Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces


CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI 
Volume 4 (4) (2017), Pages  167-175 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: 13 March 2017 Reviewed: 26 May 2017 Accepted: 30 May 2017 
DOI: http://dx.doi.org/10.18860/ca.v4i4.4100 
 

Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces 

Minanur Rohman 

Department of Islamic Studies, Faculty of Tarbiyah, STAI Ma’Had Aly Al-Hikam,  
Malang, Indonesia 

Email: miminanira@gmail.com 

ABSTRACT  

In this paper, we will discuss some applications of almost surjective -isometry mapping, one of them is in 
Lorentz space (𝐿𝑝,𝑞 -space). Furthermore, using some classical theorems of 𝑤

∗-topology and concept of 

closed subspace 𝛼-complemented, for every almost surjective -isometry mapping  f : X → Y, where Y is a 
reflexive Banach space, then there exists a bounded linear operator T : Y → X  with ‖𝑇‖ ≤ 𝛼 such that 

 
‖𝑇𝑓(𝑥) − 𝑥‖ ≤ 4  

 
for every 𝑥 ∈ 𝑋. 
 

Keywords: almost surjectivity; -isometry; Lorentz space; reflexive space. 

INTRODUCTION  

Suppose 𝑋 and 𝑌 be real Banach spaces. A mapping 𝑈: 𝑋 → 𝑌  is called an isometry if 
‖𝑈(𝑥) − 𝑈(𝑦)‖𝑌 = ‖𝑥 − 𝑦‖𝑋 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. 

 
Mazur-Ulam showed that if 𝑈 is a surjective isometry, then 𝑈 is affine. In other word, a 

surjective isometry mapping can be translated. This result lead to the following definition. 

 
Definition 1.1. Let 𝑋 and 𝑌 be real Banach spaces. A mapping 𝑓: 𝑋 → 𝑌  is called an -
isometry if there exists ≥ 0 such that  

|‖𝑓(𝑥) − 𝑓(𝑦)‖𝑌 − ‖𝑥 − 𝑦‖𝑋 | ≤ , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋. 
 

If = 0, then 0-isometry is just an isometry. There are other names of -isometry, some of 
them are nearisometry [13], approximate isometry ([1], [7]), and perturbed metric preserved [3]. 
To simplify, the norm fuction is just written as ‖∙‖ without mentioning the vector space. 

Definition 1.1 above begs a question, “for every -isometry f, does always exist an isometry 
mapping U and 𝛾 > 0 such that 

‖𝑓(𝑥) − 𝑈(𝑥)‖ ≤ 𝛾 (1.1) 
for all 𝑥 ∈ 𝑋?”. 

This problem was first investigated by Hyers-Ulam in 1945 [7] (therefore, -isometry 
problems are also called as Hyers-Ulam problems) and they found that for every surjective -
isometry mapping  f  in the Euclidean spaces, there always exists a surjective isometry mapping U 

http://dx.doi.org/10.18860/ca.v4i4.4100
mailto:miminanira@gmail.com


Almost  Surjective  Epsilon-Isometry  in  The  Reflexive  Banach  Spaces 

Minanur Rohman 168 

satisfying (1.1) with 𝛾 = 10. Some years later, D. G Bourgin studied -isometry mapping in the 
Lebesgue space with 𝛾 = 12. 

In 1983, Gevirtz delivered 𝛾 = 5 which was hold for any Banach space X and Y [6]. This 
result was sharpened by Omladić and Šemrl [10]. 

 
Theorem 1.2. Let X and Y be Banach spaces and  f : X → Y is a surjective -isometry, there 
exists a linear surjective isometry mapping U: X → Y such that 
 

‖𝑓(𝑥) − 𝑈(𝑥)‖ ≤ 2 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. 
 

There are many applications of -isometry mapping, such as Dai-Dong who use -isometry 
mapping to determine the smoothness and rotundity of Banach spaces. Šemrl and Väisälä 
proposed a definition of almost surjective mapping as follows [13]. 

 
Definition 1.3. Let  f : X → Y is a mapping, 𝑌1 is a closed subspace of Y, and 𝛿 ≥ 0. A mapping  f  is 
called almost surjective onto Y if for every 𝑦 ∈ 𝑌1, there exists 𝑥 ∈ 𝑋 with ‖𝑓(𝑥) − 𝑦‖ ≤ 𝛿 and for 
every 𝑢 ∈ 𝑋, there exists 𝑣 ∈ 𝑌1 with ‖𝑓(𝑢) − 𝑣‖ ≤ 𝛿. 

