ( p ) ) . Obviously, the inverse inclusion is true.□

As we mentioned earlier, we will see that the second-order generalized linear span is the last step in this kind of domain extension.

Proposition 5.4

If E is a bounded subset of M a and p ∈ E , then

E ⊂ GS ( E , p ) ⊂ GS ( E , p ) ¯ = GS 2 ( E , p ) = GS n ( E , p )

for any integer n ≥ 3 . Indeed, GS n ( E , p ) − p is a real Hilbert space for n ≥ 2 .

Proof

( a ) Considering Proposition 5.1, we can choose a real number r > 0 that satisfies E ⊂ B r ( p ) . Assume that x ∈ GS 3 ( E , p ) . Then, there exist some m 0 ∈ N , u i j ∈ GS 2 ( E , p ) ∩ B r ( p ) , and α i j ∈ R such that x = p + ∑ i = 1 m 0 ∑ j = 1 ∞ α i j ( u i j − p ) ∈ M a .

We define x m = p + ∑ i = 1 m 0 ∑ j = 1 m α i j ( u i j − p ) for each m ∈ N . Since u i j ∈ GS 2 ( E , p ) , there exist some m i j ∈ N , v i j k ℓ ∈ GS ( E , p ) ∩ B r ( p ) , and β i j k ℓ ∈ R such that u i j = p + ∑ k = 1 m i j ∑ ℓ = 1 ∞ β i j k ℓ ( v i j k ℓ − p ) ∈ M a . Hence, it holds that

x m = p + ∑ i = 1 m 0 ∑ j = 1 m ∑ k = 1 m i j ∑ ℓ = 1 ∞ α i j β i j k ℓ ( v i j k ℓ − p ) ∈ M a ,

which implies that x m ∈ GS 2 ( E , p ) for all m ∈ N . Thus, { x m } is a sequence in GS 2 ( E , p ) that converges to x . Therefore, x ∈ GS 2 ( E , p ) because GS 2 ( E , p ) is closed. Thus, GS 3 ( E , p ) ⊂ GS 2 ( E , p ) . The inverse inclusion is of course true due to Lemma 5.3 ( i i ) . We have proved that GS 2 ( E , p ) = GS 3 ( E , p ) .

( b ) Assume that GS 2 ( E , p ) = ⋯ = GS n ( E , p ) = GS n + 1 ( E , p ) for some integer n ≥ 2 .

( c ) If we replace GS ( E , p ) , GS 2 ( E , p ) , and GS 3 ( E , p ) in the previous part ( a ) with GS n ( E , p ) , GS n + 1 ( E , p ) , and GS n + 2 ( E , p ) , respectively, and if we consider the fact that GS n + 1 ( E , p ) = GS 2 ( E , p ) is closed in M a by Lemma 5.3 ( i i i ) and our assumption ( b ), then we arrive at the conclusion that GS n + 1 ( E , p ) = GS n + 2 ( E , p ) .

( d ) With the conclusion of mathematical induction we prove that GS n ( E , p ) = GS 2 ( E , p ) for every integer n ≥ 3 . Moreover, when n ≥ 2 , GS n ( E , p ) is complete as a closed subset of a real Hilbert space M a (ref. Remark 2.2). Therefore, GS n ( E , p ) − p is a real Hilbert space for n ≥ 2 .□

The following lemma is an extension of Lemma 3.1 for the second-order generalized linear span GS 2 ( E , p ) . Indeed, we prove that if i ∈ Λ ( GS 2 ( E , p ) ) , then the second-order generalized linear span of E contains all the lines through GS ( E , p ) in the direction e i .

Lemma 5.5

Assume that a bounded subset E of M a contains at least two elements and p ∈ E . If i ∈ Λ ( GS 2 ( E , p ) ) and p ′ ∈ GS ( E , p ) , then p ′ + α i e i ∈ GS 2 ( E , p ) for any α i ∈ R .

Proof

Let r be a positive real number with E ⊂ B r ( p ) . Assume that i ∈ Λ ( GS 2 ( E , p ) ) . Considering Lemma 5.3 ( i v ) and Proposition 5.4, if we substitute GS 2 ( E , p ) ∩ B r ( p ) for E in Lemma 3.1, then p + α i e i ∈ GS 3 ( E , p ) = GS 2 ( E , p ) for all α i ∈ R . Thus, there are some m ∈ N , w i j ∈ GS ( E , p ) ∩ B r ( p ) , and β i j ∈ R with ∑ i = 1 m ∑ j = 1 ∞ β i j ( w i j − p ) ∈ M a such that p + α i e i = p + ∑ i = 1 m ∑ j = 1 ∞ β i j ( w i j − p ) , and hence, we have

(5.5) p ′ + α i e i = p + α i e i + ( p ′ − p ) = p + ∑ i = 1 m ∑ j = 1 ∞ β i j ( w i j − p ) + ( p ′ − p ) .

Because p ′ − p belongs to GS ( E , p ) − p , which is a real vector space by Lemma 5.3 ( i ) , and B r ( p ) − p = B r ( 0 ) , we can choose some sufficiently small real number μ ≠ 0 such that

(5.6) μ ( p ′ − p ) ∈ GS ( E , p ) − p and μ ( p ′ − p ) ∈ B r ( p ) − p .

Considering (5.1), (5.5), and (5.6), if we put μ ( p ′ − p ) = w − p with a w ∈ GS ( E , p ) ∩ B r ( p ) , then we have

p ′ + α i e i = p + ∑ i = 1 m ∑ j = 1 ∞ β i j ( w i j − p ) + 1 μ ( w − p ) ∈ GS 2 ( E , p )

for all α i ∈ R .□

6 Basic cylinders and basic intervals

First, we will define the infinite dimensional intervals, which were simply defined in [14], more precisely divided into nondegenerate basic cylinders, degenerate basic cylinders, and basic intervals.

Definition 6.1

For any positive integer n , we define the infinite dimensional interval by

J = ∏ i = 1 ∞ J i , where J i = [ 0 , p 2 i ] ( for i ∈ Λ 1 ) , [ p 1 i , p 2 i ] ( for i ∈ Λ 2 ) , [ p 1 i , 1 ] ( for i ∈ Λ 3 ) , { p 1 i } ( for i ∈ Λ 4 ) , [ 0 , 1 ] ( otherwise )

for some disjoint finite subsets Λ 1 , Λ 2 , and Λ 3 of { 1 , 2 , … , n } and 0 < p 1 i < p 2 i < 1 for i ∈ Λ 1 ∪ Λ 2 ∪ Λ 3 and 0 ≤ p 1 i ≤ 1 for i ∈ Λ 4 . If Λ 4 = ∅ , then J is called a nondegenerate basic cylinder. When Λ 4 is a nonempty finite set, J is called a degenerate basic cylinder. If Λ 4 is an infinite set, then J will be called a basic interval.

