# 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

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

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

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

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 \(\phi \in M \),

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

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 \(\chi _{A} \otimes f \in M\).

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

For \(\phi =f \) one obtains

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.

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 \(L(\mu ,X)\).

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 ] .

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 \(L^{p}(\mu ,G)\).

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

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

\(C_{n}=A_{n} \cap B_{n}\), 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 \(f=\sum _{n=1}^{\infty } \chi _{D_{n}} \otimes f\).

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 \(L^{p}(\mu ,X)\).

\((2)\Rightarrow (1)\). 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

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

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

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

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

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

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

Let

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

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

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

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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