# Pointwise Coproximinality in $$L^{p}(\mu ,X)$$

May 2, 2023; accepted: June 18, 2023; published online: July 5, 2023.

Let $$X$$ be a Banach space, $$G$$ be a closed subspace of $$X$$, $$(\Omega ,\Sigma ,\mu )$$ be a $$\sigma$$-finite measure space, $$L(\mu ,X)$$ be the space of all strongly measurable functions from $$\Omega$$ to $$X$$, and $$L^{p}(\mu ,X)$$ be the space of all Bochner $$p$$-integrable functions from $$\Omega$$ to $$X$$. The purpose of this paper is to discuss the relationship between the pointwise coproximinality of $$L(\mu , G)$$ in $$L(\mu , X)$$ and the pointwise coproximinality of $$L^{p}(\mu , G)$$ in $$L^{p}(\mu , X)$$.

MSC. 41A50, 41A52, 41A655.

Keywords. Best coapproximation; coproximinal; Banach space.

$$^\ast$$Department of Mathematics, Tafila Technical University, Jordan, e-mail: abu-sirhan2@ttu.edu.jo, https://orcid.org/0000-0003-1360-5554

# 1 Introduction

In this article, $$(X,\| \cdot \| )$$ will be referred to as a Banach space over the field $$R$$ of real numbers, $$G$$ as a closed subspace of $$X$$, and $$(\Omega ,\Sigma ,\mu )$$ as a $$\sigma$$-finite nontrivial measure space, i.e. $$\Omega$$ is a countable union of measurable sets each with finite measure and there exists at least $$A\in \Sigma$$ with $$\infty {\gt}\mu (A){\gt}0$$. For $$p\geq 1$$, we suppose $$L^{p}(\mu ,X)$$ is the space of all Bochner $$p$$-integrable functions from $$\Omega$$ to $$X$$, $$L(\mu ,X)$$ is the linear space of all $$\mu$$-equivalence classes of strongly measurable functions from $$\Omega$$ to $$X$$, see [ 10 ] . If a subset $$M$$ of $$L(\mu ,X)$$ is closed with regard to the pointwise limits of sequences, we say that it is closed. For $$E \subseteq \Omega$$ and a function $$f:\Omega \rightarrow X$$, $$\chi _{E}$$ is the characteristic function of the set $$E$$ and $$\chi _{E} \otimes f$$ is the function $$\chi _{E} \otimes f(s)=\chi _{E}(s)f(s)$$. A function $$f:\Omega \rightarrow X$$ is said to be simple if

\begin{equation*} f=\sum ^{n}_{i=1}{x_{i}\chi _{E_{i}}}, \end{equation*}

where $$x_{i}\in X$$ and $$E_{i}=f^{-1}(x_{i})$$ is measurable for $$i=1,2,\ldots ,n$$. A function $$f:\Omega \rightarrow X$$ is said to be strongly measurable if there exists a sequence of simple functions $$\{ f_{n} \}$$ with

\begin{equation*} \lim _{n\to \infty }\| f_{n}(s)-f(s)\| =0, \hspace{1cm} a.e. \end{equation*}

Let $$M$$ be a closed subspace of $$L(\mu ,X)$$, we write $$L^{p}(M)$$ for the Banach space of all functions in $$M$$ such that $$\int _{\Omega } \| f(s)\| ^{p}$$ is finite. An element $$g_{0}\in G$$ is said to be a best coapproximation of $$x\in X$$ if

\begin{equation*} \| g-g_{0}\| \leq \| x-g\| \hspace{1cm}\forall g\in G. \end{equation*}

The set of all elements of best coapproximation of $$x$$ in $$G$$ will be denoted by $$R_{G} (x)$$. If $$R_{G} (x)$$ is nonempty for any $$x \in X$$, $$G$$ is said to be a coproximinal subspace of $$X$$. This new kind of approximation has been introduced by C. Franchetti and M. Furri (1972) [ 4 ] to characterize real Hilbert spaces among real reflexive Banach spaces.

