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<h1>Pointwise Best Coapproximation in the Space of Bochner Integrable Functions</h1>
<p class="authors">
<span class="author">Eyad Abu-Sirhan\(^\ast \)</span>
</p>
<p class="date">November 27, 2019; accepted: September 17, 2020; published online: February 15, 2021.</p>
</div>
<div class="abstract"><p> Let \(X\) be a Banach space, \(G\)  be a closed subset of \(X,\) and \(\left( \Omega ,\Sigma ,\mu \right) \) be a \(\sigma \)-finite measure space\(.\) In this paper we present some results on coproximinality (pointwise coproximinality) of \(L^{p}(\mu ,G),\) \(1\leq p\leq \infty ,\) in \(L^{p}(\mu ,X).\) </p>
<p><b class="bf">MSC.</b> 41A50, 41A52, 41A65. </p>
<p><b class="bf">Keywords.</b> best coapproximation, coproximinal, Banach space. </p>
</div>
<p>\(^\ast \)Math. Department, Tafila Technical University, Jordan, e-mail: <span class="tt">abu-sirhan2@ttu.</span> <span class="tt">edu.jo, eyadabusirhan@gmail.com</span>. </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(G\) be a nonempty subset of a Banach space \(X\) and let \(x\in X.\) An element \(g_{0}\) in \(G\)  satisfying </p>
<div class="displaymath" id="a0000000003">
  \[  \left\Vert x-g_{0}\right\Vert \leq \left\Vert x-g\right\Vert ,\   \text{for every }g\in G  \]
</div>
<p> is called a best approximation to \(x\) from \(G\). The set \(G\) is called proximinal in \(X\) if every element \(x\) in \(X\) has a best approximation from \(G.\) </p>
<p>Another kind of approximation, called best coapproximation was introduced by Franchetti and Furi <span class="cite">
	[
	<a href="#1" >1</a>
	]
</span>, who considered those elements \(g_{0}\in G,\) for which </p>
<div class="equation" id="f.1">
<p>
  <div class="equation_content">
    \begin{equation} \label{f.1} \left\Vert g-g_{0}\right\Vert \leq \left\Vert x-g\right\Vert ,\   \   \text{for every }g\in G. \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>An element \(g_{0}\) in \(G\) satisfying <a href="#f.1" class="eqref">1</a> is called a best coapproximation to \(x\) from \(G.\) \(G\) is called coproximinal in \(X\) if every element \(x\) in \(X\) has a best coapproximation from \(G.\) </p>
<p>Several papers have been devoted for studying when the space \(L^{p}(\mu ,G)\) is proximinal in \(L^{p}(\mu ,X)\) see for example <span class="cite">
	[
	<a href="#5" >5</a>
	]
</span>, <span class="cite">
	[
	<a href="#9" >9</a>
	]
</span>–<span class="cite">
	[
	<a href="#12" >12</a>
	]
</span>. As a counter part of this problem is the problem of coproximinality of \(L^{p}(\mu ,G)\) in \(L^{p}(\mu ,X)\) ,which has been recently studied by some authors <span class="cite">
	[
	<a href="#2" >2</a>
	]
</span>–<span class="cite">
	[
	<a href="#4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span>, and it will be the object of this paper. </p>
<p>Throughout the whole paper, we always suppose that \((X,\left\Vert \cdot \right\Vert )\) is a Banach space and \(\left( \Omega ,\Sigma ,\mu \right) \) is a given non trivial(\(\mu (\Omega )\neq \left\{  0,\infty \right\}  \)) \(\sigma \)-finite measure space. We write \(L(\mu ,X)\) to denote the space of all \(X\)-valued strongly measurable functions, \(\  L^{p}(\mu ,X),\) \(1\leq p{\lt}\infty ,\) to denote the space of \(p\)-Bochner integrable functions defined on \(\Omega \) with values in \(X\)  and, for \(p=\infty ,\) \(L^{\infty }\left( \mu ,X\right) \) to denote the Banach space of all essentially bounded strongly measurable functions on \(\Omega \) with values in \(X,\) endowed with the usual norm</p>
