Proximinality in Banach space valued Musielak-Orlicz spaces
Journal of Inequalities and Applications volume 2014, Article number: 146 (2014)
Let be a σ-finite complete measure space and let Y be a subspace of a Banach space X. Let φ be a generalized Φ-function on . Denote by and the Musielak-Orlicz spaces whose functions take values in Y and X, respectively. Firstly, let , we characterize the distance of f from . Then, if Y is weakly -analytic and proximinal in X, we show that is proximinal in . Finally, we give the connection between the proximinality of in and the proximinality of in .
It is well known that Musielak-Orlicz spaces include many spaces as special spaces, such as Lebesgue spaces, weighted Lebesgue spaces, variable Lebesgue spaces and Orlicz spaces; see . Especially, in recent decades, variable exponent function spaces, such as Lebesgue, Sobolev, Besov, Triebel-Lizorkin, Hardy, Morrey, and Herz spaces with variable exponents, have attracted much attention; see [2–16] and references therein. Cheng and the author discussed geometric properties of Banach space valued Bochner-Lebesgue and Bochner-Sobolev spaces with a variable exponent in . Very recently, Musielak-Orlicz-Hardy spaces have been systemically developed; see, for example, [18–22]. These spaces have many applications in various fields such as PDE, electrorheological fluids, and image restoration; see [6, 23–25].
In recent years, proximinality in Banach space valued Bochner-Lebesgue spaces with constant exponent have been extensively studied; see [26–33]. Proximinality in Banach space valued Bochner-Lebesgue spaces with variable exponent was discussed by the author in . In fact, we generalized those results in [29, 31] to Banach space valued Bochner-Lebesgue spaces with a variable exponent. Khandaqji, Khalil and Hussein considered proximinality in Orlicz-Bochner function spaces on the unit interval in , and Al-Minawi and Ayesh consider the same problem on finite measures in . The best simultaneous approximation in Banach space valued Orlicz spaces was discussed in [37, 38]. Micherda discussed proximinality of subspaces of vector-valued Musielak-Orlicz spaces via modular in . However, as usual, one considers the best approximation via the norm, so in this paper, we will discuss proximinality of subspaces of vector-valued Musielak-Orlicz spaces via the norm. To proceed, we need to recall some definitions. Our results will be given in the next section.
In what follows, will be a σ-finite complete measure space. Suppose D is a subset of A, let be the indicator function on D. Let be a Banach space. The dual space of X is the vector space of all continuous linear mappings from X to ℝ or ℂ. To avoid a double definition we let be either ℝ or ℂ.
Definition 1 A convex, left-continuous function with , , is called a Φ-function. It is called positive if for all .
It is easy to see that if φ is a Φ-function, then it is nondecreasing on .
Definition 2 Let be a σ-finite complete measure space. A real function is called a generalized Φ-function on if
is a Φ-function for all ,
is measurable for all .
If φ is a generalized Φ-function on , we write .
Definition 3 Let . Define
for strongly μ-measurable functions . Then the Bochner-Musielak-Orlicz space is the collection of all strongly μ-measurable functions endowed with the norm:
Definition 4 Let . The function φ is said to obey the -condition if there exists such that
for all and all .
When X is ℝ or ℂ, we simply denote by , and by . Usually, is a proper subspace of . But when the φ satisfies the -condition, they are the same. It is easy to see that is equivalent to , this means that the equality depends only on φ.
We remark that is a semimodular on the space of all X-valued strongly μ-measurable functions on A. For a semimodular, we recommend the reader reference . Let ρ be a semimodular on vector space E, , . We will use the following elementary result for a semimodular, which is Corollary 2.1.15 in .
Lemma 1 Let ρ be a semimodular on E, .
If , then .
If , then .
Let X be a Banach space and let Y be a closed subspace of X. Then Y is called proximinal in X if for any there exists such that
In this case y is called a best approximation of x in Y. If this best approximation is unique for any , then Y is said to be Chebyshev.