 
Using Definition 1.3, Šemrl and Väisälä weakened the surjective condition become almost 

surjective [13] (see also [14]). 
 
Theorem 1.4. Let E and F be Hilbert spaces and  f : E → F is an almost surjective -isometry with 
𝑓(0) = 0 that satisfies 

sup
‖𝑦‖=1

lim
|𝑡|→∞

inf ‖𝑡𝑦 −
𝑓(𝐸)

𝑡
‖ < 1 , (1.2) 

 
then there exists a linear surjective isometry 𝑈: 𝐸 → 𝐹 such that 

‖𝑓(𝑥) − 𝑈(𝑥)‖ ≤ 2 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝐸. 
 

Šemrl and Väisälä showed that this result is true for E and F are Lebesgue spaces [13]. 
Vestfrid decrease the value 1 become ½ in (1.2) and is valid for any Banach space [15]. 

On the other hand, Figiel proved that for any isometry mapping U : X → Y with U(0) = 0, 
there exists an operator 𝜙 ∶  𝑠𝑝𝑎𝑛̅̅ ̅̅ ̅̅ ̅ 𝑈(𝑥) → 𝑋 with ‖𝜙‖ = 1 such that 𝜙 ° 𝑈 = 𝐼, the identity on X 
[5]. 

Using Figiel’s theorem, Cheng et. al. Give the following lemma [2] (see also Qian [11]). 
 
Lemma 1.5. If  f : ℝ → 𝑌 is a surjective -isometry with f(0) = 0, then there exists 𝜙 ∈ 𝑌∗ with ‖𝜙‖ =
1 such that 
 

|〈𝜙, 𝑓(𝑡)〉 − 𝑡| ≤ 3 , 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑡 ∈ ℝ. 
 

Using Vestfrid’s theorem and Lemma 1.5, Minan et. al. showed that the following lemma 
is true [12]. 
 
Theorem 1.6. Let X and Y be real Banach spaces, f : X → Y is an -isometry with f(0) = 0. If the 
mapping f satisfies almost surjective condition, i. e. 
 

sup
𝑦∈𝑆𝑌

lim
|𝑡|→∞

inf ‖𝑡𝑦 −
𝑓(𝑋)

𝑡
‖ <

1

2
, 

 
then for every 𝑥∗ ∈ 𝑋∗, there exists a linear functional 𝜙 ∈ 𝑌∗ with ‖𝜙‖ = ‖𝑥∗‖ = 𝑟 such that 
 

  |〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 4 𝑟, 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋.                     



Almost  Surjective  Epsilon-Isometry  in  The  Reflexive  Banach  Spaces 

Minanur Rohman 169 

RESULTS AND DISCUSSION 

We will discuss some applications of Theorem 1.6. Therefore, this paper will be organized 
as follows. Firstly, we will discuss two applications of Theorem 1.6, one of them is to determine 
the stability of almost surjective -isometry in 𝐿𝑝,𝑞 -spaces (also called as Lorentz spaces). Next, we 

will discuss the stability of almost surjective -isometry in reflexive Banach spaces. 
The used notations and terminology are standard. 𝑤 ∗ (pronounced “weak star”) topology 

of the dual of normed space X is the smallest topology of 𝑋∗ such that for every 𝑥 ∈ 𝑋, the linear 
functional 𝑥∗ → 𝑥∗𝑥 on 𝑋∗ is continuous.  

The author is greatly indebted to anonymous referee for a number of important 
suggestions that help the author to simplify the proof. 
 
1. Applications of Theorem 1.6 

Let X and Y be Banach spaces and 𝑌 ∈ 𝔅(𝑋, 𝑌). For bounded set 𝐶 ⊂ X, is defined ‖𝑇‖𝐶 =
sup{‖𝑇𝑥‖ ∶ 𝑥 ∈ 𝐶}. If there is 𝑐 ∈ 𝐶 such that ‖𝑇𝑐‖ = ‖𝑇‖𝐶 , the T attains its supremum over C. 
Operator T is called as norm-attaining operator. 
 