<div class="displaymath" id="a0000000004">
  \[  \left\Vert f\right\Vert _{\infty }=\operatorname {ess-sup} \left\Vert f\left( s\right) \right\Vert .  \]
</div>
<p>Finally, \(\mathbb {N} \)  stands for the set of natural numbers. </p>
<p>Finite measure spaces \(\left( \Omega ,\Sigma ,\mu \right) \) in <span class="cite">
	[
	<a href="#3" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span> played an important role in obtaining results on the coproximinality of \(L^{p}(\mu ,G)\) in \(L^{p}(\mu ,X).\) The purpose of the present paper to further the topics using any \(\sigma \)-finite measure space \(\left( \Omega ,\Sigma ,\mu \right) .\) The obtained results improve those in <span class="cite">
	[
	<a href="#3" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#4" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span> and our methods are not only distinct but also seem to be simpler. </p>
<p>We start by recalling a few definitions. </p>
<p>Let \(f:\Omega \rightarrow X\) be a functions. Then </p>
<ol class="enumerate">
  <li><p>\(f\) is called simple if its range contains only finitely many points \(x_{1},x_{2},\ldots ,\) \(x_{n}\) in \(X\) and \(f^{-1}(x_{i})\) is measurable for \(i=1,2,\ldots ,n.\) In this case we write \(f=\Sigma _{i=1}^{n}x_{i}\chi _{E_{i}}\  , \)where \(\chi _{E_{i}}\) is the characteristic function of the set \(E_{i}=f^{-1}(x_{i}).\) </p>
</li>
  <li><p>\(f\) is called strongly measurable if there exists a sequence \(\left( f_{n}\right) \) of simple functions with \(\lim \left\Vert f_{n}\left( s\right) -f\left( s\right) \right\Vert =0\) for almost all \(s\in \Omega .\) </p>
</li>
</ol>
<h1 id="a0000000005">2 Best Coapproximation in \(L^{\lowercase {p}}(\mu ,X)\), \(1\leq \lowercase {p}\leq \infty .\)</h1>
<p><div class="definition_thmwrapper " id="def 2.1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1</span>
  </div>
  <div class="definition_thmcontent">
  <p> Let \(f\in L\left( \mu ,X\right) \) and \(D\subset L\left( \mu ,X\right) .\) An element \(h\) in \(D\) is called a pointwise best coapproximation to \(f\) from \(D\) if for all \(\varphi \in D,\) we have </p>
<div class="displaymath" id="a0000000006">
  \[  \left\Vert h(s)-\varphi (s)\right\Vert \leq ||f(s)-\varphi (s)||,\text{ for almost all }s\in \Omega .  \]
</div>

  </div>
</div> </p>
<p>The notation of “pointwise coproximinal" is defined accordingly. </p>
<p><div class="proposition_thmwrapper " id="Prop 2.2">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">2</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(\  G\) be a closed subspace of \(X\) , \(1\leq p\leq \infty ,\) \(f\in L^{p}\left( \mu ,X\right) ,\) and \(h\) \(\in L(\mu ,G).\) If \(h\) is a pointwise best coapproximation to \(f\), then \(h\) belongs to \(L^{p}\left( \mu ,G\right) \) and it is a best coapproximation to \(f\) from \(L^{p}\left( \mu ,G\right) \). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000007">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Since \(h(s)\) is a best coapproximation to \(f(s)\) for almost all \(s\in \Omega \), we have \(\left\Vert h(s)\right\Vert \leq ||f(s)||\) for almost all \(s\in \Omega .\) Then \(\left\Vert h\right\Vert _{p}\leq ||f||_{p}\) and so \(h\in L^{p}\left( \mu ,G\right) .\) It is clear that \(\left\Vert h-\varphi \right\Vert _{p}\leq ||f-\varphi ||_{p}\) for all \(\varphi \in L^{p}\left( \mu ,G\right) .\) <div class="proof_wrapper" id="a0000000008">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="Rem 2.3">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3</span>