For simplicity, we denote or by . For , , let
2 Main results
Firstly, we estimate .
Theorem 1 Let Y be a subspace of Banach space X. Suppose . For , define by . Then
Proof (i) Given , we see that there exists a sequence of simple functions which converges to f almost everywhere and in . Since the distance function is a continuous function of , implies that . Moreover, each function defined by is a simple function; therefore we conclude that ϕ is measurable. Now, for any and any ,
Thus, we have
This implies and, by taking an infimum on , we have
(ii) We first assume that f is a simple function. Let where are disjoint measurable subsets in A such that and for . Without loss of generality, we suppose that . Let . Since , we have for any . Then, by the dominated convergence theorem, we find that there exists such that
Now take such that for . Let . Therefore . By Lemma 1, we see that
Thus, . Since ϵ is arbitrary, we have . By Lemma 1 again, we have . This means that . Therefore, we have proved that for simple functions.
Finally, let , there exists a sequence of simple functions convergent to f μ-almost everywhere, μ-almost everywhere and as n tends to ∞. Let . From the previous proof, we have
It is easy to see that as . Since μ-almost everywhere, and μ-almost everywhere as n tends to ∞, by Lemma 2.3.16(c) in , we conclude that in . Hence, letting , we see that
which completes the proof of Theorem 1. □
Corollary 1 Let Y be a closed subspace of a Banach space X. Suppose . An element g of is a best approximation to an element f in if and only if is a best approximation in Y to for almost every . Furthermore, if φ satisfies the -condition, then an element g of is a best approximation to an element f in if and only if is a best approximation in Y to for almost every .
Corollary 2 Let Y be a Chebyshev subspace of a Banach space X. Suppose satisfies the -condition. If is proximinal in , then it is a Chebyshev subspace of .
Remark 1 Our results Theorem 1, Corollaries 1 and 2 cover the results for vector Orlicz spaces in . Indeed, in  the authors only considered vector Orlicz spaces on finite measures. Analogous results for best simultaneous approximation were obtained in [37, 38] for vector Orlicz spaces on finite measures.
Next, we transfer the proximinality of Y in X to in . To do so, we need some preliminaries.
Lemma 2 Let be a σ-finite complete measure space. Suppose . Let Y be a proximinal subspace of a Banach space X. Suppose and g is a strongly μ-measurable function such that is a best approximation to from Y for almost everywhere . Then g is a best approximation to f from .
Proof Since , it follows that μ-almost everywhere. Thus, . For each , by assumption we know that μ-almost everywhere. So for any . Thus, . This ends the proof. □
Definition 5 Let be a Polish space (i.e. a topological space which is separable and completely metrizable). A set is analytic if it is empty or if there exists a continuous mapping satisfying , where denotes the space of all infinite sequences of natural numbers endowed with the Tychonoff topology.
Definition 6 Let be a Polish space, H a topological space and denote by the smallest σ-algebra containing all analytic subsets of T. Then a mapping is said to be analytic measurable if for every , where is for the Borel sets of H.
Definition 7 Let H, T be topological spaces. Then a multifunction is said to be upper semi-continuous if for every and for every open set U satisfying , there exists an open neighborhood V of x such that .
Definition 8 A subset C of a topological space T is -analytic if it can be written as for some upper semi-continuous mapping with compact values. In the case when T is a Banach space endowed with its weak topology, C is said to be weakly -analytic.
Lemma 3 Let be a real Banach space and let Y be a proximinal, weakly -analytic convex subset of X. Then, for each closed and separable set , there exists an analytic measurable mapping such that is separable in Y and is a best approximation of x in Y for any .
Thus, following the argument of the proof of (i) → (ii) in [, p.185], we have the following conclusion, the details being omitted.
Theorem 2 Let be a σ-finite complete measure space. Suppose , and Y is a weakly -analytic linear subspace of a real Banach space X. If Y is proximinal in X, then is proximinal in .
