Proximinality in Banach space valued Musielak-Orlicz spaces
© Xu; licensee Springer. 2014
Received: 22 December 2013
Accepted: 27 March 2014
Published: 9 April 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 .
is a Φ-function for all ,
is measurable for all .
If φ is a generalized Φ-function on , we write .
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 .
If , then .
If , then .
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.
2 Main results
Firstly, we estimate .
Thus, . Since ϵ is arbitrary, we have . By Lemma 1 again, we have . This means that . Therefore, we have proved that for simple functions.
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 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.
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 .
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 .
If , then also, thus . Hence J is surjective and also.
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.
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 .
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).
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