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Proximinality in Banach space valued MusielakOrlicz spaces
Journal of Inequalities and Applications volume 2014, Article number: 146 (2014)
Abstract
Let (A,\mathcal{A},\mu ) be a σfinite complete measure space and let Y be a subspace of a Banach space X. Let φ be a generalized Φfunction on (A,\mathcal{A},\mu ). Denote by {L}^{\phi}(A,Y) and {L}^{\phi}(A,X) the MusielakOrlicz spaces whose functions take values in Y and X, respectively. Firstly, let f\in {L}^{\phi}(A,X), we characterize the distance of f from {L}^{\phi}(A,Y). Then, if Y is weakly analytic and proximinal in X, we show that {L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X). Finally, we give the connection between the proximinality of {L}^{\phi}(A,Y) in {L}^{\phi}(A,X) and the proximinality of {L}^{1}(A,Y) in {L}^{1}(A,X).
1 Introduction
It is well known that MusielakOrlicz spaces include many spaces as special spaces, such as Lebesgue spaces, weighted Lebesgue spaces, variable Lebesgue spaces and Orlicz spaces; see [1]. Especially, in recent decades, variable exponent function spaces, such as Lebesgue, Sobolev, Besov, TriebelLizorkin, 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 BochnerLebesgue and BochnerSobolev spaces with a variable exponent in [17]. Very recently, MusielakOrliczHardy 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 BochnerLebesgue spaces with constant exponent have been extensively studied; see [26–33]. Proximinality in Banach space valued BochnerLebesgue spaces with variable exponent was discussed by the author in [34]. In fact, we generalized those results in [29, 31] to Banach space valued BochnerLebesgue spaces with a variable exponent. Khandaqji, Khalil and Hussein considered proximinality in OrliczBochner function spaces on the unit interval in [35], and AlMinawi and Ayesh consider the same problem on finite measures in [36]. The best simultaneous approximation in Banach space valued Orlicz spaces was discussed in [37, 38]. Micherda discussed proximinality of subspaces of vectorvalued MusielakOrlicz spaces via modular in [39]. However, as usual, one considers the best approximation via the norm, so in this paper, we will discuss proximinality of subspaces of vectorvalued MusielakOrlicz spaces via the norm. To proceed, we need to recall some definitions. Our results will be given in the next section.
In what follows, (A,\mathcal{A},\mu ) will be a σfinite complete measure space. Suppose D is a subset of A, let {\chi}_{D} be the indicator function on D. Let (X,\parallel \cdot \parallel ) be a Banach space. The dual space of X is the vector space {X}^{\ast} 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, leftcontinuous function \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}] with \phi (0)=0, {lim}_{t\to {0}^{+}}\phi (t)=0, {lim}_{t\to \mathrm{\infty}}\phi (t)=\mathrm{\infty} is called a Φfunction. It is called positive if \phi (t)>0 for all t>0.
It is easy to see that if φ is a Φfunction, then it is nondecreasing on [0,\mathrm{\infty}).
Definition 2 Let (A,\mathcal{A},\mu ) be a σfinite complete measure space. A real function \phi :A\times [0,\mathrm{\infty})\to [0,\mathrm{\infty}] is called a generalized Φfunction on (A,\mathcal{A},\mu ) if

(a)
\phi (y,\cdot ) is a Φfunction for all y\in A,

(b)
y\mapsto \phi (y,t) is measurable for all t\u2a7e0.
If φ is a generalized Φfunction on (A,\mathcal{A},\mu ), we write \phi \in \mathrm{\Phi}(A,\mu ).
Definition 3 Let \phi \in \mathrm{\Phi}(A,\mu ). Define
for strongly μmeasurable functions f:A\to X. Then the BochnerMusielakOrlicz space {L}^{\phi}(A,X) is the collection of all strongly μmeasurable functions f:A\to X endowed with the norm:
Let
Definition 4 Let \phi \in \mathrm{\Phi}(A,\mu ). The function φ is said to obey the {\mathrm{\Delta}}_{2}condition if there exists K\ge 2 such that
for all s\in A and all t\ge 0.
When X is ℝ or ℂ, we simply denote {L}^{\phi}(A,X) by {L}^{\phi}(A), and {E}^{\phi}(A,X) by {E}^{\phi}(A). Usually, {E}^{\phi}(A,X) is a proper subspace of {L}^{\phi}(A,X). But when the φ satisfies the {\mathrm{\Delta}}_{2}condition, they are the same. It is easy to see that {E}^{\phi}(A,X)={L}^{\phi}(A,X) is equivalent to {E}^{\phi}(A)={L}^{\phi}(A), this means that the equality depends only on φ.
We remark that {\rho}_{\phi} is a semimodular on the space of all Xvalued strongly μmeasurable functions on A. For a semimodular, we recommend the reader reference [6]. Let ρ be a semimodular on vector space E, {E}_{\rho}=\{x\in E:\rho (x/\lambda )<\mathrm{\infty}\text{for some}\lambda 0\}, {\parallel x\parallel}_{\rho}=inf\{\lambda >0:\rho (x/\lambda )\le 1\}. We will use the following elementary result for a semimodular, which is Corollary 2.1.15 in [6].
Lemma 1 Let ρ be a semimodular on E, x\in {E}_{\rho}.