Remark 6.1

In order for an infinite dimensional interval J to become a basic cylinder, Λ 4 must be a finite set.
We remark that Λ 4 = N ⧹ Λ ( J ) and Λ ( J ) = N ⧹ Λ 4 . That is, N is the disjoint union of Λ ( J ) and Λ 4 .
If p = ( p 1 , p 2 , … , p i , … ) is an element of J , then J i = { p i } for each i ∉ Λ ( J ) .

We note that the basic cylinder or the basic interval J defined in Definition 6.1 can be expressed as

J = ∑ i = 1 ∞ α i 1 a i e i : α i ∈ a i J i for all i ∈ N ,

where J i is the interval defined in Definition 6.1.

Definition 6.2

Let β = { β i } i ∈ N be a complete orthonormal sequence in M a , J i the interval given in Definition 6.1, and let n be a positive integer. We define

J β = ∑ i = 1 ∞ α i β i : α i ∈ a i J i for all i ∈ N

for some disjoint finite subsets Λ 1 , Λ 2 , and Λ 3 of { 1 , 2 , … , n } ; 0 < p 1 i < p 2 i < 1 for i ∈ Λ 1 ∪ Λ 2 ∪ Λ 3 ; and 0 ≤ p 1 i ≤ 1 for i ∈ Λ 4 . If Λ 4 = ∅ , then J β is called a nondegenerate β -basic cylinder. When Λ 4 is a nonempty finite set, J β is called a degenerate β -basic cylinder. If Λ 4 is an infinite set, then J β will be called a β -basic interval.

Using Definitions 6.1 and 6.2, Remark 6.1 ( i i ) is generalized as follows:

Remark 6.2

Let β = { β i } i ∈ N be a complete orthonormal sequence in M a and let J β be a β -basic cylinder or a β -basic interval. It holds that Λ β ( J β ) = N ⧹ Λ 4 , where Λ 4 is given in Definitions 6.1 and 6.2.

Proof

In general, if i ∈ Λ 4 , then it follows from Definition 6.2 that

⟨ x , β i ⟩ a = ∑ j = 1 ∞ α j β j , β i a = α i ∈ a i J i = { a i p 1 i }

for all x ∈ J β . That is, ⟨ x , β i ⟩ a = α i = a i p 1 i for all x ∈ J β and i ∈ Λ 4 . If i ∈ Λ 4 , then ⟨ x + α β i , β i ⟩ a = ⟨ x , β i ⟩ a + α = a i p 1 i + α ≠ a i p 1 i for all x ∈ J β and α ≠ 0 , which implies that x + α β i ∉ J β . That is, in view of Definition 3.2 ( i i ) , we conclude that i ∉ Λ β ( J β ) .

We now assume that i ∉ Λ β ( J β ) . Then, by Definition 3.2 ( i i ) , it holds that

(6.1) x + α β i ∉ J β

for any x ∈ J β and α ≠ 0 . Using Definition 6.2 again, we have

(6.2) x + α β i = ∑ j ∉ Λ 4 α j β j + ∑ j ∈ Λ 4 a j p 1 j β j + α β i

for all x ∈ J β and α ≠ 0 . We assume on the contrary that i ∉ Λ 4 . In view of (6.2) and by the structure of J i ( a i J i is indeed a nondegenerate interval for i ∉ Λ 4 ), it holds that

x + α β i = ∑ j ∉ Λ 4 ∪ { i } α j β j + ( α i + α ) β i + ∑ j ∈ Λ 4 a j p 1 j β j ∈ J β

for some x ∈ J β and α ≠ 0 , which is contrary to (6.1). (We note that, for each i ∉ Λ 4 , α i ∈ a i J i and there exists an α ≠ 0 satisfying α i + α ∈ a i J i .) Therefore, we conclude that if i ∉ Λ β ( J β ) , then i ∈ Λ 4 .□

Theorem 6.1

Let β = { β i } i ∈ N be a complete orthonormal sequence in M a and let J β be either a translation of a β -basic cylinder or a translation of a β -basic interval and p ∈ J β . Then,

GS ( J β , p ) = p + ∑ i ∈ Λ β ( J β ) α i β i ∈ M a : α i ∈ R for all i ∈ Λ β ( J β ) .

Proof

Assume that x is an arbitrary element of GS ( J β , p ) . By Definition 3.1, we have

x − p = ∑ i = 1 m ∑ j = 1 ∞ ε i j ( x i j − p ) ∈ M a

for some m ∈ N , ε i j ∈ R , and x i j ∈ J β . Furthermore, since x i j , p ∈ J β , by Definition 6.2, we obtain

x i j = ∑ k = 1 ∞ γ k β k = ∑ k ∈ N ⧹ Λ 4 γ k β k + ∑ k ∈ Λ 4 a k p 1 k β k

and

p = ∑ k = 1 ∞ δ k β k = ∑ k ∈ N ⧹ Λ 4 δ k β k + ∑ k ∈ Λ 4 a k p 1 k β k

for some γ k , δ k ∈ a k J k .

Since { β i } i ∈ N is a complete orthonormal sequence in M a , it follows from Definition 6.2 and Remark 6.2 that

x − p = ∑ i = 1 m ∑ j = 1 ∞ ε i j ( x i j − p ) = ∑ i = 1 m ∑ j = 1 ∞ ε i j ∑ k ∈ N ⧹ Λ 4 ( γ k − δ k ) β k = ∑ k ∈ N ⧹ Λ 4 ω k β k = ∑ i ∈ Λ β ( J β ) ω i β i

for some real numbers ω i . Since x ∈ GS ( J β , p ) ⊂ M a , it holds that

x = p + ∑ i ∈ Λ β ( J β ) ω i β i ( ∈ M a ) ∈ p + ∑ i ∈ Λ β ( J β ) α i β i ∈ M a : α i ∈ R for all i ∈ Λ β ( J β ) ,

which implies that

GS ( J β , p ) ⊂ p + ∑ i ∈ Λ β ( J β ) α i β i ∈ M a : α i ∈ R for all i ∈ Λ β ( J β ) .

It remains to prove the reverse inclusion. According to the structure of J β given in Definition 6.2, for each i ∈ Λ β ( J β ) , there exists a real number γ i ≠ 0 such that p + γ i β i ∈ J β . In other words, for each i ∈ Λ β ( J β ) , there exists a u i ∈ J β such that γ i β i = u i − p . Thus, if we assume that

p + ∑ i ∈ Λ β ( J β ) α i β i ∈ M a

for some α i ∈ R , then

p + ∑ i ∈ Λ β ( J β ) α i β i = p + ∑ i ∈ Λ β ( J β ) α i γ i ( γ i β i ) = p + ∑ i ∈ Λ β ( J β ) α i γ i ( u i − p ) ∈ GS ( J β , p ) ,

since u i ∈ J β for all i ∈ Λ β ( J β ) , which implies that

GS ( J β , p ) ⊃ p + ∑ i ∈ Λ β ( J β ) α i β i ∈ M a : α i ∈ R for all i ∈ Λ β ( J β ) .