  </div>
  <div class="remark_thmcontent">
  <p>Let \(f\)  be an element of \(L^{\infty }\left( \mu ,G\right) \). We recall that </p>
<div class="displaymath" id="a0000000009">
  \[  \left\Vert f\right\Vert _{\infty }=\operatorname {ess-sup} \left\Vert f\left( s\right) \right\Vert  \]
</div>
<p> where </p>
<div class="displaymath" id="a0000000010">
  \[  \operatorname {ess-sup} \left\Vert f\left( s\right) \right\Vert =\inf _{E\in \aleph }\sup _{s\in E^{c}}\left\Vert f\left( s\right) \right\Vert .  \]
</div>
<p> Here we put </p>
<div class="displaymath" id="a0000000011">
  \[  \aleph =\left\{  E\in \Sigma :\mu (E)=0\right\}   \]
</div>
<p> and for \(A\subset \Omega \)  we let \(A^{c}=\Omega \setminus A.\) If \(f\)  is essentially bounded, one can show that there exists \(E\in \aleph \) such that  \(\left\Vert f\right\Vert _{\infty }=\sup _{s\in E^{c}}\left\Vert f(s)\right\Vert .\) </p>

  </div>
</div> </p>
<p>In <span class="cite">
	[
	<a href="#4" >4</a>
	, 
	Th.2.2
	]
</span>, It was shown that for a closed subspace \(G\) of \(X,\) the coproximinality of \(L^{\infty }\left( \mu ,G\right) \) in \(L^{\infty }\left( \mu ,X\right) \) implies the coproximinality of \(G\) in \(X.\) In fact there was a flaw in the proof given as follows: </p>
<p>For \(x\in X\) put  \(f_{x}\left( s\right) =x,s\in \Omega .\) Assume that for some \(h\in L^{\infty }\left( \mu ,G\right) \) and \(x\in X\) we have \(\left\Vert h-f_{g}\right\Vert _{\infty }\leq \left\Vert f_{x}-f_{g}\right\Vert _{\infty }\) for all \(g\in G\). The author considered the existence of \(s_{0}\in \Omega \) such that \(\left\Vert h\left( s_{0}\right) -g\right\Vert \leq \left\Vert x-g\right\Vert \) for all \(g\in G.\) However, it does not follow automatically from the assumption. The author should have proved the existence of such an \(s_{0}.\) So the theorem may need more conditions to be correct as shown in the following Proposition. </p>
<p><div class="proposition_thmwrapper " id="Prop 2.4">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">4</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(G\) be a separable subspace of \(X\). if \(L^{\infty }\left( \mu ,G\right) \) is coproximinal in \(L^{\infty }\left( \mu ,X\right) ,\) then \(G\) is coproximinal in \(X.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000012">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> For \(x\in X\) put  \(f_{x}\left( s\right) =x,s\in \Omega .\) Then \(f_{x}\in L^{\infty }\left( \mu ,X\right) ,\) so there exists \(h\in L^{\infty }\left( \mu ,G\right) \) such that </p>
<div class="displaymath" id="a0000000013">
  \[  \left\Vert \varphi -h\right\Vert _{\infty }\leq \left\Vert f_{x}-\varphi \right\Vert _{\infty } \]
</div>
<p> for all \(\varphi \in L^{\infty }\left( \mu ,G\right) .\) Taking \(\varphi =f_{g},\) it follows</p>
<div class="displaymath" id="a0000000014">
  \[  \left\Vert h-f_{g}\right\Vert _{\infty }\leq \left\Vert f_{x}-f_{g}\right\Vert _{\infty }=\left\Vert x-g\right\Vert  \]
</div>
<p> for all \(g\in G.\) That is, for every \(g\in G\)</p>
<div class="displaymath" id="a0000000015">
  \[  \inf _{E\in \aleph }\sup _{s\in E^{c}}\left\Vert h\left( s\right) -g\right\Vert \leq \left\Vert x-g\right\Vert .  \]
</div>
<p>Taking \(E_{g}\in \aleph \) such that </p>
