Proximinality in Banach space valued Musielak-Orlicz spaces

Abstract

Let $(A,\mathcal{A},\mu )$ 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 $(A,\mathcal{A},\mu )$. Denote by $L^{\phi }(A,Y)$ and $L^{\phi }(A,X)$ the Musielak-Orlicz 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 Musielak-Orlicz 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, 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 [17]. 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 [34]. 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 [35], and Al-Minawi 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 vector-valued Musielak-Orlicz 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 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, $(A,\mathcal{A},\mu )$ will be a σ-finite 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, left-continuous 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 Φ-function. It is called positive if $\phi (t)>0$ for all $t>0$.

It is easy to see that if φ is a Φ-function, then it is nondecreasing on $[0,\mathrm{\infty })$.

Definition 2 Let $(A,\mathcal{A},\mu )$ be a σ-finite complete measure space. A real function $\phi :A\times [0,\mathrm{\infty })\to [0,\mathrm{\infty }]$ is called a generalized Φ-function on $(A,\mathcal{A},\mu )$ if

• (a) $\phi (y,\cdot )$ is a Φ-function for all $y\in A$,

• (b) $y\mapsto \phi (y,t)$ is measurable for all $t⩾0$.

If φ is a generalized Φ-function on $(A,\mathcal{A},\mu )$, we write $\phi \in \mathrm{\Phi }(A,\mu )$.

Definition 3 Let $\phi \in \mathrm{\Phi }(A,\mu )$. Define

$${\varrho }_{\phi }(f):={\int }_A\phi (y,\parallel f(y)\parallel )\mathrm{d}\mu (y)$$

for strongly μ-measurable functions $f:A\to X$. Then the Bochner-Musielak-Orlicz space $L^{\phi }(A,X)$ is the collection of all strongly μ-measurable functions $f:A\to X$ endowed with the norm:

$${\parallel f\parallel }_{L^{\phi }(A,X)}:=inf\{\lambda >0,{\varrho }_{\phi }(f/\lambda )⩽1\}.$$

Let

$$E^{\phi }(A,X):=\{f\in L^{\phi }(A,X):{\rho }_{\phi }(\lambda f)<\mathrm{\infty }\text{ for all }\lambda >0\}.$$

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

$$\phi (s,2t)\le K\phi (s,t)$$

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 X-valued strongly μ-measurable 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

$$\parallel x-y\parallel =dist(x,Y)=inf\{\parallel x-u\parallel :u\in Y\}.$$

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

$${dist}_{\phi }(f,L^{\phi }(A,Y)):=inf\{{\parallel f-g\parallel }_{\phi }:g\in L^{\phi }(A,Y)\}.$$

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 $\lambda >0$,

$$\begin{array}{rl}{\rho }_{\phi }(\lambda (f-g))& ={\int }_A\phi (s,\lambda \parallel f(s)-g(s)\parallel )\mathrm{d}\mu (s)\\ \ge {\int }_A\phi (s,\lambda dist(f(s),Y))\mathrm{d}\mu (s)\\ ={\rho }_{\phi }(\lambda \varphi ).\end{array}$$

Thus, we have

$${\parallel f-g\parallel }_{\phi }\ge {\parallel \varphi \parallel }_{\phi }.$$

This implies $\varphi \in L^{\phi }(A)$ and, by taking an infimum on $g\in L^{\phi }(A,Y)$, we have

$${dist}_{\phi }(f,L^{\phi }(A,Y))\ge {\parallel \varphi \parallel }_{\phi }.$$

(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 $0<ϵ<1$. 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

$${\int }_{A_i}\phi (s,dist(x_i,Y)+\delta )\mathrm{d}\mu (s)\le {\int }_{A_i}\phi (s,dist(x_i,Y))\mathrm{d}\mu (s)+\frac{ϵ}{m},\mathrm{\forall }i\in \{1,\dots ,m\}.$$

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 ${\parallel f-g\parallel }_{\phi }\ge {dist}_{\phi }(f,L^{\phi }(A,Y))=1$. By Lemma 1, we see that

$$\begin{array}{rl}1& \le {\parallel f-g\parallel }_{\phi }\le {\rho }_{\phi }(f-g)={\sum }_{i=1}^m{\int }_{A_i}\phi (s,\parallel x_i-y_i\parallel )\mathrm{d}\mu (s)\\ \le {\sum }_{i=1}^m{\int }_{A_i}\phi (s,dist(x_i,Y)+\delta )\mathrm{d}\mu (s)\\ \le {\sum }_{i=1}^m({\int }_{A_i}\phi (s,dist(x_i,Y))\mathrm{d}\mu (s)+\frac{ϵ}{m})\\ ={\int }_A\phi (s,dist(f(s),Y))\mathrm{d}\mu (s)+ϵ.\end{array}$$