We end the proof in this way.□

Since in some ways, index sets have some properties of dimensions in vector space, the following theorem may seem to be obvious.

Theorem 6.2

Assume that a bounded subset E of M a contains at least two elements and p ∈ E . Then, Λ ( GS 2 ( E , p ) ) = N if and only if GS 2 ( E , p ) = M a .

Proof

Let x be an arbitrary element of M a . There exist some real numbers α i such that

(6.3) x = ∑ i = 1 ∞ α i e i ∈ M a .

If Λ ( GS 2 ( E , p ) ) = N , then it follows from Lemma 5.5 that

p + α i e i ∈ GS 2 ( E , p )

for all i ∈ N . In other words,

α i e i ∈ GS 2 ( E , p ) − p

for all i ∈ N .

By Lemma 5.3 ( i ) , we obtain

x n ≔ ∑ i = 1 n α i e i ∈ GS 2 ( E , p ) − p

for any n ∈ N . Due to Lemma 5.3 ( i i i ) and (6.3), we further obtain

x = ∑ i = 1 ∞ α i e i = lim n → ∞ x n ∈ GS 2 ( E , p ) − p ,

which implies that M a ⊂ GS 2 ( E , p ) − p , or equivalently, M a ⊂ GS 2 ( E , p ) .

The reverse inclusion is trivial.□

7 Second-order extension of isometries

It was proved in Theorem 4.2 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 , whether degenerate or nondegenerate.

Now we generalize Theorem 4.2 into the following theorem. More precisely, we prove that the domain of f can be extended to its second-order generalized linear span GS 2 ( E 1 , p ) . It follows from Lemma 5.3 ( i i i ) that GS 2 ( E 1 , p ) = GS ( E 1 , p ) ¯ . Therefore, the following theorem is a further generalization of [19, Theorem 2.2].

Although the closure span ¯ E 1 of the linear span of E 1 is a real Hilbert space, it seems difficult to extend the domain E 1 of a local isometry to the Hilbert space span ¯ E 1 . However, we can extend the domain E 1 of a local isometry to the second-order generalized linear span, as we see in the following theorem. For this reason, we use the second-order generalized linear span instead of the closure of linear span of E 1 .

In the proof, we use the fact that GS n ( E 1 , p ) − p is a real vector space.

Theorem 7.1

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 ) . The function F 2 : GS 2 ( E 1 , p ) → M a is a d a -isometry and the function T − q ∘ F 2 ∘ T p : GS 2 ( E 1 , p ) − p → M a is linear. In particular, F 2 is an extension of F.

Proof

( a ) Suppose r is a positive real number satisfying E 1 ⊂ B r ( p ) . Referring to the changes presented in the table below and following the first part of proof of Theorem 4.2, we can easily prove that F 2 is a d a -isometry.

Theorem 4.2:	E 1	GS ( E 1 , p )	f	F	Definition 4.1	Lemma 3.2
Here:	GS ( E 1 , p ) ∩ B r ( p )	GS 2 ( E 1 , p )	F	F 2	Definition 5.1	Lemma 5.2

( b ) We prove the linearity of T − q ∘ F n ∘ T p : GS n ( E 1 , p ) − p → M a in a more general setting for n ≥ 2 . Referring to the changes presented in the table below and following ( d ) of the proof of Theorem 4.2, we can easily prove that T − q ∘ F n ∘ T p is linear.

Theorem 4.2:	GS ( E 1 , p )	F	(4.4)
Here:	GS n ( E 1 , p )	F n	Lemma 5.2

( c ) According to Definition 5.1 ( i ) , for any m ∈ N , x i j ∈ GS ( E 1 , p ) ∩ B r ( p ) , and any α i j ∈ R with ∑ i = 1 m ∑ j = 1 ∞ α i j ( x i j − p ) ∈ M a , there exists a u ∈ GS 2 ( E 1 , p ) satisfying

(7.1) u − p = ∑ i = 1 m ∑ j = 1 ∞ α i j ( x i j − p ) ∈ M a .

Due to Definition 5.1 ( i i ) , we further have

(7.2) ( T − q ∘ F 2 ∘ T p ) ( u − p ) = ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ F ∘ T p ) ( x i j − p ) .

If we set α 11 = 1 , α i j = 0 for each ( i , j ) ≠ ( 1 , 1 ) , and x 11 = x in (7.1) and (7.2), we see that

(7.3) ( T − q ∘ F 2 ∘ T p ) ( x − p ) = ( T − q ∘ F ∘ T p ) ( x − p )

for all x ∈ GS ( E 1 , p ) ∩ B r ( p ) .

Let w be an arbitrary element of GS ( E 1 , p ) . Then, w − p ∈ GS ( E 1 , p ) − p . Since GS ( E 1 , p ) − p is a real vector space and B r ( p ) − p = B r ( 0 ) , there exists a (sufficiently small) real number μ ≠ 0 such that

μ ( w − p ) ∈ ( GS ( E 1 , p ) − p ) ∩ ( B r ( p ) − p ) .

Hence, by (5.1), we can choose a v ∈ GS ( E 1 , p ) ∩ B r ( p ) such that μ ( w − p ) = v − p . Since both T − q ∘ F 2 ∘ T p and T − q ∘ F ∘ T p are linear and GS ( E 1 , p ) ⊂ GS 2 ( E 1 , p ) , it follows from (7.3) that

μ ( T − q ∘ F 2 ∘ T p ) ( w − p ) = ( T − q ∘ F 2 ∘ T p ) ( μ ( w − p ) ) = ( T − q ∘ F 2 ∘ T p ) ( v − p ) = ( T − q ∘ F ∘ T p ) ( v − p ) = ( T − q ∘ F ∘ T p ) ( μ ( w − p ) ) = μ ( T − q ∘ F ∘ T p ) ( w − p ) .

Therefore, it follows that ( T − q ∘ F 2 ∘ T p ) ( w − p ) = ( T − q ∘ F ∘ T p ) ( w − p ) for all w ∈ GS ( E 1 , p ) , i.e., F 2 ( w ) = F ( w ) for all w ∈ GS ( E 1 , p ) . In other words, F 2 is an extension of F . Also, because of Theorem 4.2, we see that F 2 is obviously an extension of f .□

On account of Proposition 5.4, it holds that

GS 2 ( E 1 , p ) = ⋯ = GS n − 1 ( E 1 , p ) = GS n ( E 1 , p )

for every integer n ≥ 3 .

