Some Generalizations of Ulam-Hyers Stability Functional Equations to Riesz Algebras

Abstract

Badora (2002) proved the following stability result. Let 𝜀 and 𝛿 be nonnegative real numbers, then for every mapping 𝑓 of a ring ℛ onto a Banach algebra ℬ satisfying ||𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)||≤𝜀 and ||𝑓(𝑥⋅𝑦)−𝑓(𝑥)𝑓(𝑦)||≤𝛿 for all 𝑥,𝑦∈ℛ, there exists a unique ring homomorphism ℎ∶ℛ→ℬ such that ||𝑓(𝑥)−ℎ(𝑥)||≤𝜀,𝑥∈ℛ. Moreover, 𝑏⋅(𝑓(𝑥)−ℎ(𝑥))=0,(𝑓(𝑥)−ℎ(𝑥))⋅𝑏=0, for all 𝑥∈ℛ and all 𝑏 from the algebra generated by ℎ(ℛ). In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.

1. Introduction

The approximation of solution of the Cauchy's equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) lying near to some solution has received a lot of attention from mathematicians in the areas of modern analysis and applied mathematics. Any solution 𝑓 of this equation is called an additive function. Let 𝐸 and 𝐸′ be Banach spaces, and let 𝛿 be a positive number. A function 𝑓 of 𝐸 into 𝐸′ is called 𝛿-additive if ‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖<𝛿 for all 𝑥,𝑦∈𝐸. In the 1940s, Ulam [1] proposed the following stability problem of this equation. Does there exist for each 𝜀>0 a 𝛿>0 such that, to each 𝛿-additive function 𝑓 of 𝐸 into 𝐸′ there, corresponds an additive function 𝑙 of 𝐸 into 𝐸′ satisfying the inequality ‖𝑓(𝑥)−𝑙(𝑥)‖≤𝜀 for each 𝑥∈𝐸? In 1941, Hyers [2] answered this question in the affirmative way and showed that 𝛿 may be taken equal to 𝜀. The answer of Hyers is presented in a great number of articles and books. There are several definitions and critics of the notion of this stability in the literature (see, e.g., [3, 4]). In 1949, Bourgin [5] generalized Hyers' results to the ring homomorphisms and proved the following.

Theorem 1.1. Let 𝜀 and 𝛿 be nonnegative real numbers. Then every mapping 𝑓 of a Banach algebra 𝒜 with an identity element onto a Banach algebra ℬ with an identity element satisfying 𝑓(𝑥)𝑦𝑓(𝑥)𝑓𝑦𝑥+𝑦−𝑓−𝑓≤𝜀,𝑥⋅𝑦−𝑓≤𝛿, (1.1) for all 𝑥,𝑦∈𝒜, is a ring homomorphism of 𝒜 onto ℬ, that is, 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) and 𝑓(𝑥⋅𝑦)=𝑓(𝑥)𝑓(𝑦) for all 𝑥,𝑦∈𝒜.

Finally, Badora [6] proved the following theorem on the Bourgin's result related to stability problem (Theorem 1.1) without additional assumptions.

Theorem 1.2. Let ℛ be a ring, let ℬ be a Banach algebra, and let 𝜀 and 𝛿 be nonnegative real numbers. Assume that 𝑓∶ℛ→ℬ satisfies (1.1) for all 𝑥,𝑦∈𝐸. Then there exists a unique ring homomorphism ℎ∶ℛ→ℬ such that 𝑓(𝑥)−ℎ(𝑥)≤𝜀,𝑥∈ℛ. (1.2) Moreover, 𝑏⋅𝑓(𝑥)−ℎ(𝑥)=0,𝑓(𝑥)−ℎ(𝑥)⋅𝑏=0, (1.3) for all 𝑥∈ℛ and all 𝑏 from the algebra generated by ℎ(𝑅).

The present paper is in essence a revised and extended compilation of Hyers' result and Theorem 1.2 to the Riesz algebras with extended norms. After outlining the basic information on Riesz space theory, we present the main definitions and facts concerning approximate Riesz algebra (with an extended norm)-valued ring homomorphisms.