Theorem 2.1. Let 𝑋 and 𝑌 be real Banach spaces, f : X → Y is an almost surjective -isometry with 
𝑓(0) = 0 satisfies 

sup
𝑦∈𝑆𝑌

lim
|𝑡|→∞

inf ‖𝑡𝑦 −
𝑓(𝑋)

𝑡
‖ <

1

2
,                                                (2.1) 

then 

‖∑ 𝜆𝑖 𝑓(𝑥𝑖 )‖ + 4 ≥ ‖∑ 𝜆𝑖 𝑥𝑖 ‖ 

 
for every 𝑥1, …  , 𝑥𝑛 ∈ 𝑋 and every 𝜆1, …  , 𝜆𝑛 ∈ ℝ , where ∑ 𝜆𝑖

𝑛
𝑖=1 = 1. 

 
Proof. Let 𝑥1, …  , 𝑥𝑛 ∈ 𝑋 and 𝑋𝑛 = 𝑠𝑝𝑎𝑛 (𝑥1, …  , 𝑥𝑛 ). Banach spaces X and Y that satisfy (2.1) are 
strictly convex [12]. In addition, 𝑋∗ and 𝑌∗ are 𝑤 ∗-compact convex, so that for every 𝑥∗ ∈ 𝑆𝑋𝑛∗ , 

there exists a linear functional 𝜑𝑥∗ ∈ 𝑆𝑌∗  such that 
 

|〈𝜑𝑥∗ , 𝑓(𝑥)〉 − 〈𝑥
∗, 𝑥〉| ≤ 4 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋𝑛 . 

 
Since  f  is an almost surjective -isometry, then for every 𝑦 ∈ 𝑆𝑌, there exists 𝑥0 ∈ 𝑋 such 

that f(𝑥0) = 𝑦. Therefore, 
 

‖𝑦 −
𝑓(𝑋)

𝑡
‖ = ‖𝑓(𝑥0) −

𝑓(𝑡𝑥)

𝑡
‖ 

≥ |‖𝜑𝑥∗ ‖‖𝑓(𝑥0)‖ − ‖𝜑𝑥∗ ‖ ‖
𝑓(𝑡𝑥)

𝑡
‖| − ‖𝑥∗‖‖𝑥0 − 𝑥‖ + ‖𝑥

∗‖‖𝑥0 − 𝑥‖ 

 
≥ |‖𝜑𝑥∗ ‖‖𝑓(𝑥0)‖ − ‖𝜑𝑥∗ ‖‖𝑓(𝑥)‖| − ‖𝑥

∗‖‖𝑥0 − 𝑥‖ + ‖𝑥
∗‖‖𝑥0 − 𝑥‖ 

 
≥ sup

𝑥∗∈𝑆𝑋𝑛
∗

|〈𝑥 ∗, 𝑥0 − 𝑥〉| − |〈𝑥
∗, 𝑥0 − 𝑥〉| + |〈𝜑𝑥∗ , 𝑓(𝑥0)〉 − 〈𝜑𝑥∗ , 𝑓(𝑥)〉| 

 
= sup

𝑥∗∈𝑆𝑋𝑛
∗

|〈𝑥 ∗, 𝑥0 − 𝑥〉| − |〈𝑥
∗, 𝑥0 − 𝑥〉| + |〈𝜑𝑥∗ , 𝑓(𝑥0) − 𝑓(𝑥)〉| 

 
≥ sup

𝑥∗∈𝑆𝑋𝑛
∗

|〈𝑥 ∗, 𝑥0 − 𝑥〉| − |〈𝑥
∗, 𝑥0 − 𝑥〉 − 〈𝜑𝑥∗ , 𝑓(𝑥0) − 𝑓(𝑥)〉| 

 



Almost  Surjective  Epsilon-Isometry  in  The  Reflexive  Banach  Spaces 

Minanur Rohman 170 

Recall that 𝑋 and 𝑌 are strictly convex, so 𝑓(𝑥0) − 𝑓(𝑥) = ∑ |𝜆𝑖 |𝑓(𝑥𝑖 )
𝑛
𝑖=1  and      𝑥0 − 𝑥 =

∑ |𝜆𝑖 |𝑥𝑖
𝑛
𝑖=1 . Since 𝑥

∗ ∈ 𝑆𝑋𝑛∗ , i. e. ‖𝑥
∗‖ = 1, then 𝑥∗ is norm attaining functional. Hence, 

sup
𝑥∗∈𝑆𝑋𝑛

∗

|〈𝑥∗, ∑ |𝜆𝑖 |𝑥𝑖
𝑛
𝑖=1 〉| = ‖𝑥

∗(∑ |𝜆𝑖 |𝑥𝑖
𝑛
𝑖=1 )‖ = ‖∑ |𝜆𝑖 |𝑥𝑖

𝑛
𝑖=1 ‖. Therefore, 

 