Theorem 7.2

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 ) . Then, F n is identically the same as F 2 for any integer n ≥ 3 , where F 2 and F n are defined in Definition 5.1.

Proof

Let r be a fixed positive real number satisfying E 1 ⊂ B r ( p ) . We assume that F 2 ≡ F 3 ≡ ⋯ ≡ F n − 1 on GS 2 ( E 1 , p ) . Let x be an arbitrary element of GS n ( E 1 , p ) . Then, in view of (5.1), there exist a real number μ ≠ 0 and an element u of GS n ( E 1 , p ) ∩ B r ( p ) such that

u − p = μ ( x − p ) ∈ ( GS n ( E 1 , p ) − p ) ∩ ( B r ( p ) − p ) .

If we put α 11 = 1 , α i j = 0 for all ( i , j ) ≠ ( 1 , 1 ) , and x 11 = v in Definition 5.1 ( i i ) , then we obtain

(7.4) ( T − q ∘ F n ∘ T p ) ( v − p ) = ( T − q ∘ F n − 1 ∘ T p ) ( v − p )

for all v ∈ GS n − 1 ( E 1 , p ) ∩ B r ( p ) = GS n ( E 1 , p ) ∩ B r ( p ) by Proposition 5.4.

Since T − q ∘ F n ∘ T p is linear by ( b ) in the proof of Theorem 7.1, it follows from (7.4) and our assumption that

μ ( T − q ∘ F n ∘ T p ) ( x − p ) = ( T − q ∘ F n ∘ T p ) ( u − p ) = ( T − q ∘ F n − 1 ∘ T p ) ( u − p ) = ( T − q ∘ F 2 ∘ T p ) ( u − p ) = μ ( T − q ∘ F 2 ∘ T p ) ( x − p ) ,

i.e., F n ( x ) = F 2 ( x ) for every x ∈ GS n ( E 1 , p ) = GS 2 ( E 1 , p ) . By mathematical induction, we conclude that F n is identically the same as F 2 for every integer n ≥ 3 .□

Assume that J is either a translation of a basic cylinder or a translation of a basic interval, and p is an element of J . Due to Definition 6.1, Remark 6.1, and Theorem 6.1, GS ( J , p ) is a closed subset of M a .

Remark 7.1

GS ( J , p ) is a closed subset of M a .

Proof

Assume that p = ( p 1 , p 2 , … , p i , … ) is a fixed element of J , where J is a translation of a basic cylinder or a translation of a basic interval. In view of Definition 3.1 and Remark 6.1 ( i i i ) , we note that x i = p i for each x = ( x 1 , x 2 , … , x i , … ) ∈ GS ( J , p ) and each i ∉ Λ ( J ) .

Assume that { z n } n ∈ N is a sequence of elements in GS ( J , p ) , which converges to an element z = ( z 1 , z 2 , … , z i , … ) of M a . Let us denote by z n i the i th component of z n for any i , n ∈ N . Since z n ∈ GS ( J , p ) for every n ∈ N , the previous argument implies that z n i = p i for each i ∉ Λ ( J ) . Thus, we conclude that z i = p i for each i ∉ Λ ( J ) . This fact and Theorem 6.1 with β = 1 a i e i i ∈ N imply that z ∈ GS ( J , p ) . Therefore, we conclude that GS ( J , p ) is a closed subset of M a .□

We note that 1 a i e i i ∈ N is a complete orthonormal sequence in M a . On account of Theorem 6.1 with β = 1 a i e i i ∈ N , we note that Λ ( J ) = Λ ( GS ( J , p ) ) .

Remark 7.2

GS 2 ( J , p ) = GS ( J , p ) .

Proof

Referring to the changes presented in the table below

Proposition 5.4:	GS ( E , p ) ∩ B r ( p )	GS 2 ( E , p )	GS 3 ( E , p )	x	x m
Here:	J	GS ( J , p )	GS 2 ( J , p )	u	u m

and following the part ( a ) in the proof of Proposition 5.4, we can easily show that GS 2 ( J , p ) = GS ( J , p ) .□

Hence, by Theorem 6.1 with β = 1 a i e i i ∈ N and Remark 7.2, we have

(7.5) u − p = ∑ i = 1 ∞ u − p , 1 a i e i a 1 a i e i = ∑ i = 1 ∞ a i ( u i − p i ) 1 a i e i = ∑ i ∈ Λ ( J ) a i ( u i − p i ) 1 a i e i = ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i e i

for all u ∈ GS 2 ( J , p ) = GS n ( J , p ) , where n ∈ N .

Using a similar approach to the proof of [14, Theorem 2.4], we can apply Lemma 5.2 to prove the following theorem.

Theorem 7.3

Assume that J is either a translation of a basic cylinder or a translation of a basic interval, K is a subset of M a , and that there exists a surjective d a -isometry f : J → K . Suppose p is an element of J and q is an element of K with q = f ( p ) . For any n ∈ N , the d a -isometry F n : GS n ( J , p ) → M a given in Definition 5.1 satisfies

( T − q ∘ F n ∘ T p ) ( u − p ) = ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i ( T − q ∘ F n ∘ T p ) ( e i )

for all u ∈ GS n ( J , p ) .

Proof

First, we have

( T − q ∘ F n ∘ T p ) ( u − p ) − ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i ( T − q ∘ F n ∘ T p ) ( e i ) , ( T − q ∘ F n ∘ T p ) ( u − p ) − ∑ j ∈ Λ ( J ) u − p , 1 a j e j a 1 a j ( T − q ∘ F n ∘ T p ) ( e j ) a = ⟨ ( T − q ∘ F n ∘ T p ) ( u − p ) , ( T − q ∘ F n ∘ T p ) ( u − p ) ⟩ a − ∑ j ∈ Λ ( J ) u − p , 1 a j e j a 1 a j ⟨ ( T − q ∘ F n ∘ T p ) ( u − p ) , ( T − q ∘ F n ∘ T p ) ( e j ) ⟩ a − ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i ⟨ ( T − q ∘ F n ∘ T p ) ( e i ) , ( T − q ∘ F n ∘ T p ) ( u − p ) ⟩ a + ∑ i ∈ Λ ( J ) ∑ j ∈ Λ ( J ) u − p , 1 a i e i a u − p , 1 a j e j a 1 a i a j ⟨ ( T − q ∘ F n ∘ T p ) ( e i ) , ( T − q ∘ F n ∘ T p ) ( e j ) ⟩ a

for all u ∈ GS n ( J , p ) .

