On non--linear perturbations of random isometries in random normed modules
© Zhao et al.; licensee Springer. 2014
Received: 3 September 2014
Accepted: 26 November 2014
Published: 12 December 2014
The purpose of this paper is to study non--linear perturbations of random isometries in random normed modules. Let be a probability space, K the scalar field R of real numbers or C of complex numbers, the equivalence classes of K-valued ℱ-measurable random variables on Ω, and random normed modules over K with base . In this paper, we first establish the Mazur-Ulam theorem in random normed modules. Making use of this theorem and the relations between random normed modules and classical normed spaces, we show that if is a surjective random ε-isometry with and has the local property, where and , then there is a surjective -linear random isometry such that , for all . We do not obtain a sharp estimate as the classical result, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.
MSC:46A22, 46A25, 46H25.
Random metric theory originated from the theory of probabilistic metric spaces . The random distance between two points in an original random metric space (briefly, an RM space) is a nonnegative random variable defined on some probability space, similarly, the random norm of a vector in an original random normed space (briefly, an RN space) is a nonnegative random variable defined on some probability space. The development of RN spaces in the direction of functional analysis led Guo to present a new version of RM and RN spaces in , where the random distances or random norms are defined to be the equivalence classes of nonnegative random variables according to the new versions. Based on the new version of an RN space, Guo presented a definitive definition of the random conjugate space for an RN space. Along with the deep development of the theory of random conjugate spaces, Guo established the notion of a random normed module (briefly, an RN module) in . In the past ten years, as the central part of random metric theory, random normed modules and random locally convex modules (briefly, RLC modules) together with their random conjugate spaces have been deeply studied under the -topology in the direction of functional analysis, cf. [4–19] and the related references in these papers.
The purpose of this paper is to study non--linear perturbations of random isometries in random normed modules. For the readers’ convenience, let us first recall some classical results as follows.
f is surjective and ;
f is surjective and .
A celebrated result, known as the Mazur-Ulam theorem , is a perfect answer to case (1).
Theorem 1.1 (Mazur-Ulam)
Let X and Y be two Banach spaces, a surjective isometry with . Then f is linear.
For case (2), after many efforts of a number of mathematicians, the following sharp estimate was finally obtained by Omladič-Šemrl .
Theorem 1.2 (Omladič-Šemrl)
K: the scalar field R of real numbers or C of complex numbers.
: a probability space.
= the algebra of equivalence classes of K-valued F-measurable random variables on .
= the set of equivalence classes of extended real-valued F-measurable random variables on .
In order to introduce the main results of this paper, we need some notation and terminology as follows:
As usual, is partially ordered by iff for P-almost all (briefly, a.s.), where and are arbitrarily chosen representatives of ξ and η, respectively. Then is a complete lattice, ⋁H and ⋀H denote the supremum and infimum of a subset H, respectively. is a conditionally complete lattice. Please refer to  or [, p.3026] for the rich properties of the supremum and infimum of a set in .
Besides, always denotes the equivalence class of , where and is the characteristic function of A. When denotes the equivalence class of A (), namely (here, ), we also use for .
If , then the mapping f is called a random isometry; and it is said to be a surjective random ε-isometry if, in addition, .
Now, we give the main results of this paper, namely Theorems 1.4 and 1.5 below. For Theorem 1.4, it is easy to see that it has the same shape as the classical Mazur-Ulam theorem, but it is not trivial since we must make full use of the relations between random normed modules and classical normed spaces in the process of the proof. For Theorem 1.5, we do not get a sharp estimate as the classical result, namely Theorem 1.2, since the complicated stratification structure in the random setting needs to be considered, which is the essential difference between random normed modules and classical normed spaces.
Theorem 1.4 Let and be two complete random normed modules over K with base , a surjective random isometry. Then f is an -linear function.
The remainder of this paper is organized as follows: in Section 2 we will briefly collect some necessary well-known facts; in Section 3 we will give the proofs of the main results in this paper.
(RN-1) , and ;
(RN-2) implies (the null element of E);
(RN-3) , .
Here is called the random norm on E and the random norm of (if only satisfies (RN-1) and (RN-3) above, it is called a random seminorm on E).
Furthermore, if, in addition, E is a left module over the algebra (briefly, an -module) such that
(RNM-1) , and .
Then is called a random normed module (briefly, an RN module) over K with base , the random norm with the property (RNM-1) is also called an -norm on E (a mapping only satisfying (RN-3) and (RNM-1) above is called an -seminorm on E).
Definition 2.2 ()
Let be an RN space over K with base . A linear operator f from E to is said to be an a.s. bounded random linear functional if there is such that , . Denote by the linear space of a.s. bounded random linear functionals on E, define by for all , then it is easy to check that is also an RN module over K with base , called the random conjugate space of E.
Definition 2.3 Let and be two RN modules over K with base , a module homomorphism is said to be -linear.
Example 2.4 ()
Let be the -module of equivalence classes of ℱ-random variables (or, strongly ℱ-measurable functions) from to a normed space over K. induces an -norm (still denoted by ) on by := the equivalence class of for all , where is a representative of x. Then is an RN module over K with base . Specially, is an RN module, the -norm on is still denoted by .
Definition 2.5 ()
Let be an RN space over K with base . For any positive numbers ε and λ with , let , then forms a local base at θ of some Hausdorff linear topology on E, called the -topology induced by .
From now on, we always denote by the -topology for every RN space if there is no possible confusion. Clearly, the -topology for the special class of RN modules is exactly the ordinary topology of convergence in measure, and is a topological algebra over K. It is also easy to check that is a topological module over when is an RN module over K with base , namely the module multiplication operation is jointly continuous.
Let E be an -module. A sequence in E is countably concatenatable in E with respect to a countable partition of Ω to ℱ if there is such that for each , in which case we define as x. A subset G of E is said to have the countable concatenation property if each sequence in G is countably concatenatable in E with respect to an arbitrary countable partition of Ω to ℱ and . It is easy to see that a complete RN module E under has the countable concatenation property.
The following definition is very important for the main results of this paper.
Definition 2.6 ()
for any and .
3 Proofs of main results
In order to give the proof of Mazur-Ulam theorem on random normed modules, we need the following lemmas and readers can find the proofs of them in .
Lemma 3.1 ()
Let E be a left module over the algebra , a random linear functional and an -linear function such that , . Then f is an -linear function. If R is replaced by C and p is an -seminorm such that , , then f is also an -linear function.
Lemma 3.2 ()
for all .
Then is a normed space and is -dense in E.
Remark 3.3 It is easy to see that if is complete under the -topology, then is also complete, for .
With the above preparations, we can give the proof of Theorem 1.4.
Proof of Theorem 1.4 Since is a random isometry with , we see that f is random norm preserving and is a mapping from to . It is clear that and are two Banach spaces and is a surjective isometry with . By classical Mazur-Ulam theorem, we see that is linear. Since is dense in and f is continuous under , it is clear that f is a random linear functional. Since , we see that f is an -linear function from Lemma 3.1. □
Making use of Theorem 1.4 and the relations between random normed modules and classical normed spaces, we give the proof of Theorem 1.5.
for any .
for any . By Theorem 1.4, U is a surjective -linear random isometry. It completes the proof. □
Remark 3.4 In Theorem 1.5, we do not obtain a sharp estimate as the classical result, namely Theorem 1.2, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.
This work was supported by National Natural Science Foundation of China, grant 11401399, and Beijing Natural Science Foundation, grant 1144008.
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