<div class="displaymath" id="a0000000016">
  \[  \left\Vert h-f_{g}\right\Vert _{\infty }=\sup _{s\in E_{g}^{c}}\left\Vert h\left( s\right) -g\right\Vert ,  \]
</div>
<p> it follows that </p>
<div class="displaymath" id="a0000000017">
  \[  \left\Vert h\left( s\right) -g\right\Vert \leq \left\Vert x-g\right\Vert \text{ for all }s\in E_{g}^{c}\  .  \]
</div>
<p>Thus for all \(g\in G,\) there exists \(E_{g}\in \aleph ,\) \(\left\Vert h\left( s\right) -g\right\Vert \leq \left\Vert x-g\right\Vert \) for all \(s\in E_{g}^{c}\  .\) Let \(G^{\prime }\) be a countable dense subset of \(G.\) Then for every \(g^{\prime }\in G^{\prime },\) there exists \(E_{g^{\prime }}\in \aleph \) such that \(\left\Vert h\left( s\right) -g^{\prime }\right\Vert \leq \left\Vert x-g^{\prime }\right\Vert \) for all \(s\in E_{g^{\prime }}^{c}\  .\) If \(\cap \big\{  E_{g^{\prime }}^{c}:g^{\prime }\in G^{\prime }\big\}  \) is empty, then \(\cup \left\{  E_{g^{\prime }}:g^{\prime }\in G^{\prime }\right\}  =\Omega \) and so \(\mu (\Omega )=0\) which is false, since we have supposed the measure space is non-trivial. Thus there exists \(s_{0}\in \cap \big\{  E_{g^{\prime }}^{c}:g^{\prime }\in G^{\prime }\big\}  \) such that \(\left\Vert h\left( s_{0}\right) -g^{\prime }\right\Vert \leq \left\Vert x-g^{\prime }\right\Vert \) for all \(g^{\prime }\in G^{\prime }.\) Therefore \(\left\Vert h\left( s_{0}\right) -g\right\Vert \leq \left\Vert x-g\right\Vert \) for all \(g\in G\) as \(G^{\prime }\) is dense in \(G.\) <div class="proof_wrapper" id="a0000000018">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="proposition_thmwrapper " id="Prop 2.5">
  <div class="proposition_thmheading">
    <span class="proposition_thmcaption">
    Proposition
    </span>
    <span class="proposition_thmlabel">5</span>
  </div>
  <div class="proposition_thmcontent">
  <p>Let \(\left( \Omega ,\Sigma ,\mu \right) \) be a measure space such that there exists \(A\in \Sigma \) with \(0{\lt}\mu (A){\lt}\infty ,\) \(X\) be a Banach space, \(G\) be a closed subspace of \(X\) and \(1\leq p{\lt}\infty .\) If \(L^{p}\left( \mu ,G\right) \) is coproximinal in \(L^{p}\left( \mu ,X\right) ,\) then \(G\) is coproximinal in \(X.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000019">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(x\in X\) .  Define  \(f_{x}\in L^{p}\left( \mu ,X\right) \) by</p>
<div class="displaymath" id="a0000000020">
  \[  \  f_{x}=\mu (A)^{\frac{1}{p}-1}x\   \chi _{A}.  \]
</div>
<p>By hypothesis, there exists \(h\in L^{p}\left( \mu ,G\right) \) such that </p>
<div class="equation" id="f.2">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert h-\varphi \right\Vert _{p}\leq \left\Vert f_{x}-\varphi \right\Vert _{p}\text{ for all }\varphi \in L^{p}\left( \mu ,G\right) . \label{f.2}\end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>  For \(g\in G\) let \(\varphi _{g}\in L^{p}\left( \mu ,G\right) \) be given by </p>
<div class="displaymath" id="a0000000021">
  \[  \varphi _{g}=\mu (A)^{\frac{1}{p}-1}g\   \chi _{A}.  \]
</div>
<p>Then</p>
<div class="displaymath" id="a0000000022">
  \begin{align}  \left\Vert f_{x}-\varphi _{g}\right\Vert _{p}\   &  =\left( {\displaystyle \int \nolimits _{\Omega }} \left\Vert \  f_{x}(s)-\varphi _{g}(s)\right\Vert ^{p}d\mu (s)\right) ^{1/p}\nonumber \\ &  =\mu (A)^{\frac{1}{p}-1}\left( {\displaystyle \int \nolimits _{\Omega }} \left\Vert \  (x-g)\right\Vert ^{p}\chi _{A}(s)d\mu (s)\right) ^{1/p}\label{f.3}\\ &  =\mu (A)^{\frac{2}{p}-1}\left\Vert \  (x-g)\right\Vert .\nonumber \end{align}