Since p + e i ∈ GS n ( J , p ) for each i ∈ Λ ( J ) , it follows from Lemma 5.2 that

(7.6) ( T − q ∘ F n ∘ T p ) ( u − p ) − ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i ( T − q ∘ F n ∘ T p ) ( e i ) , ( T − q ∘ F n ∘ T p ) ( u − p ) − ∑ j ∈ Λ ( J ) u − p , 1 a j e j a 1 a j ( T − q ∘ F n ∘ T p ) ( e j ) a = ⟨ u − p , u − p ⟩ a − ∑ j ∈ Λ ( J ) u − p , 1 a j e j a u − p , 1 a j e j a − ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i e i , u − p a + ∑ i ∈ Λ ( J ) ∑ j ∈ Λ ( J ) u − p , 1 a i e i a u − p , 1 a j e j a 1 a i e i , 1 a j e j a = ⟨ u − p , u − p ⟩ a − ∑ j ∈ Λ ( J ) u − p , 1 a j e j a u − p , 1 a j e j a

for all u ∈ GS n ( J , p ) , since 1 a i e i is an orthonormal sequence in M a .

Furthermore, we note that each u ∈ GS n ( J , p ) has the expression given in (7.5). Hence, if we replace u − p in (7.6) with the expression (7.5), then we have

( T − q ∘ F n ∘ T p ) ( u − p ) − ∑ i ∈ Λ ( J ) u − p , 1 a i e i a 1 a i ( T − q ∘ F n ∘ T p ) ( e i ) a 2 = 0

for all u ∈ GS n ( J , p ) , which implies the validity of our assertion.□

According to the following theorem, the image of the first-order generalized linear span of E 1 with respect to p under the d a -isometry F is just the first-order generalized linear span of F ( E 1 ) with respect to F ( p ) . This assertion holds also for the second-order generalized linear span and F 2 .

Theorem 7.4

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 . Suppose p is an element of E 1 and q is an element of E 2 with q = f ( p ) . If F n : GS n ( E 1 , p ) → M a is the extension of f defined in Definition 5.1, then GS n ( E 2 , q ) = F n ( GS n ( E 1 , p ) ) for every n ∈ N .

Proof

( a ) First, we prove that our assertion is true for n = 1 , i.e., we prove that GS ( E 2 , q ) = F ( GS ( E 1 , p ) ) . Let r be a fixed positive real number satisfying E 1 ⊂ B r ( p ) .

( b ) Due to Definition 3.1, for any y ∈ F ( GS ( E 1 , p ) ) , there exists an element x ∈ GS ( E 1 , p ) with

y = F ( x ) = F p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p )

for some m ∈ N , u i j ∈ E 1 ∩ B r ( p ) , and α i j ∈ R with x = p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) ∈ M a .

On the other hand, by Definition 4.1, we have

( T − q ∘ F ∘ T p ) ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) = ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ f ∘ T p ) ( u i j − p ) ,

which is equivalent to

F ( x ) − q = F p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) − q = ∑ i = 1 m ∑ j = 1 ∞ α i j ( f ( u i j ) − q ) .

Since u i j ∈ E 1 = E 1 ∩ B r ( p ) for all i and j , it holds that f ( u i j ) ∈ f ( E 1 ) = E 2 for each i and j . Moreover, since u i j ∈ E 1 ∩ B r ( p ) for all i and j , it follows from Lemma 3.2 that

‖ f ( u i j ) − q ‖ a 2 = ‖ ( T − q ∘ f ∘ T p ) ( u i j − p ) ‖ a 2 = ⟨ ( T − q ∘ f ∘ T p ) ( u i j − p ) , ( T − q ∘ f ∘ T p ) ( u i j − p ) ⟩ a = ⟨ u i j − p , u i j − p ⟩ a = ‖ u i j − p ‖ a 2 < r 2

for all i and j . Hence, f ( u i j ) ∈ E 2 ∩ B r ( q ) for all i and j .

Furthermore, it follows from Lemma 3.2 that

∑ i = 1 m ∑ j = 1 ∞ α i j ( f ( u i j ) − q ) a 2 = ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ f ∘ T p ) ( u i j − p ) a 2 = ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ f ∘ T p ) ( u i j − p ) , ∑ k = 1 m ∑ ℓ = 1 ∞ α k ℓ ( T − q ∘ f ∘ T p ) ( u k ℓ − p ) a = ∑ i = 1 m ∑ k = 1 m ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ α k ℓ ⟨ ( T − q ∘ f ∘ T p ) ( u i j − p ) , ( T − q ∘ f ∘ T p ) ( u k ℓ − p ) ⟩ a = ∑ i = 1 m ∑ k = 1 m ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ α k ℓ ⟨ u i j − p , u k ℓ − p ⟩ a = ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) , ∑ k = 1 m ∑ ℓ = 1 ∞ α k ℓ ( u k ℓ − p ) a = ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) a 2 < ∞ ,

since ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) = x − p ∈ M a .

Thus, on account of Remark 2.1, we see that ∑ i = 1 m ∑ j = 1 ∞ α i j ( f ( u i j ) − q ) ∈ M a .

Therefore, in view of Definition 3.1, we obtain

y = F ( x ) = q + ∑ i = 1 m ∑ j = 1 ∞ α i j ( f ( u i j ) − q ) ∈ GS ( E 2 , q )

and we conclude that F ( GS ( E 1 , p ) ) ⊂ GS ( E 2 , q ) .

( c ) Now we assume that y ∈ GS ( E 2 , q ) . By Definition 3.1, there exist some m ∈ N , v i j ∈ E 2 ∩ B r ( q ) , and α i j ∈ R such that y − q = ∑ i = 1 m ∑ j = 1 ∞ α i j ( v i j − q ) ∈ M a . Since f : E 1 → E 2 is surjective, there exists a u i j ∈ E 1 satisfying v i j = f ( u i j ) for any i and j . Moreover, by Lemma 3.2, we have

‖ u i j − p ‖ a 2 = ⟨ u i j − p , u i j − p ⟩ a = ⟨ ( T − q ∘ f ∘ T p ) ( u i j − p ) , ( T − q ∘ f ∘ T p ) ( u i j − p ) ⟩ a = ⟨ f ( u i j ) − q , f ( u i j ) − q ⟩ a = ⟨ v i j − q , v i j − q ⟩ a = ‖ v i j − q ‖ a 2 < r 2

for any i and j . So we conclude that u i j ∈ E 1 ∩ B r ( p ) and v i j = f ( u i j ) for all i and j .