P.L. Papini and I. Singer (1979) [ 8 ] then went into greater depth on the best coapproximation. It has lately been studied in $$L^{p}(\mu , X)$$; for example, see [ 1 , 2 , 6 , 7 , 5 ] . With finite measure spaces, though, [ 5 , 6 , 7 ] have dealt. This paper’s goal is to demonstrate the connection between the pointwise coproximinality of $$L^{p}(\mu , G)$$ in $$L^{p}(\mu , X)$$ and the pointwise coproximinality of $$L(\mu , G)$$ in $$L(\mu , X)$$. Pointwise coproximinality is a counterpart of pointwise proximinality, it should be noted, see [ 9 , 2 ] .

# 2 Pointwise Coproximinality

Definition 1

Let $$M$$ be a subset of $$L(\mu ,X)$$ and $$f \in L(\mu ,X)$$. An element $$\phi _{0}$$ in $$M$$ is said to be a best pointwise coapproximation of $$f$$ from $$M$$ if for any $$\phi \in M$$,

$\| \phi _{0}(s)-\phi (s) \| \leq \| f(s)-\phi (s) \| \hspace{1cm} a.e.$

$$M$$ is said to be pointwise coproximinal in $$L(\Omega ,X)$$ if each element of $$L(\Omega ,X)$$ has a best pointwise coapproximation from $$M$$.

Lemma 1

Let $$M$$ be a subset of $$L(\mu ,X)$$, $$f\in M$$, and $$A\in \Sigma$$. If $$M$$ is pointwise coproximinal in $$L(\mu ,X)$$, then $$\chi _{A} \otimes f \in M$$.

Proof â–¼
Assume that $$M$$ is pointwise coproximinal and there exist $$A\in \Sigma$$, $$f \in M$$, such that $$\chi _{A} \otimes f \notin M$$. By assumption there exists $$\phi _{A}$$ in $$M$$ such that
\begin{equation*} \| \phi _{A}(s) -\phi (s) \| \leq \| \chi _{A} \otimes f(s) - \phi (s) \| \hspace{1cm} a.e. \end{equation*}

and for all $$\phi \in M.$$ Taking $$\phi =0$$ one obtains

\begin{align} \| \phi _{A}(s)\| & \leq \| \chi _{A} \otimes f(s)\| \nonumber \\ & = \chi _{A}(s) \| f(s)\| \hspace{1cm} a.e.\hspace{1cm} \label{eq2.1} \end{align}

For $$\phi =f$$ one obtains

\begin{align} \| \phi _{A}(s)-f(s)\| & \leq \| \chi _{A} \otimes f(s) -f(s)\| \nonumber \\ & = \chi _{A^{c}}(s) \| f(s)\| \hspace{1cm} a.e.\hspace{1cm}\label{eq2.2} \end{align}

By 1, $$\phi _{A}(s)=0 \hspace{0.4cm} a.e. \hspace{0.2cm} s \in A^{c}$$ and by 2 $$\phi _{A}(s)=f(s) \hspace{0.4cm} a.e. \hspace{0.2cm} s \in A$$, that is, $$\chi _{A} \otimes f=\phi _{A} \in M$$. The proof is completed by the contradiction.

Proof â–¼

Corollary 1

Let $$M$$ be a closed subspace of $$L(\mu , X)$$. If $$M$$ is pointwise coproximinal in $$L(\mu ,X)$$, then $$M$$ is an $$L(\mu ,R)$$-submodule of $$L(\mu ,X)$$.

Proof â–¼
The proof follows from lemma 1 and the fact that the set of all simple functions in $$L(\mu , X)$$ is dense in $$L(\mu , X)$$.

Let $$\mathcal{A}=\{ A_{n}:n \in \mathbb {N}\}$$ be a family of countable subset of $$\Omega$$. if $$A_{n} \in \Sigma$$ for any $$n \in \mathbb {N}$$, $$A_{i}\cap A_{j} = \phi$$ for $$i \neq j$$, and $$\mu (\Omega -\cup _{n \in \mathbb {N}} A_{n} )=0$$, then the set $$\mathcal{A}$$ is said to be measurable partition of $$\Omega$$ [ 3 ] .