Theorem 3 Let be a σ-finite measure space. Suppose such that . Let Y be a linear subspace of a real Banach space X. If is proximinal in , then Y is proximinal in X.
Proof Since the measure μ is σ-finite, let us choose positive measure set Q such that . For any , let , . Then . By the assumption, we know that there is a g in which is a best approximation element of f. Consequently, is a best approximation to in Y for almost every by Corollary 1. Therefore, there is a best approximation element of x in Y. Thus, Y is proximinal in X. □
From Theorems 2 and 3, we deduce the following corollary.
Corollary 3 Let be a σ-finite complete measure space. Suppose such that . Let Y be a weakly -analytic linear subspace of a real Banach space X. Then the following conditions are equivalent:
Y is proximinal in X;
is proximinal in .
Remark 2 An analog to Corollary 3 in terms of a modular was obtained in .
Finally, we give a characterization of proximinity of in via the proximinity of in . When is a Bochner-Lebesgue space, which was obtained in  and [32, 33] on finite measure spaces and σ finite measure spaces, respectively. is a Bochner-Orlicz space, which was discussed in .
Theorem 4 Let be a σ-finite complete measure space. Suppose such that the set of simple functions, , satisfies , where is the conjugate function of φ (see ). Let Y be a closed subspace of a Banach space X. If is proximinal in , then is proximinal in .
Proof Since A is σ-finite, we may write , where is a sequence of disjoint measurable sets each of finite measure. Let . For any , since , then . Thus, by the norm conjugate formula (see Corollary 2.7.5 in ), we find that . By assumption, we know that there exists such that
By Corollary 1, we have, for all ,
μ-almost everywhere. Therefore μ-almost every . Let . Since , it follows that for all ,
μ-almost everywhere. Because , it follows that . Thus, and
for all . This finishes the proof. □
Theorem 5 Let be a σ-finite complete measure space. Suppose satisfies and, for each , is strictly increasing. Let Y be a closed subspace of a Banach space X. If is proximinal in , then is proximinal in .
Proof We use the idea from . Indeed in  the authors only considered Banach space valued Orlicz spaces on the unit interval. Since, for each , is strictly increasing, let be its inverse function, which means, for each , . Define the map by setting
Then . Therefore . So J is injective. Moreover, if , let
Then and . Thus, . In addition, for ,
If , then also, thus . Hence J is surjective and also.
Now, let . Without loss of generality we may suppose that μ-almost everywhere, for otherwise we can restrict our measure to the support of f. Since , by the assumption, we know that there exists some such that
for all . By Corollary 1, we see that, for all ,
μ-almost everywhere. Multiplying both sides of the last inequality by , we obtain, for all ,
Let . Since is a best approximation of in Y, and , it follows that . Therefore, . Thus, for all ,
μ-almost everywhere. Thus, by Corollary 1 h is a best approximation of f in . This finishes the proof. □
From Theorems 4 and 5, we deduce the following corollary.
Corollary 4 Let be a σ-finite complete measure space. Suppose such that , for each , is strictly increasing and the set of simple functions satisfies . Let Y be a closed subspace of a Banach space X. Then the following conditions are equivalent:
is proximinal in ;
is proximinal in .
Remark 3 When is a finite measure and φ is a Orlicz function that satisfies the -condition, the result of Corollary 4 was obtained in . While is the unit interval and φ is a Young function that satisfies the -condition, the result of Corollary 4 was obtained in .
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The author would like to thank the referee for carefully reading which made the presentation more readable and for his or her suggestion for references [36–38]. The author was supported by the National Natural Science Foundation of China (Grant No. 11361020) and the National Natural Science Foundation of Hainan Providence (113004).
The author declare that he has no competing interests.
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Xu, J. Proximinality in Banach space valued Musielak-Orlicz spaces. J Inequal Appl 2014, 146 (2014). https://doi.org/10.1186/1029-242X-2014-146
- Musielak-Orlicz space
- best approximation
- weakly -analytic