‖∑ |𝜆𝑖 |𝑓(𝑥𝑖 )
𝑛

𝑖=1
‖ ≥ ‖∑ |𝜆𝑖 |𝑥𝑖

𝑛

𝑖=1
‖ − |〈𝑥∗, ∑ |𝜆𝑖 |𝑥𝑖

𝑛

𝑖=1
〉 − 〈𝜑𝑥∗ , ∑ |𝜆𝑖 |𝑓(𝑥𝑖 )

𝑛

𝑖=1
〉| 

= ‖∑ |𝜆𝑖 |𝑥𝑖
𝑛

𝑖=1
‖ − ∑ |𝜆𝑖 |

𝑛

𝑖=1
|〈𝑥∗, ∑ 𝑥𝑖

𝑛

𝑖=1
〉 − 〈𝜑𝑥∗ , ∑ 𝑓(𝑥𝑖 )

𝑛

𝑖=1
〉| 

= ‖∑ |𝜆𝑖 |𝑥𝑖
𝑛

𝑖=1
‖ − |〈𝑥∗, ∑ 𝑥𝑖

𝑛

𝑖=1
〉 − 〈𝜑𝑥∗ , ∑ 𝑓(𝑥𝑖 )

𝑛

𝑖=1
〉| 

≥ ‖∑ |𝜆𝑖 |𝑥𝑖
𝑛

𝑖=1
‖ − 4  

 
or 
 

‖∑ |𝜆𝑖 |𝑓(𝑥𝑖 )
𝑛

𝑖=1
‖ + 4 ≥ ‖∑ |𝜆𝑖 |𝑥𝑖

𝑛

𝑖=1
‖ .                                            ∎ 

Next, we will discuss an application of Theorem 1.6 in 𝐿𝑝,𝑞 -spaces. Let 𝐿(Ω) is an algebra 

of equivalence classes of real valued measured function over (Ω, ∑, 𝜇). The distribution 𝛿𝑓 (𝑠) for 

𝑓 ∈ 𝐿(Ω) is defined as 
𝛿𝑓 (𝑠) ∶= 𝜇(|𝑓| > 𝑠) 

 
𝐿𝑝,𝑞-spaces are collections of all measured function over Ω such that 

 

‖𝑓‖𝑝,𝑞 ∶= (
𝑞

𝑝
∫ 𝑓 ∗(𝑡)𝑞 𝑡

(
𝑞

𝑝
) − 1

𝑑𝑡

∞

0

)

1

𝑞

< ∞ 

 
where 𝑓 ∗(𝑡) ∶= inf{𝑠 > 0 ∶  𝛿𝑓 (𝑠) < 𝑡} , 𝑡 > 0. If p = q, then 𝐿𝑝,𝑝(Ω) is just Lebesgue space 

𝐿𝑝(Ω). The following theorem shows that there exists an isomorphism operator [8] in 𝐿𝑝,𝑞 -spaces.  

 
Theorem 2.2. Suppose that 1 ≤ 𝑝,𝑞 ≤ ∞, 𝑝 ≠ 𝑞, 𝑝 ≠ 1, 𝑝 ≠ 2. Then there exists no weakly sequence 
{𝑓𝑘 }𝑘=1

∞  in 𝐿𝑝,𝑞 (0,1)  such that for some 𝐶 > 0 and for any subsequence {𝑓𝑘
′}𝑘=1

∞  of {𝑓𝑘 }𝑘=1
∞ , the 

estimate  
 

𝐶−1𝑁1/𝑝 ≤ ‖∑ 𝑓𝑘
′

𝑁

𝑘=1

‖

𝑝,𝑞

≤ 𝐶𝑁1/𝑝 

 
holds. 

Now, using Theorem 2.2, the author will show that the following theorem is true.  
 
Theorem 2.3. Let X =𝐿𝑝,𝑞 and Y be Banach space. If  f : X → Y is an almost surjective -isometry with 

f(0) = 0, then there exists an operator S : Y → X  with ‖𝑆‖ = 1 such that 
 

‖𝑆𝑓(𝑥) − 𝑥‖ ≤ 4 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. 
 

Proof. From Theorem 2.2, then 𝐵𝑋∗  is a 𝑤
∗-closed convex hull of 𝑤 ∗-exposed point in 𝐿𝑝,𝑞 . 