2. Preliminaries

A real Banach space 𝐹=(𝐹,+,⋅,‖⋅‖) endowed with a (partial) order ≤ is called a Banach lattice whenever(1)the order ≤ agrees with the linear operations, that is, 𝑥≤𝑦⇒𝛼𝑥+𝑧≤𝛼𝑦+𝑧 for all 𝑧∈𝐹 and 0≤𝛼∈ℝ;(2)the order ≤ makes 𝐹 a lattice, that is, for all 𝑥,𝑦∈𝐹, the supremum 𝑥∨𝑦 and infimum 𝑥∧𝑦 exist in 𝐹 (hence, the modulus |𝑥|∶=𝑥∨(−𝑥) exists for each 𝑥∈𝐹);(3)the norm ‖⋅‖ is monotonous with respect to the order ≤, that is, for all 𝑥,𝑦∈𝐹, |𝑥|≤|𝑦| implies ‖𝑥‖≤‖𝑦‖ (hence, ‖𝑥‖=‖|𝑥|‖ for all 𝑥∈𝐹).

Recall that a (partially) ordered vector space 𝐹 satisfying (1) and (2) above is called a Riesz space. 𝐶(𝐾) the spaces of real valued continuous functions on a compact Hausdorff space 𝐾, 𝑙_𝑝-spaces, 𝑐 the spaces of convergent sequences, and 𝑐₀ the spaces of sequences converging to zero are natural examples of Riesz spaces under the pointwise ordering. A Riesz space 𝐹 is called Archimedean if 0≤𝑢,𝑣∈𝐹, and 𝑛𝑢≤𝑣 for each 𝑛∈ℕ imply 𝑢=0. Throughout the present paper, all the Riesz spaces are assumed to be Archimedean. A subset 𝑆 in a Riesz space 𝐹 is said to be solid if it follows from |𝑢|≤|𝑣| in 𝐹 and 𝑣∈𝑆 that 𝑢∈𝑆. A solid linear subspace of a Riesz space 𝐹 is called an ideal. Every subset 𝐷 of a Riesz space 𝐹 is included in a smallest ideal 𝐹_𝐷, called ideal generated by 𝐷. A principal ideal of a Riesz space 𝐹 is any ideal generated by a singleton {𝑢}. This ideal will be denoted by 𝐼_𝑢. It is easy to see that𝐼_𝑢={𝑣∈𝐹∶𝜆≥0suchthat|𝑣|≤𝜆|𝑢|}. (2.1)

We assume that 𝑢 is a fixed positive element in the Riesz space 𝐹. First of all, we present the following definition.

Definition 2.1. (1) It is said that the sequence (𝑥_𝑛) in 𝐹 converges 𝑢-uniformly to the element 𝑥∈𝐹 whenever, for every 𝜀>0, there exists 𝑛₀ such that |𝑥_𝑛0+𝑘−𝑥|≤𝜀𝑢 holds for each 𝑘.
(2) It is said that the sequence (𝑥_𝑛) in 𝐹 converges relatively uniformly to 𝑥 whenever 𝑥_𝑛 converges 𝑢-uniformly to 𝑥 for some 0≤𝑢∈𝐹.

When dealing with relative uniform convergence in an Archimedean Riesz space 𝐹, it is natural to associate with every positive element 𝑢∈𝐹 an extended norm ‖⋅‖_𝑢 in 𝐹 by the formula ‖𝑥‖_𝑢=inf{𝜆≥0∶|𝑥|≤𝜆𝑢}(𝑥∈𝐹). (2.2)

Note that ‖𝑥‖_𝑢<∞ if and only if 𝑥∈𝐼_𝑢, the ideal generated by 𝑢. Also |𝑥|≤𝛿𝑢 if and only if ‖𝑥‖_𝑢≤𝛿.

The sequence (𝑥_𝑛) in 𝐹 is called an extended 𝑢-normed Cauchy sequence, if for every 𝜀>0 there exists 𝑘 such that ‖𝑥_𝑛+𝑘−𝑥_𝑚+𝑘‖_𝑢<𝜀 for all 𝑚,𝑛. If every extended 𝑢-normed Cauchy sequence is convergent in 𝐹, then 𝐹 is called an extended 𝑢-normed Banach lattice.

A Riesz space 𝐹 is called a Riesz algebra or a lattice-ordered algebra if there exists in 𝐹 an associative multiplication with the usual algebra properties such that 𝑢𝑣≥0 for all 0≤𝑢,𝑣∈𝐹.

For more detailed information about Riesz spaces, the reader can consult the book “Riesz Spaces” by Luxemburg and Zaanen [7].