(i)
If {\parallel x\parallel}_{\rho}\le 1, then \rho (x)\le {\parallel x\parallel}_{\rho}.

(ii)
If 1<{\parallel x\parallel}_{\rho}, then {\parallel x\parallel}_{\rho}\le \rho (x).
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 x\in X there exists y\in Y such that
In this case y is called a best approximation of x in Y. If this best approximation is unique for any x\in X, then Y is said to be Chebyshev.
For simplicity, we denote {\parallel \cdot \parallel}_{{L}^{\phi}(A,X)} or {\parallel \cdot \parallel}_{{L}^{\phi}(A)} by {\parallel \cdot \parallel}_{\phi}. For f\in {L}^{\phi}(A,X), Y\subset X, let
2 Main results
Firstly, we estimate {dist}_{\phi}(f,{L}^{\phi}(A,Y)).
Theorem 1 Let Y be a subspace of Banach space X. Suppose \phi \in \mathrm{\Phi}(A,\mu ). For f\in {L}^{\phi}(A,X), define \varphi :A\to \mathbb{R} by \varphi (s):=dist(f(s),Y). Then

(i)
\varphi \in {L}^{\phi}(A) and {dist}_{\phi}(f,{L}^{\phi}(A,Y))\ge {\parallel \varphi \parallel}_{\phi};