Let 𝑡 ∈ (0,1) and 𝑥∗ ∈ 𝑆𝑋∗ . Theorem 1.6 implies that there exists 𝜙𝑡 ∈ 𝑆𝑌∗  such that 
 

|〈𝜙𝑡 , 𝑓(𝑥)〉 −  〈𝑥𝑡
∗, 𝑥〉| ≤ 4 , 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ 𝑋. 



Almost  Surjective  Epsilon-Isometry  in  The  Reflexive  Banach  Spaces 

Minanur Rohman 171 

Let S : Y → X  is defined by S(y) = 〈𝜙𝑡 , 𝑓(𝑥)〉𝑒𝑡 = 𝑥0. Clearly ‖𝑆‖ = 1 and 
 

‖𝑆𝑓(𝑥) − 𝑥‖𝑝,𝑞 =  (
𝑞

𝑝
∫(𝑥0(𝑡)

𝑞 − 𝑥(𝑡)𝑞 )𝑡
(

𝑞

𝑝
) − 1

𝑑𝑡

1

0

)

1

𝑞

 

 
≤ sup

𝑡∈(0,1)
|〈𝜙𝑡 , 𝑓(𝑥)〉 −  〈𝑥𝑡

∗, 𝑥〉| ≤ 4 .                                 ∎ 

 
2. Almost Surjective 𝜺-Isometry in The Reflexive Banach Spaces 
 

A normed space is reflexive if the natural mapping 𝜋(𝑥) from 𝑋 into 𝑋∗∗ which is defined 
by 𝜋(𝑥) : f → f(x) where 𝑓 ∈ 𝑋∗, is onto 𝑋∗∗, or 𝑋 ≈ 𝑋∗∗. Therefore, every reflexive normed space 
is Banach space. In addition, every closed subspace of  reflexive Banach space is  reflexive [4]. 

 
Proposition 3.1 (Megginson [9], Proposition 1.11.11). Let X be reflexive normed space. Every 
functional 𝑥∗ ∈ 𝑋∗ is norm-attaining functional. 
 

The following definitions are classic (for more detail see [4]). Let X be Banach space and 
𝑀 ⊂ 𝑋. The polar set of M is 𝑀∘ ∶= { 𝑥∗ ∈ 𝑋∗ ∶  〈𝑥∗, 𝑥〉 ≤ 1, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑀}. Conversely, 

𝑀° ∶= { 𝑥 ∈ 𝑋 ∶  〈𝑥∗, 𝑥〉 ≤ 1, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥∗ ∈ 𝑀°}° . The annihilator of M is 𝑀⊥ ∶= { 𝑥∗ ∈ 𝑋∗ ∶  〈𝑥∗, 𝑥〉 =

0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑀}, and preannihilator 𝑀⊥ is 𝑀⊥ ∶= { 𝑥 ∈ 𝑋: 〈𝑥∗, 𝑥〉 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 ⊥     𝑥 ∗ ∈ 𝑀⊥}. 
For an almost surjective -isometry mapping f : X → Y with f(0) = 0 and > 0, 𝐶(𝑓) is a 

closed convex hull of f(X), E is an annihilator of subspace F⊂ 𝑌∗ where F is a set of all bounded 
functionals over 𝐶(𝑓), and 

 
𝑀𝜀  ∶=  {𝜙 ∈ 𝑌

∗: 𝜙 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝛽  𝑜𝑣𝑒𝑟 𝐶(𝑓), 𝑓𝑜𝑟 𝛽 > 0}. 
 

Obviously, 𝐶(𝑓) is symmetric, then 𝑀𝜀 is linear subspace of 𝑌
∗ with 𝑀𝜀 = ⋃ 𝑛𝐶(𝑓)

°∞
𝑛=1 . 

Therefore, 𝐸 = ⋂{𝑘𝑒𝑟𝜙 ∶  𝜙 ∈ 𝑀}. The following lemma is easy to be proved (see [2]). 
 

Lemma 3.2. Let Y be Banach space, the following statements are equicalence. 
(1) 𝐶(𝑓) ⊂ 𝐸 + 𝐹 for a bounded 𝐵 ⊂ 𝑌; 

(2) 𝑀𝜀 is 𝑤
∗-closed; 

(3) 𝑀𝜀 is closed. 

For every almost surjective -isometry f, is defined a linear mapping ℓ : 𝑋∗ → 2𝑌
∗
by 

 
ℓ𝑥∗ ∶= {𝜙 ∈ 𝑌∗: |〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝛽 > 0 𝑎𝑛𝑑 𝑎𝑙𝑙 𝑥 ∈ 𝑋}. 