3. Main Results

We begin with the following theorem concerning stability of the functional equation 𝐻𝑜𝑓𝑜𝐺=𝑓. For a function 𝐺∶𝐾→𝐾, let us denote by 𝐺⁰(𝑥)=𝑥 for 𝑥∈𝐾, 𝐺² the composition of 𝐺 by itself and in general let 𝐺^𝑖=𝐺𝑜(𝐺^𝑖−1) for 𝑖=1,2,….

The theorem can easily be obtained from [8] or [9]. We give the proof here for the benefit of the reader.

Theorem 3.1. Let (𝑌,𝑑) be a complete metric space, 𝐾 a nonempty set and 𝜆∈[0,∞) such that 𝐺∶𝐾→𝐾 and 𝐻∶𝑌→𝑌 are two given functions. Assume that 𝑓∶𝐾→𝑌 is a function satisfying 𝑑𝐻𝑜𝑓𝑜𝐺(𝑥),𝑓(𝑥)≤ℎ(𝑥), (3.1) for each 𝑥∈𝐾 and for some function ℎ∶𝐾→[0,∞). If the function 𝐻∶𝑌→𝑌 satisfies the inequality 𝑑(𝐻(𝑢),𝐻(𝑣))≤𝜆𝑑(𝑢,𝑣),𝑢,𝑣∈𝑌, (3.2) and the series ^∞_𝑖=0𝜆^𝑖ℎ𝐺^𝑖(𝑥) (3.3) is convergent for each 𝑥∈𝐾, then for each integer 𝑛, one has
(1) 𝑑𝐻^𝑛+1𝑜𝑓𝑜𝐺^𝑛+1(𝑥),𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥)≤𝜆^𝑛ℎ𝐺^𝑛(𝑥),𝑥∈𝐾, (3.4)
(2) 𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥),(𝑥∈𝐾) is a Cauchy sequence. 𝐹(𝑥)=lim_𝑛→∞𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥) exists for every 𝑥∈𝐾, and 𝐹∶𝐾→𝑌 is the unique function satisfying 𝐻𝑜𝐹𝑜𝐺=𝐹 and the inequality 𝑑𝐹(𝑥),𝑓(𝑥)≤^∞_𝑖=0𝜆^𝑖ℎ𝐺^𝑖(𝑥). (3.5)

Proof. (1) Replacing 𝑥 by 𝐺(𝑥) in (3.1), we get 𝑑𝐻𝑜𝑓𝑜𝐺²(𝑥),𝑓(𝐺(𝑥))≤ℎ(𝐺(𝑥)). (3.6) Then by (3.2), we obtain 𝑑𝐻²𝑜𝑓𝑜𝐺²(𝑥),𝐻𝑜𝑓𝑜𝐺(𝑥)≤𝜆𝑑𝐻𝑜𝑓𝑜𝐺²(𝑥),𝑓𝑜𝐺(𝑥)≤𝜆ℎ(𝐺(𝑥)). (3.7) The proof follows by induction.
(2) Let 𝑚>𝑛, then 𝑑𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥),𝐻^𝑚𝑜𝑓𝑜𝐺^𝑚(𝑥)≤^𝑚−1_𝑖=𝑛𝑑𝐻^𝑖+1𝑜𝑓𝑜𝐺^𝑖+1(𝑥),𝐻^𝑖𝑜𝑓𝑜𝐺^𝑖(𝑥≤_{𝑚−1𝑖=𝑛})𝜆^𝑖ℎ𝐺^𝑖(𝑥), (3.8) thus 𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥) is a Cauchy sequence for each 𝑥∈𝐾 and it is convergent as (𝑌,𝑑) is complete. Let 𝐹(𝑥)=lim_𝑛→∞𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥) for each 𝑥∈𝐾.
By using (3.4), we get 𝑑𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝑥),𝑓(𝑥)≤^𝑛_𝑖=1𝑑𝐻^𝑖𝑜𝑓𝑜𝐺^𝑖(𝑥),𝐻^𝑖−1𝑜𝑓𝑜𝐺^𝑖−1(𝑥≤_𝑛𝑖=1)𝜆^𝑖−1ℎ𝐺^𝑖−1(𝑥)=^𝑛_𝑖=0𝜆^𝑖ℎ𝐺^𝑖(𝑥), (3.9) taking the limit as 𝑛 goes to infinity, then we obtain (3.5). By continuity of 𝐻, we have 𝐻𝑜𝐹𝑜𝐺(𝑥)=𝐻lim_𝑛→∞𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝐺(𝑥))=lim_𝑛→∞𝐻𝑜𝐻^𝑛𝑜𝑓𝑜𝐺^𝑛(𝐺(𝑥))=lim_𝑛→∞𝐻^𝑛+1𝑜𝑓𝑜𝐺^𝑛+1(𝑥)=𝐹(𝑥). (3.10) Suppose that another function 𝐹∶𝐾→𝑌 satisfies 𝐻𝑜𝐹𝑜𝐺(𝑥)=𝐹 and (3.5). By induction it is easy to show that 𝐻^𝑛𝑜𝐹𝑜𝐺^𝑛(𝑥)=𝐹 and 𝐻^𝑛𝑜𝐹𝑜𝐺^𝑛(𝑥)=𝐹. Hence for 𝑥∈𝐾, 𝑑𝐹(𝑥),𝐹(𝑥)=𝑑𝐻^𝑛𝑜𝐹𝑜𝐺^𝑛(𝑥),𝐻^𝑛𝑜𝐹𝑜𝐺^𝑛(𝑥)≤𝜆^𝑛𝑑𝐹𝑜𝐺^𝑛(𝑥),𝐹𝑜𝐺^𝑛(𝑥)≤𝜆^𝑛𝑑𝐹𝑜𝐺^𝑛(𝑥),𝑓𝑜𝐺^𝑛(𝑥)+𝜆^𝑛𝑑𝑓𝑜𝐺^𝑛(𝑥),𝐹𝑜𝐺^𝑛(𝑥)≤2^∞_𝑖=𝑛𝜆^𝑛ℎ𝐺^𝑖(𝑥). (3.11) Since for every 𝑥∈𝐾, ∑^∞_𝑖=𝑛ℎ(𝐺^𝑖(𝑥))→0 with 𝑛→∞, this completes the proof.