Thus, ${\rho }_{\phi }(\varphi )\ge 1-ϵ$. 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 μ-almost everywhere, $\parallel g_n(s)\parallel \le \parallel f(s)\parallel$ μ-almost 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

$${\parallel {\varphi }_n\parallel }_{\phi }={dist}_{\phi }(g_n,L^{\phi }(A,Y)).$$

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 ${\varphi }_n(s)\le \parallel g_n(s)\parallel \le \parallel f(s)\parallel$ μ-almost everywhere, and ${\varphi }_n(s)\to \varphi (s)$ μ-almost 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

$${\parallel \varphi \parallel }_{\phi }={dist}_{\phi }(f,L^{\phi }(A,Y)),$$

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 σ-finite 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 μ-measurable 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 $\parallel g(s)\parallel \le 2\parallel f(s)\parallel$ μ-almost everywhere. Thus, $g\in L^{\phi }(A,Y)$. For each $h\in L^{\phi }(A,Y)$, by assumption we know that $\parallel f(s)-g(s)\parallel \le \parallel f(s)-h(s)\parallel$ μ-almost everywhere. So ${\rho }_{\phi }(\lambda (f-g))\le {\rho }_{\phi }(\lambda (f-h))$ for any $\lambda >0$. Thus, ${\parallel f-g\parallel }_{\phi }\le {\parallel f-h\parallel }_{\phi }$. 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 σ-algebra 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 semi-continuous 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 semi-continuous 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 σ-finite 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 σ-finite 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 σ-finite, 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 σ-finite 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 Bochner-Lebesgue 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 Bochner-Orlicz space, which was discussed in [35].

Theorem 4 Let $(A,\mathcal{A},\mu )$ be a σ-finite 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 σ-finite, 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

$${\parallel f{\chi }_{A_n}-g_n\parallel }_{L^1}\le {\parallel f{\chi }_{A_n}-h\parallel }_{L^1},\mathrm{\forall }h\in L^1(A,Y).$$

By Corollary 1, we have, for all $y\in Y$,

$$\parallel f(t){\chi }_{A_n}-g_n(t)\parallel \le \parallel f(t){\chi }_{A_n}-y\parallel$$

μ-almost everywhere. Therefore $g_n(t)=0$ μ-almost 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 $h\in L^{\phi }(A,Y)$,

$$\parallel f(t)-g(t)\parallel \le \parallel f(t)-h(t)\parallel$$

μ-almost 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

$${\parallel f-g\parallel }_{\phi }\le {\parallel f-h\parallel }_{\phi }$$

for all $h\in L^{\phi }(A,Y)$. This finishes the proof. □

Theorem 5 Let $(A,\mathcal{A},\mu )$ be a σ-finite 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

$$J(f)(t):=\{\begin{array}{cc}\frac{{\phi }^{-1}(t,\parallel f(t)\parallel )}{\parallel f(t)\parallel }f(t),\hfill & f(t)\ne 0;\hfill \\ 0,\hfill & f(t)=0.\hfill \end{array}$$

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

$$f(t):=\{\begin{array}{cc}\frac{\phi (t,\parallel g(t)\parallel )}{\parallel g(t)\parallel }g(t),\hfill & g(t)\ne 0;\hfill \\ 0,\hfill & g(t)=0.\hfill \end{array}$$

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 $g(t)\ne 0$,

$$J(f)(t)=\frac{{\phi }^{-1}(t,\phi (t,\parallel g(t)\parallel ))}{\phi (t,\parallel g(t)\parallel )}f(t)=\frac{\parallel g(t)\parallel }{\phi (t,\parallel g(t)\parallel )}f(t)=g(t).$$

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 $f(t)\ne 0$ μ-almost 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

$${\parallel J(f)-J(g)\parallel }_{\phi }\le {\parallel J(f)-J(v)\parallel }_{\phi }$$

for all $v\in L^1(A,Y)$. By Corollary 1, we see that, for all $y\in Y$,

$$\parallel J(f)(t)-J(g)(t)\parallel \le \parallel J(f)(t)-y\parallel$$

μ-almost 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 $y\in Y$,

$$\parallel f(t)-\frac{\parallel f(t)\parallel }{{\phi }^{-1}(t,\parallel f(t)\parallel )}\frac{{\phi }^{-1}(t,\parallel g(t)\parallel )}{\parallel g(t)\parallel }g(t)\parallel \le \parallel f(t)-y\parallel .$$

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 $w\in L^1(A,Y)$,

$$\parallel f(t)-h(t)\parallel \le \parallel f(t)-w(t)\parallel$$

μ-almost 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 σ-finite 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].