 
Since ℓ is a linear mapping, the definition of 𝑀𝜀  implies 

 
ℓ0  = {𝜙 ∈ 𝑌∗: |〈𝜙, 𝑓(𝑥)〉 − 〈0, 𝑥〉| ≤ 𝛽 } 

            = {𝜙 ∈ 𝑌∗: |〈𝜙, 𝑓(𝑥)〉| ≤ 𝛽 } 
                            = {𝜙 ∈ 𝑌∗: 𝜙 𝑖𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝛽  𝑜𝑣𝑒𝑟 𝐶(𝑓)} = 𝑀𝜀 

and 
 

ℓ𝑥 ∗ = ℓ(𝑥∗ + 0) = ℓ𝑥∗ +  ℓ0 ⊃ 𝜙 + ℓ0 = 𝜙 + 𝑀𝜀. 
 
Lemma 3.3. If 𝑥∗ ≠ 𝑦∗, then ℓ𝑥∗ ∩ ℓ𝑦∗ = ∅. 
Proof. Suppose that ℓ𝑥∗ ∩ ℓ𝑦∗ is nonempty, then there exists 𝜑 ∈ ℓ𝑥∗ ∩ ℓ𝑦∗. Assume 

𝜑𝑥∗ = {𝜑 ∈ 𝑌
∗: |〈𝜑, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋) 

and 



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Minanur Rohman 172 

𝜑𝑦∗ = {𝜑 ∈ 𝑌
∗: |〈𝜑, 𝑓(𝑥)〉 − 〈𝑦∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋). 

 
For 𝑥∗ ≠ 𝑦∗, there exists 𝜑 ∈ ℓ𝑥∗ ∩ ℓ𝑦∗ such that 𝜑𝑥∗ = 𝜑𝑦∗ . This is impossible because 𝜑 

is a linear functional and the proof is complete.         ∎ 
 

Let 𝑈 ∈ 𝔅(𝑋, 𝑌). Then 𝑈∗ ∈ 𝔅(𝑌∗, 𝑋∗) is called an adjoint operator such that 〈𝑈𝑥, 𝑓〉 =
〈𝑥, 𝑈∗𝑓〉 for all 𝑥 ∈ 𝑋. Two results below are classic. 

 
Proposition 3.4 (Fabian et. al. [4], Proposition 2.6). Let M is a closed subspace of Banach space X. 
Then (𝑋/𝑌)∗ is isometric with 𝑀⊥ and 𝑀∗ is isometric with 𝑋∗/𝑀⊥. 
 
Theorem 3.5 (Megginson [9], Proposition 3.1.11). Let X and Y be normed spaces. If 𝑈 ∈ 𝔅(𝑋, 𝑌), 
then 𝑈∗ is 𝑤 ∗-to-𝑤 ∗ continuous. Conversely, if 𝑄 is 𝑤 ∗-to-𝑤 ∗ linear continuous operator from 𝑌∗ to 
𝑋∗, then there exists 𝑈 ∈ 𝔅(𝑋, 𝑌) such that 𝑈∗ = 𝑄.  

 
Now, using Proposition 3.4 and Theorem 3.5, we will show that the following theorem is 

true. 
 

Theorem 3.6. Let X and Y be Banach spaces, f : X → Y is an almost surjective -isometry with f(0) = 
0, and 𝑀 = ℓ̅0. Then 
 

(1) 𝑄: 𝑋∗ → 𝑌∗/𝑀, which is defined by 𝑄𝑥∗ = ℓ𝑥∗ + 𝑀, is linear isometry. 

(2) If M is 𝑤 ∗-closed, then 𝑄 is an adjoint operator of 𝑈: 𝐸 → 𝑋 with ‖𝑈‖ = 1. 

 
Proof. (1) From Lemma 3.3 and definition of ℓ, it is clear that 𝑄 is linear.  Lemma 3.2 implies 𝑀 =
ℓ̅0 = 𝑀𝜀̅̅ ̅̅ . For every 𝑥

∗ ∈ 𝑋∗, 
 

‖𝑄𝑥∗‖ = inf{‖𝜙 − 𝑚‖ : 𝜙 ∈ ℓ𝑥∗, 𝑚 ∈ 𝑀} 
               = inf{‖𝜙 − 𝑚‖ : 𝜙 ∈ ℓ𝑥∗, 𝑚 ∈ 𝑀𝜀 }. 