On the other hand, using Lemma 3.2, we have

∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) a 2 = ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) , ∑ k = 1 m ∑ ℓ = 1 ∞ α k ℓ ( u k ℓ − p ) a = ∑ i = 1 m ∑ k = 1 m ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ α k ℓ ⟨ u i j − p , u k ℓ − p ⟩ a = ∑ i = 1 m ∑ k = 1 m ∑ j = 1 ∞ α i j ∑ ℓ = 1 ∞ α k ℓ ⟨ ( T − q ∘ f ∘ T p ) ( u i j − p ) , ( T − q ∘ f ∘ T p ) ( u k ℓ − p ) ⟩ a = ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ f ∘ T p ) ( u i j − p ) , ∑ k = 1 m ∑ ℓ = 1 ∞ α k ℓ ( T − q ∘ f ∘ T p ) ( u k ℓ − p ) a = ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ f ∘ T p ) ( u i j − p ) a 2 = ∑ i = 1 m ∑ j = 1 ∞ α i j ( f ( u i j ) − q ) a 2 = ∑ i = 1 m ∑ j = 1 ∞ α i j ( v i j − q ) a 2 < ∞

since ∑ i = 1 m ∑ j = 1 ∞ α i j ( v i j − q ) = y − q ∈ M a . Thus, Remark 2.1 implies that ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) ∈ M a .

Hence, it follows from Definition 4.1 that

y = q + ∑ i = 1 m ∑ j = 1 ∞ α i j ( f ( u i j ) − q ) = q + ∑ i = 1 m ∑ j = 1 ∞ α i j ( T − q ∘ f ∘ T p ) ( u i j − p ) = q + ( T − q ∘ F ∘ T p ) ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) = F p + ∑ i = 1 m ∑ j = 1 ∞ α i j ( u i j − p ) ∈ F ( GS ( E 1 , p ) ) .

Thus, we conclude that GS ( E 2 , q ) ⊂ F ( GS ( E 1 , p ) ) .

( d ) Similarly, referring to the changes presented in the tables below and following the previous parts ( b ) and ( c ) in this proof, we can prove that GS 2 ( E 2 , q ) = F 2 ( GS 2 ( E 1 , p ) ) .

The case n = 1 :	E 1	E 2	GS ( E 1 , p )	GS ( E 2 , q )	f	F
The case n = 2 :	GS ( E 1 , p )	GS ( E 2 , q )	GS 2 ( E 1 , p )	GS 2 ( E 2 , q )	F	F 2

The case n = 1 :	Definition 3.1	Definition 4.1	Lemma 3.2
The case n = 2 :	Definition 5.1 ( i )	Definition 5.1 ( i i )	(4.4)

( e ) Finally, according to Proposition 5.4, Theorem 7.2, and ( d ), we further have

GS n ( E 2 , q ) = GS 2 ( E 2 , q ) = F 2 ( GS 2 ( E 1 , p ) ) = F n ( GS n ( E 1 , p ) )

for any integer n ≥ 3 .□

8 Extension of isometries to the entire space

Let I ω = ∏ i = 1 ∞ I be the Hilbert cube, where I = [ 0 , 1 ] is the unit closed interval. From now on, we assume that E 1 and E 2 are nonempty subsets of I ω . They are bounded, of course.

In our main theorem (Theorem 8.1), we will prove that the domain of a local d a -isometry f : E 1 → E 2 can be extended to any real Hilbert space including the domain E 1 .

Definition 8.1

Let E 1 be a nonempty subset of I ω that is d a -isometric to a subset E 2 of I ω via a surjective d a -isometry f : E 1 → E 2 . Let p be an element of E 1 and q be an element of E 2 with q = f ( p ) . Assume that 1 a i e i i ∈ Λ α is a complete orthonormal sequence in the Hilbert space GS 2 ( E 1 , p ) − p , where Λ α is a nonempty proper subset of N . Moreover, assume that { β i } i ∈ N is a complete orthonormal sequence in the Hilbert space M a such that β i = 1 a i ( T − q ∘ F 2 ∘ T p ) ( e i ) for each i ∈ Λ α , where F 2 : GS 2 ( E 1 , p ) → M a is defined in Definition 5.1. Let p i be the i th component of p , i.e., p = ∑ i = 1 ∞ p i e i . For any set Λ satisfying Λ α ⊂ Λ ⊂ N , we define a basic cylinder or a basic interval J ˜ by

J ˜ = ∏ i = 1 ∞ J ˜ i , where J ˜ i = [ 0 , 1 ] ( for i ∈ Λ ) , { p i } ( for i ∉ Λ ) .

Moreover, referring to Theorem 7.3, we define the function G 2 : GS 2 ( J ˜ , p ) → M a by

(8.1) ( T − q ∘ G 2 ∘ T p ) ( u − p ) = ∑ i ∈ Λ ( J ˜ ) u − p , 1 a i e i a β i

for all u ∈ GS 2 ( J ˜ , p ) .

The following theorem states that the domain of a local d a -isometry can be extended to any real Hilbert space including the domain of the local d a -isometry.

Theorem 8.1

Let E 1 be a bounded subset of I ω that contains at least two elements. Suppose E 1 is d a -isometric to a subset E 2 of I ω via a surjective d a -isometry f : E 1 → E 2 . Let p and q be elements of E 1 and E 2 satisfying q = f ( p ) . Assume that 1 a i e i i ∈ Λ α is a complete orthonormal sequence in the Hilbert space GS 2 ( E 1 , p ) − p , where Λ α is a nonempty proper subset of N . Moreover, assume that { β i } i ∈ N is a complete orthonormal sequence in the Hilbert space M a such that β i = 1 a i ( T − q ∘ F 2 ∘ T p ) ( e i ) for each i ∈ Λ α . Let Λ be a set satisfying Λ α ⊂ Λ ⊂ N and let J ˜ be defined as in Definition 8.1. Then, the function G 2 : GS 2 ( J ˜ , p ) → M a is a d a -isometry and the function T − q ∘ G 2 ∘ T p : GS 2 ( J ˜ , p ) − p → M a is linear. In particular, G 2 is an extension of F 2 .

Proof

( a ) First, we assert that the function T − q ∘ G 2 ∘ T p : GS 2 ( J ˜ , p ) − p → M a preserves the inner product. Assume that u and v are arbitrary elements of GS 2 ( J ˜ , p ) . Since Λ = Λ ( J ˜ ) , it follows from (7.5), (8.1), and the orthonormality of 1 a i e i i ∈ N and { β i } i ∈ N that

⟨ ( T − q ∘ G 2 ∘ T p ) ( u − p ) , ( T − q ∘ G 2 ∘ T p ) ( v − p ) ⟩ a = ∑ i ∈ Λ u − p , 1 a i e i a β i , ∑ j ∈ Λ v − p , 1 a j e j a β j a = ∑ i ∈ Λ u − p , 1 a i e i a ∑ j ∈ Λ v − p , 1 a j e j a ⟨ β i , β j ⟩ a = ∑ i ∈ Λ u − p , 1 a i e i a ∑ j ∈ Λ v − p , 1 a j e j a 1 a i e i , 1 a j e j a

= ∑ i ∈ Λ u − p , 1 a i e i a 1 a i e i , ∑ j ∈ Λ v − p , 1 a j e j a 1 a j e j a = ⟨ u − p , v − p ⟩ a

for all u , v ∈ GS 2 ( J ˜ , p ) , i.e., T − q ∘ G 2 ∘ T p preserves the inner product.