Proof â–¼

Lemma 2 [ 2 ]

Let $$G$$ be a closed subspace of $$X$$, $$1\leq p \leq \infty$$, $$f \in L^{p}(\mu ,X)$$, and $$h \in L(\mu ,X)$$. If $$h$$ is pointwise coapproximation to $$f$$, then $$h \in L^{p}(\mu ,G)$$ and it is a best coapproximation to $$f$$ from $$L^{p}(\mu ,G)$$.

Lemma 3

Let $$M$$ be a closed subspace of $$L(\mu , X)$$ and $$f \in M$$. Then there exists a measurable partition $$\{ D_{n} , n \in \mathbb {N} \}$$ such that

$f(s)=\sum _{n \in \mathbb {N}} \chi _{D_{n}} \otimes f(s)$

and $$\chi _{D_{n}} \otimes f \in L^{p}(M)$$, $$1\leq p \leq \infty$$, for all $$n$$ in $$\mathbb {N}$$ .

Proof â–¼
Since $$(\Omega ,\Sigma ,\mu )$$ is $$\sigma$$-finite, we may assume that $$\Omega =\bigcup _{n \in \mathbb {N} } A_{n}$$, $$A_{n}$$ is measurable, $$A_{n} \subseteq A_{n+1}$$, and $$\mu (A_{n}) {\lt} \infty$$ for any $$n \in \mathbb {N}$$. For each $$n \in \mathbb {N}$$, let
$B_{n}= \lbrace s \in \Omega : \| f(s)\| \leq n \rbrace ,$

$$C_{n}=A_{n} \cap B_{n}$$, and $$D_{n}=C_{n}-C_{n-1}.$$ Then $$\{ D_{n} , n \in \mathbb {N} \}$$ is a measurable partition, $$\chi _{D_{n}} \otimes f \in L^{p}(M)$$ by lemma 1, and $$f=\sum _{n=1}^{\infty } \chi _{D_{n}} \otimes f$$.

Proof â–¼

Theorem 1

Let $$G$$ be a closed subspace of $$X$$. Then the following are equivalent

$$(1)$$ $$L(\mu ,G)$$ is pointwise coproximinal in $$L(\mu ,X)$$.

$$(2)$$ $$L^{p}(\mu ,G)$$ is pointwise coproximinal in $$L^{p}(\mu ,X)$$.

Proof â–¼
$$(1)\Rightarrow (2)$$. Let$$f\in L^{p}(\mu ,X)$$. Then $$f \in L(\mu ,X)$$ and it has a best pointwise coapproximation $$f_{0}$$ from $$L(\mu ,G)$$. By lemma 2, $$f_{0}$$ is a best pointwise coapproximation to $$f$$ from $$L^{p}(\mu ,G)$$.

$$(2)\Rightarrow (1)$$. Let $$f\in L(\mu ,X)$$. By lemma 3, there exists $$\lbrace D_{n} : n \in \Bbb {N} \rbrace$$ measurable partition of $$\Omega$$ such that

\begin{equation*} f=\sum _{n \in \Bbb {N}} \chi _{D_{n}} \otimes f \end{equation*}

and $$\chi _{D_{n}} \otimes f \in L^{p} (\mu , X)$$ for any $$n$$ in $$\Bbb {N}$$ and $$1\leq p \leq \infty$$. By assumption, there exists $$\phi _{n} \in L^{p} (\mu , G)$$ such that

$$\| \phi _{n}(s)-\phi (s)\| \leq \| \chi _{D_{n}} \otimes f(s) -\phi (s)\| \hspace{1cm} a.e.$$
3

for all $$\phi \in L(\mu ,G)$$. Taking $$\phi =0$$ yields,

\begin{equation*} \| \phi _{n}(s)\| \leq \| \chi _{D_{n}} \otimes f(s) \| \hspace{1cm} a.e. \end{equation*}