Let 𝑌 be a linear space over either complex or real numbers. The operation of addition of elements 𝑥,𝑦∈𝑌 will be denoted, as usual, by 𝑥+𝑦. The operation of multiplication of an element 𝑥∈𝑌 by a scalar 𝑡 will be denoted by 𝑡𝑥. Suppose that in the linear space 𝑌, we are given a metric 𝑑. The space (𝑌,𝑑) is called a metric linear space if the operations of addition and multiplication by numbers are continuous with respect to the metric 𝑑. A metric linear space (𝑌,𝑑) is called complete if every Cauchy sequence (𝑥_𝑛) converges to an element 𝑥₀∈𝑌, that is, lim_𝑛→∞𝑑(𝑥_𝑛,𝑥₀)=0.

We now give the following corollary in [9] which will be useful in the sequel.

Corollary 3.2. Let (𝑌,𝑑) a complete metric linear space and 𝐾 be a linear space. Suppose that there exists 𝜉,𝜂∈[0,∞) such that 𝜉𝜂<1, 𝑑121𝑥,2𝑦𝜒≤𝜉𝑑𝑥,𝑦for𝑥,𝑦∈𝑌,2𝑥,2𝑦≤𝜂𝜒𝑥,𝑦for𝑥,𝑦∈𝐾, (3.12)(3.13) where 𝜒∶𝐾×𝐾→[0,∞). Let 𝜑∶𝐾→𝑌 satisfy 𝑑𝜑𝑥+𝑦,𝜑(𝑥)+𝜑𝑦≤𝜒𝑥,𝑦for𝑥,𝑦∈𝐾. (3.14) Then there is a unique solution 𝐹∶𝐾→𝑌 of 𝐹(𝑥+𝑦)=𝐹(𝑥)+𝐹(𝑦) with 𝑑𝜑(𝑥)(𝑥≤(),𝐹)𝜉𝜒𝑥,𝑥1−𝜉𝜂for𝑥∈𝐾. (3.15)