( b ) We assert that G 2 is a d a -isometry. Let u and v be arbitrary elements of GS 2 ( J ˜ , p ) . Since T − q ∘ G 2 ∘ T p preserves the inner product by ( a ), we have

d a ( G 2 ( u ) , G 2 ( v ) ) 2 = ∥ ( T − q ∘ G 2 ∘ T p ) ( u − p ) − ( T − q ∘ G 2 ∘ T p ) ( v − p ) ∥ a 2 = ⟨ ( T − q ∘ G 2 ∘ T p ) ( u − p ) − ( T − q ∘ G 2 ∘ T p ) ( v − p ) ,

( T − q ∘ G 2 ∘ T p ) ( u − p ) − ( T − q ∘ G 2 ∘ 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 2 ( J ˜ , p ) , i.e., G 2 : GS 2 ( J ˜ , p ) → M a is a d a -isometry.

( c ) Now, we assert that the function T − q ∘ G 2 ∘ T p : GS 2 ( J ˜ , p ) − p → M a is linear. Assume that u and v are arbitrary elements of GS 2 ( J ˜ , p ) and α and β are real numbers. Since GS 2 ( J ˜ , p ) − p is a real vector space, it holds that α ( u − p ) + β ( v − p ) ∈ GS 2 ( J ˜ , p ) − p . Thus, α ( u − p ) + β ( v − p ) = w − p for some w ∈ GS 2 ( J ˜ , p ) . Hence, referring to the changes presented in the table below and following ( d ) of the proof of Theorem 4.2, we can easily prove that T − q ∘ G 2 ∘ T p is linear.

Theorem 4.2:	GS ( E 1 , p )	F	(4.4)
Here:	GS 2 ( J ˜ , p )	G 2	( a )

( d ) Finally, we assert that G 2 is an extension of F 2 . Let J ˆ be either a basic cylinder or a basic interval defined by

J ˆ = ∏ i = 1 ∞ J ˆ i , where J ˆ i = [ 0 , 1 ] ( for i ∈ Λ α ) , { p i } ( for i ∉ Λ α ) .

We see that p = ( p 1 , p 2 , … ) ∈ J ˆ ∩ E 1 and Λ ( J ˆ ) = Λ α = Λ ( GS 2 ( E 1 , p ) ) .

According to Lemma 5.5, if i ∈ Λ ( GS 2 ( E 1 , p ) ) , then α i e i ∈ GS 2 ( E 1 , p ) − p for all α i ∈ R . Since GS 2 ( E 1 , p ) − p is a real vector space, if we set Λ n = { i ∈ Λ ( GS 2 ( E 1 , p ) ) : i < n } , then we have

∑ i ∈ Λ n α i e i ∈ GS 2 ( E 1 , p ) − p

for all n ∈ N and all α i ∈ R . For now, with all α i fixed, we define x n = p + ∑ i ∈ Λ n α i e i for any n ∈ N . Then, { x n } is a sequence in GS 2 ( E 1 , p ) . When { x n } converges in M a , it holds that

p + ∑ i ∈ Λ ( GS 2 ( E 1 , p ) ) α i e i = lim n → ∞ x n ∈ GS 2 ( E 1 , p ) ,

because GS 2 ( E 1 , p ) is closed by Lemma 5.3 ( i i i ) . That is,

p + ∑ i ∈ Λ ( GS 2 ( E 1 , p ) ) α i e i ∈ M a : α i ∈ R for all i ∈ Λ ( GS 2 ( E 1 , p ) ) ⊂ GS 2 ( E 1 , p ) .

Hence, by the previous inclusion and Theorem 6.1 with β = 1 a i e i i ∈ N and J β = J ˆ , we obtain

GS ( J ˆ , p ) − p = ∑ i ∈ Λ ( J ˆ ) α i e i ∈ M a : α i ∈ R for all i ∈ Λ ( J ˆ ) = ∑ i ∈ Λ ( GS 2 ( E 1 , p ) ) α i e i ∈ M a : α i ∈ R for all i ∈ Λ ( GS 2 ( E 1 , p ) ) ⊂ GS 2 ( E 1 , p ) − p .

So, we have

J ˆ ∩ B r ( p ) ⊂ GS ( J ˆ , p ) ∩ B r ( p ) ⊂ GS 2 ( E 1 , p ) ∩ B r ( p )

for some real number r > 0 , and hence, we further have

GS ( J ˆ , p ) ⊂ GS 2 ( J ˆ , p ) ⊂ GS 3 ( E 1 , p ) = GS 2 ( E 1 , p ) .

Moreover, by Remark 7.2, we know that GS 2 ( J ˆ , p ) = GS ( J ˆ , p ) . Hence, we have

GS ( J ˆ , p ) = GS 2 ( J ˆ , p ) ⊂ GS 2 ( E 1 , p ) .

On the other hand, since 1 a i e i i ∈ Λ α is a complete orthonormal sequence in GS 2 ( E 1 , p ) − p , it follows from Theorem 6.1 with β = 1 a i e i i ∈ N that

x = ∑ i ∈ Λ α x , 1 a i e i a 1 a i e i ∈ GS ( J ˆ , p ) − p = GS 2 ( J ˆ , p ) − p

for all x ∈ GS 2 ( E 1 , p ) − p , which implies that GS 2 ( E 1 , p ) = GS 2 ( J ˆ , p ) = GS ( J ˆ , p ) .