and hence $$\phi _{n}=\chi _{D_{n}} \otimes \phi _{n}$$. If we let $$\Bar {\phi }=\sum _{ n\in \Bbb {N}} \phi _{n}$$, then $$\Bar {\phi } \in L(\mu ,G)$$. We claim that

\begin{equation*} \| \Bar {\phi }(s)-\phi (s)\| \leq \| f(s) -\phi (s)\| \hspace{1cm} a.e. \end{equation*}

for all $$\phi \in L(\mu ,G)$$. If the claim is incorrect, then $$\phi _{0} \in L(\mu ,G)$$ exists such that

\begin{equation*} \mu \lbrace s: \| \Bar {\phi }(s) - \phi _{0}(s) \| {\gt} \| f(s) - \phi _{0}(s) \| \rbrace {\gt}0 \end{equation*}

and let $$A_{0}=\lbrace s: \| \Bar {\phi }(s) - \phi _{0}(s) \| {\gt} \| f(s) - \phi _{0}(s) \| \rbrace$$. By lemma 3, there exists a measurable partition $$\lbrace C_{n} : n \in \Bbb {N} \rbrace$$ of $$\Omega$$ such that

\begin{equation*} \phi _{0}=\sum _{n \in \Bbb {N}} \chi _{C_{n}} \otimes \phi _{0} \end{equation*}

and $$\chi _{C_{n}} \otimes \phi _{0} \in L^{p} (\mu , G)$$ for any $$n$$ in $$\Bbb {N}$$ and $$1\leq p \leq \infty$$. Then $$\lbrace C_{n} \cap D_{m} : n,m \in \Bbb {N} \rbrace$$ is a measurable partition of $$\Omega$$. Hence there exist $$n_{0},m_{0} \in \Bbb {N}$$ such that $$\mu (A_{0} \cap C_{m_{0}} \cap D_{n_{0}}){\gt}0.$$ If $$s \in A_{0} \cap C_{m_{0}} \cap D_{n_{0}}$$, then

\begin{align*} \| \Bar {\phi }(s) - \phi _{0}(s) \| {\gt} & \| f(s) - \phi _{0}(s) \| , \\ \| \chi _{D_{n_{0}}} \otimes \phi _{n}(s) - \chi _{C_{m_{0}}} \otimes \phi _{0}(s) \| {\gt} & \| \chi _{D_{n_{0}}} \otimes f(s) - \chi _{C_{m_{0}}} \otimes \phi _{0}(s) \| . \end{align*}

Let

\begin{equation*} B_{0}=\big\{ s:\| \chi _{D_{n_{0}}} \otimes \phi _{n}(s) - \chi _{C_{m_{0}}} \otimes \phi _{0}(s) \| {\gt} \| \chi _{D_{n_{0}}} \otimes f(s) - \chi _{C_{m_{0}}} \otimes \phi _{0}(s) \| \big\} . \end{equation*}

Then $$A_{0} \cap C_{m_{0}} \cap D_{n_{0}} \subseteq B_{0}$$ and $$\mu (B_{0}){\gt}0$$, which contradicts 1.

Proof â–¼

Theorem 2 [ 2 ]

If $$L^{p}(\mu ,G)$$ is pointwise coproximinal in $$L^{p}(\mu ,X)$$, then $$G$$ is coproximinal in $$X$$.

Corollary 2

If $$L(\mu ,G)$$ is coproximinal in $$L(\mu ,X)$$, then $$G$$ is coproximinal in $$X$$.

Proof â–¼
It is obvious that pointwise coproximinality of $$L(\mu , G)$$ in $$L(\mu , x)$$ follows from coproximinality of $$L(\mu , G)$$ in $$L(\mu , x)$$. Then $$L^{p}(\mu ,G)$$ is pointwise coproximinal in $$L^{p}(\mu ,X)$$ by theorem 1. Therefore $$L^{p}(\mu ,G)$$ is coproximinal in $$L^{p}(\mu ,X)$$ and the result follows from theorem 2.
Proof â–¼

Acknowledgements

The author would like to thank the referee who provided useful and detailed comments on a earlier version of the manuscript.

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