Proof. From (3.12) and (3.14), we get 𝑑12𝜑(2𝑥),𝜑(𝑥)≤𝜉𝑑𝜑(2𝑥),2𝜑(𝑥)≤𝜉𝜒(𝑥,𝑥) (3.16) for 𝑥∈𝐾. By using Theorem 3.1 with 𝑓=𝜑,𝐻(𝑧)=(1/2)𝑧,𝐺(𝑥)=2𝑥, 𝜆=𝜉, and ℎ(𝑥)=𝜉𝜒(𝑥,𝑥), the limit function 𝐹(𝑥) exists for each 𝑥∈𝐾 and 𝑑𝑓(𝑥),𝐹(𝑥)≤^∞_𝑖=0𝜉^𝑖ℎ𝐺^𝑖(𝑥). (3.17)
As 𝐺^𝑖(𝑥)=2^𝑖𝑥 and ℎ(𝐺^𝑖(𝑥))≤𝜉𝜂^𝑖𝜒(𝑥,𝑥) for every 𝑥∈𝐾, we get 𝑑𝜑(𝑥),𝐹(𝑥)=𝑑𝑓(𝑥),𝐹(𝑥)≤𝜉𝜒(𝑥,𝑥)^∞_𝑖=0𝜉𝜂^𝑖=𝜉𝜒(𝑥,𝑥)1−𝜉𝜂. (3.18) Next, by (3.14), for every 𝑥,𝑦∈𝐾 we have 𝑑12^𝑛𝜑2^𝑛,1𝑥+𝑦2^𝑛𝜑2^𝑛𝑥+12^𝑛𝜑2^𝑛𝑥≤𝜂^𝑛𝜒𝑥,𝑦 (3.19) for 𝑛∈ℕ, so letting 𝑛→∞ we obtain 𝐹(𝑥+𝑦)=𝐹(𝑥)+𝐹(𝑦).
Suppose 𝐹₀∶𝐾→𝑌 is also a solution of 𝐹(𝑥+𝑦)=𝐹(𝑥)+𝐹(𝑦) and 𝑑𝜑(𝑥),𝐹₀(𝑥)≤^∞_𝑖=0𝜉^𝑖ℎ𝐺^𝑖(𝑥),forevery𝑥∈𝐾. (3.20) Then 𝐻𝑜𝐹₀𝑜𝐺=𝐹₀, whence, by Theorem 3.1, we have 𝐹=𝐹₀ which implies the uniqueness of 𝐹.

The following theorem is an extended application of Hyers' result to the Riesz spaces.

Theorem 3.3. Let 𝐸 a linear space, 𝐹 be a Riesz space equipped with an extended norm ‖⋅‖_𝑢 such that the space (𝐹,‖⋅‖_𝑢) is complete. If, for some 𝛿>0, a map 𝑓∶𝐸→(𝐹,‖⋅‖_𝑢) is 𝛿-additive, then limit 𝑙(𝑥)=𝑙𝑖𝑚_𝑛→∞𝑓(2^𝑛𝑥)/2^𝑛 exists for each 𝑥∈𝐸. 𝑙(𝑥) is the unique additive function satisfying the inequality ‖𝑓(𝑥)−𝑙(𝑥)‖_𝑢≤𝛿 for all 𝑥∈𝐸.

Now, if 𝐹 is a Banach space or extended 𝑢-normed Banach lattice, then we can take 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖ or 𝑑(𝑥,𝑦)=‖𝑥−𝑦‖_𝑢, 𝜒(𝑥,𝑥)=𝜀, 𝜉=1/2, and 𝜂=1. We may obtain the classical Hyers' result [2] and Theorem 3.3 with such 𝜒(𝑥,𝑥), 𝜉 and 𝜂 by using Corollary 3.2.

Finally, we give the following theorem which is an extended application of Badora's result (Theorem 1.2) to Riesz algebras with extended norms. For a proof, we use Theorem 3.3 and the similar techniques of Badora [6] with suitable modifications.

Theorem 3.4. Let 𝐸 be a linear algebra, and let 𝐹 be a Riesz algebra with an extended norm ‖⋅‖_𝑢 such that (𝐹,‖⋅‖_𝑢) is complete. Also, let ‖⋅‖_𝑣 be another extended norm in 𝐹 weaker than ‖⋅‖_𝑢 such that whenever(1)𝑥_𝑛→𝑥 and 𝑥_𝑛⋅𝑦→𝑧 in ‖⋅‖_𝑣, then 𝑧=𝑥⋅𝑦;(2)𝑦_𝑛→𝑦 and 𝑥⋅𝑦_𝑛→𝑧 in ‖⋅‖_𝑣, then 𝑧=𝑥⋅𝑦.Let 𝜀 and 𝛿 be nonnegative real numbers. Assume that a map 𝑓∶𝐸→𝐹 satisfies 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦_𝑢𝑓(𝑥)𝑓𝑦≤𝜀,𝑥⋅𝑦−𝑓_𝑣≤𝛿 (3.21)(3.22) for all 𝑥,𝑦∈𝐸. Then there exists a unique ring homomorphism ℎ∶𝐸→𝐹 such that ||𝑓(𝑥)−ℎ(𝑥)||_𝑢≤𝜀, 𝑥∈𝐸. Moreover, 𝑏⋅𝑓(𝑥)−ℎ(𝑥)=0,𝑓(𝑥)−ℎ(𝑥)⋅𝑏=0, (3.23) for all 𝑥∈𝐸 and all 𝑏 from the algebra generated by ℎ(𝐸).