(ii)
{dist}_{\phi}(f,{L}^{\phi}(A,Y))={\parallel \varphi \parallel}_{\phi} for f\in {E}^{\phi}(A,X).
Proof (i) Given f\in {L}^{\phi}(A,X), we see that there exists a sequence of simple functions \{{f}_{n}\} which converges to f almost everywhere and in {L}^{\phi}(A,X). Since the distance function d(x,Y) is a continuous function of x\in X, \parallel {f}_{n}(s)f(s)\parallel \to 0 implies that dist({f}_{n}(s),Y)dist(f(s),Y)\to 0. Moreover, each function {\varphi}_{n}:A\to \mathbb{R} defined by {\varphi}_{n}(s):=dist({f}_{n}(s),Y) is a simple function; therefore we conclude that ϕ is measurable. Now, for any g\in {L}^{\phi}(A,Y) and any \lambda >0,
Thus, we have
This implies \varphi \in {L}^{\phi}(A) and, by taking an infimum on g\in {L}^{\phi}(A,Y), we have
(ii) We first assume that f is a simple function. Let f(s):={\sum}_{i=1}^{m}{\chi}_{{A}_{i}}{x}_{i} where {\{{A}_{i}\}}_{i=1}^{m} are disjoint measurable subsets in A such that 0<\mu ({A}_{i})<\mathrm{\infty} and 0\ne {x}_{i}\in X for i\in \{1,\dots ,m\}. Without loss of generality, we suppose that {dist}_{\phi}(f,{L}^{\phi}(A,Y))=1. Let 0<\u03f5<1. Since \varphi (s)\le \parallel f(s)\parallel, we have {\rho}_{\phi}(\lambda \varphi )\le {\rho}_{\phi}(\lambda f)<\mathrm{\infty} for any \lambda >0. Then, by the dominated convergence theorem, we find that there exists \delta >0 such that
Now take {y}_{i}\in Y such that \parallel {x}_{i}{y}_{i}\parallel <dist({x}_{i},Y)+\delta for i\in \{1,\dots ,m\}. Let g(s)={\sum}_{i=1}^{m}{\chi}_{{A}_{i}}{y}_{i}. Therefore {\parallel fg\parallel}_{\phi}\ge {dist}_{\phi}(f,{L}^{\phi}(A,Y))=1. By Lemma 1, we see that
Thus, {\rho}_{\phi}(\varphi )\ge 1\u03f5. Since ϵ is arbitrary, we have {\rho}_{\phi}(\varphi )\ge 1. By Lemma 1 again, we have {\parallel \varphi \parallel}_{\phi}\ge 1. This means that {\parallel dist(f(\cdot ),Y)\parallel}_{\phi}\ge {dist}_{\phi}(f,{L}^{\phi}(A,Y)). Therefore, we have proved that {\parallel dist(f(\cdot ),Y)\parallel}_{\phi}={dist}_{\phi}(f,{L}^{\phi}(A,Y)) for simple functions.
Finally, let f\in {E}^{\phi}(A,X), there exists a sequence of simple functions {\{{g}_{n}\}}_{n\in \mathbb{N}} convergent to f μalmost everywhere, \parallel {g}_{n}(s)\parallel \le \parallel f(s)\parallel μalmost everywhere and {\parallel f{g}_{n}\parallel}_{\phi}\to 0 as n tends to ∞. Let {\varphi}_{n}(s)=dist({g}_{n}(s),Y). From the previous proof, we have
It is easy to see that {dist}_{\phi}({g}_{n},{L}^{\phi}(A,Y))\to {dist}_{\phi}(f,{L}^{\phi}(A,Y)) as n\to \mathrm{\infty}. Since {\varphi}_{n}(s)\le \parallel {g}_{n}(s)\parallel \le \parallel f(s)\parallel μalmost everywhere, and {\varphi}_{n}(s)\to \varphi (s) μalmost everywhere as n tends to ∞, by Lemma 2.3.16(c) in [6], we conclude that {\varphi}_{n}\to \varphi in {L}^{\phi}(A). Hence, letting n\to \mathrm{\infty}, we see that
which completes the proof of Theorem 1. □
Corollary 1 Let Y be a closed subspace of a Banach space X. Suppose \phi \in \mathrm{\Phi}(A,\mu ). An element g of {L}^{\phi}(A,Y) is a best approximation to an element f in {E}^{\phi}(A,X) if and only if g(s) is a best approximation in Y to f(s) for almost every s\in A. Furthermore, if φ satisfies the {\mathrm{\Delta}}_{2}condition, then an element g of {L}^{\phi}(A,Y) is a best approximation to an element f in {L}^{\phi}(A,X) if and only if g(s) is a best approximation in Y to f(s) for almost every s\in A.
Corollary 2 Let Y be a Chebyshev subspace of a Banach space X. Suppose \phi \in \mathrm{\Phi}(A,\mu ) satisfies the {\mathrm{\Delta}}_{2}condition. If {L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X), then it is a Chebyshev subspace of {L}^{\phi}(A,X).
Remark 1 Our results Theorem 1, Corollaries 1 and 2 cover the results for vector Orlicz spaces in [36]. Indeed, in [36] 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 {L}^{\phi}(A,Y) in {L}^{\phi}(A,X). To do so, we need some preliminaries.