From definition of  ℓ𝑥∗ and 𝑀𝜀, we have 
 

‖𝑄𝑥∗‖ = inf{‖𝜙‖ : 𝜙 ∈ ℓ𝑥∗}. 
 

Theorem 1.6 gives 
     inf{‖𝜙‖ : 𝜙 ∈ ℓ𝑥∗} ≤ ‖𝑥∗‖.                                             (3.1) 

 
To show the conversely, from definition of ℓ, there exists 𝛽 > 0 such that 

 
|〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 𝛽 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. 

For any 𝛿 > 0, we can choose 𝑥0 ∈ 𝑆𝑋 such that 
 

〈𝑥∗, 𝑥0〉 > ‖𝑥
∗‖ − 𝛿. 

For all 𝑛 ∈ ℕ, we get 
 

|〈𝜙,
𝑓(𝑛𝑥0)

𝑛
〉 − 〈𝑥∗, 𝑥0〉| ≤

𝛽

𝑛
 

As 𝑛 → ∞,  

‖𝜙‖ ≥ lim sup
𝑛

|〈𝜙,
𝑓(𝑛𝑥0)

𝑛
〉| = 〈𝑥∗, 𝑥0〉 > ‖𝑥

∗‖ − 𝛿. 

 
 
 



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Minanur Rohman 173 

Since 𝛿 is arbitrary, then 
‖𝜙‖ ≥ ‖𝑥∗‖.                                                        (3.2) 

From (3.1) and (3.2), then ‖𝜙‖ = ‖𝑥∗‖ which shows that 𝑄 is an isometry. 
(2) Since 𝑀 is 𝑤 ∗-closed, then 𝑀 = { 𝑀⊥ }⊥ = { 𝑀𝜀

⊥ }⊥ = 𝐸⊥. Therefore, 𝑌∗/𝑀 = 𝑌∗/𝐸⊥. 

From definitions of 𝐸 and 𝑀𝜀, Proposition 3.4 implies 𝑌
∗/𝐸⊥ = 𝐸∗. 

We will prove that 𝜙 is 𝑤 ∗-to-𝑤 ∗ continuous over the unit ball 𝐵𝑋∗ . Let 𝐼 is an index set and 
𝛼 ∈ 𝐼. Suppose there exists net (𝑥∗) ⊂ 𝐵𝑋∗  which is 𝑤

∗-converging to 𝑥∗ ∈ 𝑋∗. Using Theorem 1.6, 
there exists net (𝜙𝛼 ) ⊂ 𝑌

∗ with ‖𝜙𝛼 ‖ = ‖𝑥𝛼
∗ ‖ = 𝑟𝛼 ≤ 1 such that 

 
|〈𝜙𝛼 , 𝑓(𝑥)〉 − 〈𝑥𝛼

∗ , 𝑥〉| ≤ 4 𝑟𝛼 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. 
 

Since (𝜙𝛼 ) is 𝑤
∗-compact, then there exists 𝑤 ∗-limit point ∈ 𝑌∗ such that 

 
|〈𝜙, 𝑓(𝑥)〉 − 〈𝑥∗, 𝑥〉| ≤ 4 𝑟, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋 

 
Therefore, 𝑄𝑥𝛼

∗ = ℓ𝑥𝛼
∗ + 𝑀 = 𝜙𝛼 + 𝑀 is 𝑤

∗-convergence to 𝜙 + 𝑀 = 𝑄𝑥∗. This result 
shows that 𝑄: 𝐵𝑋∗ → 𝐸

∗ is 𝑤 ∗-to-𝑤 ∗ continuous. From Theorem 3.5, there exists 𝑈: 𝐸 → 𝑋 such 
that 𝑈∗ = 𝑄. Definition of 𝐸 shows that 𝑈 is a surjective mapping. Since   𝑈∗ = 𝑄 is a linear 
isometry, then ‖𝑈‖ = 1.             ∎ 

 
Before discussing the stability of an almost surjective -isometry in reflexive Banach 

space, we need some resuls below. 
 

Proposition 3.7 (Fabian et. al. [4], Proposition 3.114). Let Y be reflexive Banach space. If 𝑀 is a 
closed subspace of Y, then 𝑀 is Banach space. 
 
Corollary 3.8. If Y be reflexive Banach space, then the mapping 𝑄 in Theorem 3.6 is an adjoint 
operator. 
 