Proof. From Theorem 3.3, it follows that there exists a unique additive function ℎ∶𝐸→𝐹 such that 𝑓(𝑥)−ℎ(𝑥)_𝑢≤𝜀,(𝑥∈𝐸). (3.24) Hence, it is enough to show that ℎ is a multiplicative function. Using the additivity of ℎ, it follows that 1𝑛𝑓(𝑛𝑥)−ℎ(𝑥)_𝑢≤1𝑛𝜀,(𝑥∈𝐸,𝑛∈ℕ), (3.25) which means that ℎ(𝑥)=lim_𝑛→∞1𝑛𝑓(𝑛𝑥),(𝑥∈𝐸), (3.26) with respect to ‖⋅‖_𝑢 norm.
Let 𝑟𝑥,𝑦=𝑓𝑥⋅𝑦−𝑓(𝑥)𝑓𝑦,𝑥,𝑦∈𝐸. (3.27) Then using inequality (3.22), we get lim_𝑛→∞1𝑛𝑟𝑛𝑥,𝑦=0,𝑥,𝑦∈𝐸, (3.28) with respect to ‖⋅‖_𝑣 norm.
Applying (3.26) and (3.28), we have ℎ𝑥⋅𝑦=lim_𝑛→∞1𝑛𝑓𝑛𝑥⋅𝑦=lim_𝑛→∞1𝑛𝑓(𝑛𝑥)⋅𝑦=lim_𝑛→∞1𝑛𝑓(𝑛𝑥)𝑓𝑦+𝑟𝑛𝑥,𝑦=ℎ(𝑥)𝑓𝑦, (3.29) for all 𝑥,𝑦∈𝐸, since ‖⋅‖_𝑣 is weaker than ‖⋅‖_𝑢. Hence, we get the following functional equation: ℎ𝑥⋅𝑦=ℎ(𝑥)𝑓𝑦,𝑥,𝑦∈𝐸. (3.30) From this equation and the additivity of ℎ, we have ℎ(𝑥)𝑓()()𝑓𝑦(𝑥)𝑓𝑦,.𝑛𝑦=ℎ𝑥⋅𝑛𝑦=ℎ𝑛𝑥⋅𝑦=ℎ𝑛𝑥=𝑛ℎ𝑥,𝑦∈𝐸,𝑛∈ℕ (3.31) Therefore, ℎ(𝑥)1𝑛𝑓𝑛𝑦=ℎ(𝑥)𝑓𝑦,𝑥,𝑦∈𝐸,𝑛∈ℕ. (3.32) Sending 𝑛 to infinity, by (3.26), we see that ℎ(𝑥)ℎ𝑦=ℎ(𝑥)𝑓𝑦,𝑥,𝑦∈𝐸. (3.33) Combining this equation with (3.30), we see that ℎ is a multiplicative function.
Moreover, from (3.22) we get lim_𝑛→∞1𝑛𝑟𝑥,𝑛𝑦=0,𝑥,𝑦∈𝐸, (3.34) with respect to ‖⋅‖_𝑣 norm.
Thus, by (3.26) and the fact that ‖⋅‖_𝑣 is weaker than ‖⋅‖_𝑢, we get that ℎ𝑥⋅𝑦=lim_𝑛→∞1𝑛𝑓𝑛𝑥⋅𝑦=lim_𝑛→∞1𝑛𝑓𝑥⋅𝑛𝑦=lim_𝑛→∞𝑓(𝑥)1𝑛𝑓+1𝑛𝑦𝑛𝑟𝑥,𝑛𝑦=𝑓(𝑥)ℎ𝑦, (3.35) for all 𝑥,𝑦∈𝐸. Hence, by (3.33), 𝑓(𝑥)ℎ𝑦=ℎ𝑥⋅𝑦=ℎ(𝑥)ℎ𝑦=ℎ(𝑥)𝑓𝑦,𝑥,𝑦∈𝐸, (3.36) so that ℎ(𝑥)𝑓𝑦−ℎ𝑦=0,𝑓(𝑥)−ℎ(𝑥)ℎ𝑦=0,𝑥,𝑦∈𝐸 (3.37) which completes the proof.