</div>
<p>Now, let</p>
<div class="displaymath" id="a0000000023">
  \[  g_{0}={\displaystyle \int \nolimits _{\Omega }} h(s)\chi _{A}(s)d\mu (s)\in G.  \]
</div>
<p>If \(1{\lt}p{\lt}\infty ,\) then </p>
<div class="displaymath" id="a0000000024">
  \begin{align*} & \left\Vert g-\mu (A)^{-1/p}g_{0}\right\Vert = \\ &  =\left\Vert \mu (A_{k_{0}})^{-1}{\displaystyle \int \nolimits _{\Omega }} g\   \chi _{A}(s)d\mu (s)-\mu (A)^{-1/p}{\displaystyle \int \nolimits _{\Omega }} h(s)\  d\mu (s)\right\Vert \\ &  =\left\Vert \mu (A)^{-1/p}{\displaystyle \int \nolimits _{\Omega }} \left( \varphi _{g}(s)-h(s)\right) \chi _{A}(s)d\mu (s)\right\Vert \\ &  \leq \mu (A)^{-1/p}{\displaystyle \int \nolimits _{\Omega }} \left\Vert \varphi _{g}(s)-h(s)\   \right\Vert \chi _{A}(s)d\mu (s)\\ &  \leq \mu (A)^{-1/p}\left( {\displaystyle \int \nolimits _{\Omega }} \left\Vert \varphi _{g}(s)-h(s) \right\Vert ^{p}d\mu (s)\right) ^{1/p}\mu (A)^{1-1/p}\   \   \   \   \   \  (\text{H\"{o}lder Inequality})\\ &  =\mu (A)^{1-2/p}\left\Vert \varphi _{g}-h\right\Vert \\ &  \leq \mu (A)^{1-2/p}\left\Vert \varphi _{g}-f_{x}\right\Vert \   \   \   \   \   \   \   \   (\text{by (2)})\   \   \   \   \   \   \   \   \\ &  =\left\Vert x-g\right\Vert , \  \   \   \   \   (\text{by (3)})\end{align*}
</div>
<p> which show that \(\mu (A)^{-1/p}g_{0}\) is a best coapproximation element to \(x\) in \(G.\) </p>
<p>If \(p=1,\) then, instead of Hőlder Inequality, the following inequality ca be used: </p>
<div class="displaymath" id="a0000000025">
  \begin{align*}  \left\Vert g-\mu (A)^{-1}g_{0}\right\Vert &  \leq \mu (A)^{-1}{\displaystyle \int \limits _{\Omega }} \left\Vert \varphi _{g}(s)-h(s)\   \right\Vert \chi _{A}(s)d\mu (s)\\ &  \leq \mu (A)^{-1}{\displaystyle \int \limits _{\Omega }} \left\Vert \varphi _{g}(s)-h(s)\   \right\Vert d\mu (s)\\ &  =\mu (A)^{-1}\left\Vert \varphi _{g}-h\right\Vert \leq \mu (A)^{-1}\left\Vert \varphi _{g}-f_{x}\right\Vert \\ &  =\left\Vert x-g\right\Vert . \end{align*}
</div>
<p><div class="remark_thmwrapper " id="Rem 2.6">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">6</span>
  </div>
  <div class="remark_thmcontent">
  <p>If by a trivial space one understands that \(\mu (\Sigma )\neq \left\{  0,\infty \right\}  ,\) <i class="it">i.e.</i> \(\mu \) takes only the values \(0\) and \(\infty ,\) then in a non-trivial measure space \(\left( \Omega ,\Sigma ,\mu \right) \) there exists always \(A\in \Sigma \) with \(0{\lt}\mu (A){\lt}\infty .\) </p>

  </div>
</div> </p>
<p><div class="corollary_thmwrapper " id="Cor 2.7">
  <div class="corollary_thmheading">
    <span class="corollary_thmcaption">
    Corollary
    </span>
    <span class="corollary_thmlabel">7</span>
  </div>
  <div class="corollary_thmcontent">
  <p>Let \(G\) be a separable subspace of \(X\) and \(1\leq p\leq \infty .\) If \(L^{p}\left( \mu ,G\right) \) is coproximinal in \(L^{p}\left( \mu ,X\right) ,\) then \(G\) is coproximinal in \(X.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof follows from <a href="#Prop 2.4">proposition 4</a> and <a href="#Prop 2.5">proposition 5</a>. <div class="proof_wrapper" id="a0000000027">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><span class="scshape">C. Franchetti, M. Furi</span>, <i class="itshape">Some characteristic properties of real Hilbert spaces</i>, Rev. Romaine Math. Pures Appl., <b class="bfseries">17</b> (1972), pp. 1045–1048. </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><a href ="https://doi.org/10.12988/ijma.2019.9847"> <span class="scshape">E. Abu-Sirhan</span>, <i class="itshape">A remark on best coapproximation in</i> \(L^{\infty }(\mu ,X),\) Intern. J. Math. Anal., <b class="bfseries">13</b> (2019) no. 9, pp. 449–458. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s00009-007-0131-0"> <span class="scshape">H. Mazaheri, S. Javad Jesmani</span>, <i class="itshape">Some results on best coapproximation in</i> \(L^{1}(\mu ,X)\), Mediterr. J. Math., <b class="bfseries">4</b> (2007) no. 4, pp. 497–503. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><span class="scshape">J. Jawdat</span>, <i class="itshape">Best coapproximation in</i> \(L^{\infty }(\mu ,X),\) TWMS J. App. Eng. Math., <b class="bfseries">8</b> (2018) no. 2, pp. 448–453. </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><a href ="https://doi.org/10.1006/jath.1997.3163"> <span class="scshape">J. Mendoza</span>, <i class="itshape">Proximinality in</i> \(L^{p}(\mu ,X)\), J. Approx. Theory, <b class="bfseries">93</b> (1998), pp. 331–343. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><span class="scshape">K. Kuratowiski, C. Ryll-Nardzewski</span>, <i class="itshape">A general theorem on selector</i>, Bull. Acad. Polonaise Science, Series Math. Astr. Phys., <b class="bfseries">13</b> (1965), pp. 379–403. </p>
</dd>
  <dt><a name="7">7</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s10496-010-0069-0"> <span class="scshape">M. R. Haddadi, N. Hejazjpoor, H. Mazaheri</span>, <i class="itshape">Some result about best coapproximation in</i> \(L^{p}(S,X),\) Anal. Theory Appl., <b class="bfseries">26</b> (2010) no. 1, pp. 69–75. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="8">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/bf01305855"> <span class="scshape">P.L. Papini, I. Singer</span>, <i class="itshape">Best coapproximation in normed linear spaces</i>, Mh. Math., <b class="bfseries">88</b> (1979), pp. 27–44. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="9">9</a></dt>
  <dd><p><a href ="https://doi.org/10.1017/s0305004100061120"> <span class="scshape">R. Khalil</span>, <i class="itshape">Best approximation in</i> \(L^{p}(\mu ,X)\), Math. Proc. Cambridge Philos. Soc., <b class="bfseries">94</b> (1983), pp. 277–279. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="10">10</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0021-9045(89)90094-4"> <span class="scshape">R. Khalil, W. Deeb</span>, <i class="itshape">Best approximation in</i> \(L^{p}(\mu ,X)\), II, J. Approx. Theory, <b class="bfseries">59</b> (1989), pp. 296–299. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="11">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1216/rmj-1989-19-1-251"> <span class="scshape">W.A. Light</span>, <i class="itshape">Proximinality in</i> \(L^{p}(\mu ,X)\), Rocky Mountain J. Math., <b class="bfseries">19</b> (1989), pp. 251–259. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="12">12</a></dt>
  <dd><p><a href ="https://doi.org/10.1006/jath.1994.1081"> <span class="scshape">Y. Zhao-Yong, G. Tie-Xin</span>, <i class="itshape">Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in</i> \(L^{p}(\mu ,X)\), J. Approx. Theory, <b class="bfseries">78</b> (1994), pp. 314–320. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
</dl>


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