Let u be an arbitrary element of GS 2 ( E 1 , p ) . Then, by (7.5) with J ˆ instead of J , we have

(8.2) u − p = ∑ i ∈ Λ ( J ˆ ) u − p , 1 a i e i a 1 a i e i ,

and since T − q ∘ F 2 ∘ T p is linear and continuous, we use (8.1), (8.2), and the facts GS 2 ( E 1 , p ) = GS 2 ( J ˆ , p ) = GS ( J ˆ , p ) , and Λ ( J ˆ ) = Λ α = Λ ( GS 2 ( E 1 , p ) ) to have

( T − q ∘ G 2 ∘ T p ) ( u − p ) = ∑ i ∈ Λ ( J ˜ ) u − p , 1 a i e i a β i = ∑ i ∈ Λ ( J ˆ ) u − p , 1 a i e i a β i = ∑ i ∈ Λ ( J ˆ ) u − p , 1 a i e i a 1 a i ( T − q ∘ F 2 ∘ T p ) ( e i ) = lim n → ∞ ∑ i ∈ Λ n ( J ˆ ) u − p , 1 a i e i a 1 a i ( T − q ∘ F 2 ∘ T p ) ( e i ) = lim n → ∞ ( T − q ∘ F 2 ∘ T p ) ∑ i ∈ Λ n ( J ˆ ) u − p , 1 a i e i a 1 a i e i = ( T − q ∘ F 2 ∘ T p ) ∑ i ∈ Λ ( J ˆ ) u − p , 1 a i e i a 1 a i e i = ( T − q ∘ F 2 ∘ T p ) ( u − p ) ,

where we set Λ n ( J ˆ ) = { i ∈ Λ ( J ˆ ) : i < n } for every n ∈ N .

Therefore, it follows that ( T − q ∘ G 2 ∘ T p ) ( u − p ) = ( T − q ∘ F 2 ∘ T p ) ( u − p ) for all u ∈ GS 2 ( E 1 , p ) , i.e., G 2 ( u ) = F 2 ( u ) for all u ∈ GS 2 ( E 1 , p ) . In other words, G 2 is an extension of F 2 .□

9 Applications

Given an integer n > 0 , let a = { a i } i ∈ N be a sequence of positive real numbers that satisfies

a 1 = ⋯ = a n = 1 and ∑ i = n + 1 ∞ a i 2 < ∞ .

We know that the n dimensional real vector space R n can be identified with a subspace of M a . More precisely, it holds that R n ≃ { ( x 1 , … , x n , 0 , 0 , … ) : x i ∈ R for 1 ≤ i ≤ n } .

We can define an inner product ⟨ ⋅ , ⋅ ⟩ a for R n by

⟨ x , y ⟩ a = ∑ i = 1 ∞ a i 2 x i y i = ∑ i = 1 n x i y i

for all x , y ∈ R n , with which ( R n , ⟨ ⋅ , ⋅ ⟩ a ) becomes a real Hilbert space. This inner product induces the Euclidean norm in the natural way as follows:

‖ x ‖ a = ⟨ x , x ⟩ a = ∑ i = 1 n x i 2

for all x ∈ R n , and ( R n , ‖ ⋅ ‖ a ) becomes a real Banach space.

If we replace M a , N , and I ω with R n , { 1 , 2 , … , n } , and [ 0 , 1 ] n , respectively, in the definitions and theorems in the previous sections, it is not difficult to see that they also hold for the n dimensional Euclidean space R n .

The following theorem is the finite dimensional real Hilbert space version of Theorem 8.1, the main theorem of this article. More specifically, Theorem 8.1, which applies to infinite dimensional real Hilbert spaces, is also applicable to finite dimensional real Hilbert spaces.

Theorem 9.1

Let E 1 be a bounded subset of [ 0 , 1 ] n that contains at least two elements. Suppose E 1 is d a -isometric to a subset E 2 of [ 0 , 1 ] n via a surjective d a -isometry f : E 1 → E 2 . Let p and q be elements of E 1 and E 2 satisfying q = f ( p ) . Assume that 1 a i e i i ∈ Λ α is a complete orthonormal sequence in the Hilbert space GS 2 ( E 1 , p ) − p , where Λ α is a nonempty proper subset of { 1 , 2 , … , n } . Moreover, assume that { β i } i ∈ { 1 , 2 , … , n } is a complete orthonormal sequence in the Hilbert space R n such that β i = 1 a i ( T − q ∘ F 2 ∘ T p ) ( e i ) for each i ∈ Λ α . Let Λ be a set satisfying Λ α ⊂ Λ ⊂ { 1 , 2 , … , n } and let J ˜ be defined as in Definition 8.1. Then, the function G 2 : GS 2 ( J ˜ , p ) → R n is a d a -isometry and the function T − q ∘ G 2 ∘ T p : GS 2 ( J ˜ , p ) − p → R n is linear. In particular, G 2 is an extension of F 2 .

10 Discussion

The pair ( X , Y ) of Hilbert spaces is said to have the isometric extension property if for every isometry f from an arbitrary subset S of X into Y , there exists an isometry F of X into Y such that the restriction of F to S coincides with f .

The following theorem is a well-known result due to [20, Theorem 11.4].

Theorem 10.1

(Wells and Williams) If H is a Hilbert space, then ( H , H ) has the isometric extension property if and only if H is finite dimensional. In general, if S ⊂ H and f : S → H is an isometry, then f can be extended as an isometry to the closed linear span of S .

We note that Theorem 10.1 does not imply Theorem 8.1. For example, let E 1 and E 2 be subsets of the Hilbert cube I ω . Then, GS 2 ( J ˜ , p ) − p is a proper subspace of the real Hilbert space M a , and GS 2 ( E 1 , p ) − p is a proper subspace of GS 2 ( J ˜ , p ) − p . Nevertheless, it follows from Theorem 8.1 that every surjective isometry f : E 1 → E 2 can be extended to an isometry G 2 : GS 2 ( J ˜ , p ) → M a . On the other hand, we cannot expect to obtain this result using Theorem 10.1, since the closed linear span of E 1 is a proper subset of GS 2 ( J ˜ , p ) − p , which implies that Theorem 8.1 is not only different from Theorem 10.1 but also has a number of advantages.

Moreover, for any bounded subset S of I ω , it is clear that span ¯ S ⊂ GS 2 ( S , p ) . But, it is not yet clear whether span ¯ S = GS 2 ( S , p ) , where span ¯ S denotes the closed linear span of S . If span ¯ S ≠ GS 2 ( S , p ) is correct, Theorem 8.1 has more advantages than Theorem 10.1.

According to Theorem 8.1, the domain of a local d a -isometry can be extended to any real Hilbert space containing that domain.

11 Conclusion

In view of Theorem 8.1, the domain of a local d a -isometry can be extended to any real Hilbert space containing that domain. The domain of a local d a -isometry does not need to be a convex body or an open set required by [1], it just needs to be bounded and contain at least two elements. Therefore, the coverage of our result is wider than that of previous results. This is the biggest advantage of this article compared to the previous results.

Acknowledgement

The author sincerely thanks the anonymous reviewers for their very detailed and kind review of his article.

Funding information: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01049560).
Conflict of interest: The author states no conflict of interest.

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Received: 2022-01-05

Revised: 2022-10-01

Accepted: 2022-10-04

Published Online: 2022-10-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0518

Keywords for this article

isometry; extension of isometry; generalized linear span

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