Lemma 2 Let (A,\mathcal{A},\mu ) be a σfinite complete measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ). Let Y be a proximinal subspace of a Banach space X. Suppose f\in {L}^{\phi}(A,X) and g is a strongly μmeasurable function such that g(s) is a best approximation to f(s) from Y for almost everywhere s\in A. Then g is a best approximation to f from {L}^{\phi}(A,Y).
Proof Since 0\in Y, it follows that \parallel g(s)\parallel \le 2\parallel f(s)\parallel μalmost everywhere. Thus, g\in {L}^{\phi}(A,Y). For each h\in {L}^{\phi}(A,Y), by assumption we know that \parallel f(s)g(s)\parallel \le \parallel f(s)h(s)\parallel μalmost everywhere. So {\rho}_{\phi}(\lambda (fg))\le {\rho}_{\phi}(\lambda (fh)) for any \lambda >0. Thus, {\parallel fg\parallel}_{\phi}\le {\parallel fh\parallel}_{\phi}. This ends the proof. □
Definition 5 Let (T,\tau ) be a Polish space (i.e. a topological space which is separable and completely metrizable). A set Q\subset T is analytic if it is empty or if there exists a continuous mapping f:{\mathbb{N}}^{\mathbb{N}}\to T satisfying f({\mathbb{N}}^{\mathbb{N}})=Q, where {\mathbb{N}}^{\mathbb{N}} denotes the space of all infinite sequences of natural numbers endowed with the Tychonoff topology.
Definition 6 Let (T,\tau ) be a Polish space, H a topological space and denote by \sigma (\mathcal{A}) the smallest σalgebra containing all analytic subsets of T. Then a mapping f:T\to H is said to be analytic measurable if {f}^{1}(C)\in \sigma (\mathcal{A}) for every C\in \mathcal{B}(H), where \mathcal{B}(H) is for the Borel sets of H.
Definition 7 Let H, T be topological spaces. Then a multifunction F:H\to {2}^{T} is said to be upper semicontinuous if for every x\in H and for every open set U satisfying F(x)\subset U, there exists an open neighborhood V of x such that F(V)\subset U.
Definition 8 A subset C of a topological space T is analytic if it can be written as C={\bigcup}_{\sigma \in {\mathbb{N}}^{\mathbb{N}}}F(\sigma ) for some upper semicontinuous mapping F:{\mathbb{N}}^{\mathbb{N}}\to {2}^{T} with compact values. In the case when T is a Banach space endowed with its weak topology, C is said to be weakly analytic.
For the theory of analytic sets, we recommend [40]. Specially, all reflexive and all separable Banach spaces are weakly analytic. The following lemma is just Theorem 3.3 in [31].
Lemma 3 Let (X,\parallel \cdot \parallel ) be a real Banach space and let Y be a proximinal, weakly analytic convex subset of X. Then, for each closed and separable set M\subset X, there exists an analytic measurable mapping h:M\to Y such that h(M) is separable in Y and h(x) is a best approximation of x in Y for any x\in M.
Thus, following the argument of the proof of (i) → (ii) in [[39], p.185], we have the following conclusion, the details being omitted.
Theorem 2 Let (A,\mathcal{A},\mu ) be a σfinite complete measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ), and Y is a weakly analytic linear subspace of a real Banach space X. If Y is proximinal in X, then {L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X).
Theorem 3 Let (A,\mathcal{A},\mu ) be a σfinite measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ) such that {E}^{\phi}(A)={L}^{\phi}(A). Let Y be a linear subspace of a real Banach space X. If {L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X), then Y is proximinal in X.
Proof Since the measure μ is σfinite, let us choose positive measure set Q such that {\chi}_{Q}\in {L}^{\phi}(A). For any x\in X, let f(t):={\chi}_{Q}(t)\cdot x, t\in A. Then f\in {L}^{\phi}(A,X). By the assumption, we know that there is a g in {L}^{\phi}(A,Y) which is a best approximation element of f. Consequently, g(s) is a best approximation to f(s) in Y for almost every s\in A 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 (A,\mathcal{A},\mu ) be a σfinite complete measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ) such that {E}^{\phi}(A)={L}^{\phi}(A). Let Y be a weakly analytic linear subspace of a real Banach space X. Then the following conditions are equivalent:

(i)
Y is proximinal in X;

(ii)
{L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X).
Remark 2 An analog to Corollary 3 in terms of a modular was obtained in [39].
Finally, we give a characterization of proximinity of {L}^{\phi}(A,Y) in {L}^{\phi}(A,X) via the proximinity of {L}^{1}(A,Y) in {L}^{1}(A,X). When {L}^{\phi}(A,X) is a BochnerLebesgue space, which was obtained in [29] and [32, 33] on finite measure spaces and σ finite measure spaces, respectively. {L}^{\phi}(A,X) is a BochnerOrlicz space, which was discussed in [35].
Theorem 4 Let (A,\mathcal{A},\mu ) be a σfinite complete measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ) such that the set of simple functions, S(A,\mu ), satisfies S(A,\mu )\subset {L}^{{\phi}^{\ast}}(A,\mu ), where {\phi}^{\ast} is the conjugate function of φ (see [6]). Let Y be a closed subspace of a Banach space X. If {L}^{1}(A,Y) is proximinal in {L}^{1}(A,X), then {L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X).
Proof Since A is σfinite, we may write A={\bigcup}_{i=1}^{\mathrm{\infty}}{A}_{i}, where \{{A}_{i}\} is a sequence of disjoint measurable sets each of finite measure. Let f\in {L}^{\phi}(A,X). For any n\in \mathbb{N}, since \mu ({A}_{n})<\mathrm{\infty}, then {\chi}_{{A}_{n}}\in {L}^{{\phi}^{\ast}}(A). Thus, by the norm conjugate formula (see Corollary 2.7.5 in [6]), we find that f{\chi}_{{A}_{n}}\in {L}^{1}(A,X). By assumption, we know that there exists {g}_{n}\in {L}^{1}(A,Y) such that
By Corollary 1, we have, for all y\in Y,
μalmost everywhere. Therefore {g}_{n}(t)=0 μalmost every t\in {A}_{n}^{c}. Let g={\sum}_{n=1}^{\mathrm{\infty}}{g}_{n}. Since f={\sum}_{n=1}^{\mathrm{\infty}}f{\chi}_{{A}_{n}}, it follows that for all h\in {L}^{\phi}(A,Y),
μalmost everywhere. Because 0\in Y, it follows that \parallel g(t)\parallel \le 2\parallel f(t)\parallel. Thus, g\in {L}^{\phi}(A,Y) and
for all h\in {L}^{\phi}(A,Y). This finishes the proof. □
Theorem 5 Let (A,\mathcal{A},\mu ) be a σfinite complete measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ) satisfies {E}^{\phi}(A)={L}^{\phi}(A) and, for each t\in A, \phi (t,\cdot ) is strictly increasing. Let Y be a closed subspace of a Banach space X. If {L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X), then {L}^{1}(A,Y) is proximinal in {L}^{1}(A,X).
Proof We use the idea from [35]. Indeed in [35] the authors only considered Banach space valued Orlicz spaces on the unit interval. Since, for each t\in A, \phi (t,\cdot ) is strictly increasing, let {\phi}^{1}(t,\cdot ) be its inverse function, which means, for each s\in [0,\mathrm{\infty}), \phi (t,{\phi}^{1}(t,s))=s. Define the map J:{L}^{1}(A,X)\to {L}^{\phi}(A,X) by setting
Then \parallel J(f)(t)\parallel ={\phi}^{1}(t,\parallel f(t)\parallel ). Therefore {\rho}_{\phi}(J(f))={\parallel f\parallel}_{{L}^{1}}. So J is injective. Moreover, if g\in {L}^{\phi}(A,X), let
Then f(t)\in X and \parallel f(t)\parallel =\phi (t,\parallel g(t)\parallel ). Thus, f\in {L}^{1}(A,X). In addition, for g(t)\ne 0,
If g(t)=0, then f(t)=0 also, thus J(f)(t)=0=g(t). Hence J is surjective and J({L}^{1}(A,Y))={L}^{\phi}(A,Y) also.
Now, let f\in {L}^{1}(A,X). Without loss of generality we may suppose that f(t)\ne 0 μalmost everywhere, for otherwise we can restrict our measure to the support of f. Since J(f)\in {L}^{\phi}(A,X), by the assumption, we know that there exists some g\in {L}^{1}(A,Y) such that
for all v\in {L}^{1}(A,Y). By Corollary 1, we see that, for all y\in Y,
μalmost everywhere. Multiplying both sides of the last inequality by \frac{\parallel f(t)\parallel}{{\phi}^{1}(t,\parallel f(t)\parallel )}, we obtain, for all y\in Y,
Let h(t)=\frac{\parallel f(t)\parallel}{{\phi}^{1}(t,\parallel f(t)\parallel )}\frac{{\phi}^{1}(t,\parallel g(t)\parallel )}{\parallel g(t)\parallel}g(t). Since h(t) is a best approximation of f(t) in Y, and 0\in Y, it follows that \parallel h(t)\parallel \le 2\parallel f(t)\parallel. Therefore, h\in {L}^{1}(A,Y). Thus, for all w\in {L}^{1}(A,Y),
μalmost everywhere. Thus, by Corollary 1 h is a best approximation of f in {L}^{1}(A,Y). This finishes the proof. □
From Theorems 4 and 5, we deduce the following corollary.
Corollary 4 Let (A,\mathcal{A},\mu ) be a σfinite complete measure space. Suppose \phi \in \mathrm{\Phi}(A,\mu ) such that {E}^{\phi}(A)={L}^{\phi}(A), for each t\in A, \phi (t,\cdot ) is strictly increasing and the set of simple functions S(A,\mu ) satisfies S(A,\mu )\subset {L}^{{\phi}^{\ast}}(A,\mu ). Let Y be a closed subspace of a Banach space X. Then the following conditions are equivalent:

(i)
{L}^{1}(A,Y) is proximinal in {L}^{1}(A,X);

(ii)
{L}^{\phi}(A,Y) is proximinal in {L}^{\phi}(A,X).
Remark 3 When (A,\mu ) is a finite measure and φ is a Orlicz function that satisfies the {\mathrm{\Delta}}_{2}condition, the result of Corollary 4 was obtained in [36]. While (A,\mu ) is the unit interval and φ is a Young function that satisfies the {\mathrm{\Delta}}_{2}condition, the result of Corollary 4 was obtained in [35].
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Acknowledgements
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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Xu, J. Proximinality in Banach space valued MusielakOrlicz spaces. J Inequal Appl 2014, 146 (2014). https://doi.org/10.1186/1029242X2014146
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DOI: https://doi.org/10.1186/1029242X2014146
Keywords
 proximinality
 MusielakOrlicz space
 best approximation
 weakly analytic