Proof. Since Y is reflexive, from Proposition 3.7, 𝑀 is 𝑤 ∗-closed. The proof can be completed by 
Theorem 3.6. 

 
Definition 3.9. Let X be Banach space and 0 ≤ α < ∞. A closed subspace 𝑀 ⊂ 𝑋 is called 𝛼-
complemented in X if there is a closed subspace 𝑁 ⊂ 𝑋 with 𝑀 ∩ 𝑁 = {0} and projection 𝑃: 𝑋 →
𝑀 𝑎𝑙𝑜𝑛𝑔 𝑁 such that 𝑋 = 𝑀 + 𝑁 and ‖𝑃‖ ≤ 𝛼. 
 
Theorem 3.10. Let X and Y be Banach spaces where Y is reflexive, and f : X → Y is an almost surjective 

-isometry, then there exists a bounded linear operator T : Y → X and ‖𝑇‖ ≤ 𝛼 such that 
 

‖𝑇𝑓(𝑥) − 𝑥‖ ≤ 4 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋. 
 

Proof. Since Y is reflexive, from Corollary 3.8 and Theorem 3.6, there exists a surjective operator 
𝑈: 𝐸 → 𝑋 with ‖𝑈‖ = 1 such that 𝑄 = 𝑈∗. Since 𝐸 is 𝛼-complemented in Y, there is a closed 
subspace 𝐹 ⊂ 𝑌 and 𝐸 ∩ 𝐹 = {0} such that 𝐸 + 𝐹 = 𝑌 and projection 𝑃: 𝑌 → 𝐸 𝑎𝑙𝑜𝑛𝑔 𝐹 satisfies 
‖𝑃‖ ≤ 𝛼. Let 𝑇 = 𝑈𝑃, then 
 

‖𝑇‖ = ‖𝑈𝑃‖ ≤ ‖𝑈‖‖𝑃‖ = ‖𝑃‖ ≤ 𝛼. 
 
Furthermore, 
 

〈𝑄𝑥∗, 𝑃𝑦〉 = 〈𝑄𝑥∗, 𝑦〉 
 
for all 𝑥∗ ∈ 𝑋∗ and 𝑦 ∈ 𝑌. Therefore, 
 



Almost  Surjective  Epsilon-Isometry  in  The  Reflexive  Banach  Spaces 

Minanur Rohman 174 

|〈𝑄𝑥∗, 𝑃𝑓(𝑋)〉 − 〈𝑥∗, 𝑥〉| = |〈𝑄𝑥∗, 𝑓(𝑋)〉 − 〈𝑥∗, 𝑥〉| ≤ 4 ‖𝑥∗‖,                  (3.3) 
 
for all 𝑥 ∈ 𝑋 and 𝑥∗ ∈ 𝑋∗. Definition of adjoint operator gives 
 

〈𝑄𝑥∗, 𝑃𝑓(𝑋)〉 = 〈𝑥∗, 𝑄∗𝑃𝑓(𝑋)〉 
= 〈𝑥∗, 𝑈𝑃𝑓(𝑋)〉 = 〈𝑥∗, 𝑇𝑓(𝑋)〉.                     (3.4) 

 
Substitute (3.4) into (3.3), then 
 

|〈𝑥∗, 𝑇𝑓(𝑋)〉 − 〈𝑥∗, 𝑥〉| = |〈𝑥∗, 𝑇𝑓(𝑋) − 𝑥〉| ≤ 4 ‖𝑥∗‖ 
 
for all 𝑥 ∈ 𝑋 and 𝑥∗ ∈ 𝑋∗. In addition, 
 

|〈𝑥∗, 𝑇𝑓(𝑋) − 𝑥〉| ≤ ‖𝑥∗‖‖𝑇𝑓(𝑋) − 𝑥‖ 
              ≤ 4 ‖𝑥∗‖ 

 
or 
 

‖𝑇𝑓(𝑋) − 𝑥‖ ≤ 4  
 
for all 𝑥 ∈ 𝑋.                 ∎ 

 

CONCLUSION 

In this paper, the author has shown simple applications of Theorem 1.6. In addition, almost 
surjective -isometries are remain stable in the reflexive Banach spaces. 
 
Open Problem 1. This paper has shown that almost surjective -isometries are stable in Lorentz 
spaces when the conditions in Theorem 2.2 are satisfied. Is it possible to generalize these restrictions? 
 
Open Problem 2. According to the results of this paper, is it possible to determine the stability of 
almost surjective -isometries in the higher dual of Banach spaces? 

 

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