Pointwise Best Coapproximation in the Space of Bochner Integrable Functions

Eyad Abu-Sirhan$^\ast $

November 27, 2019; accepted: September 17, 2020; published online: February 15, 2021.

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

MSC. 41A50, 41A52, 41A65.

Keywords. best coapproximation, coproximinal, Banach space.

$^\ast $Math. Department, Tafila Technical University, Jordan, e-mail: abu-sirhan2@ttu. edu.jo, eyadabusirhan@gmail.com.

1 Introduction

Let $G$ be a nonempty subset of a Banach space $X$ and let $x\in X.$ An element $g_{0}$ in $G$ satisfying

\[ \left\Vert x-g_{0}\right\Vert \leq \left\Vert x-g\right\Vert ,\ \text{for every }g\in G \]

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 $G.$

Another kind of approximation, called best coapproximation was introduced by Franchetti and Furi [ 1 ] , who considered those elements $g_{0}\in G,$ for which

\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}

An element $g_{0}$ in $G$ satisfying 1 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 $G.$

Several papers have been devoted for studying when the space $L^{p}(\mu ,G)$ is proximinal in $L^{p}(\mu ,X)$ see for example [ 5 ] , [ 9 ] – [ 12 ] . 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 [ 2 ] – [ 4 ] , [ 7 ] , and it will be the object of this paper.

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

\[ \left\Vert f\right\Vert _{\infty }=\operatorname {ess-sup} \left\Vert f\left( s\right) \right\Vert . \]

Finally, $\mathbb {N} $ stands for the set of natural numbers.

Finite measure spaces $\left( \Omega ,\Sigma ,\mu \right) $ in [ 3 ] , [ 4 ] , [ 7 ] 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 [ 3 ] , [ 4 ] , [ 7 ] and our methods are not only distinct but also seem to be simpler.

We start by recalling a few definitions.

Let $f:\Omega \rightarrow X$ be a functions. Then

$f$ 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 $E_{i}=f^{-1}(x_{i}).$
$f$ 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 $s\in \Omega .$

2 Best Coapproximation in $L^{\lowercase {p}}(\mu ,X)$, $1\leq \lowercase {p}\leq \infty .$

Definition 1

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

\[ \left\Vert h(s)-\varphi (s)\right\Vert \leq ||f(s)-\varphi (s)||,\text{ for almost all }s\in \Omega . \]

The notation of “pointwise coproximinal" is defined accordingly.

Proposition 2

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) $.

Proof â–¼

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 $\varphi \in L^{p}\left( \mu ,G\right) .$

Proof â–¼

Remark 3

Let $f$ be an element of $L^{\infty }\left( \mu ,G\right) $. We recall that

\[ \left\Vert f\right\Vert _{\infty }=\operatorname {ess-sup} \left\Vert f\left( s\right) \right\Vert \]

where

\[ \operatorname {ess-sup} \left\Vert f\left( s\right) \right\Vert =\inf _{E\in \aleph }\sup _{s\in E^{c}}\left\Vert f\left( s\right) \right\Vert . \]

Here we put

\[ \aleph =\left\{ E\in \Sigma :\mu (E)=0\right\} \]

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 $\left\Vert f\right\Vert _{\infty }=\sup _{s\in E^{c}}\left\Vert f(s)\right\Vert .$

In [ 4 , Th.2.2 ] , 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:

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.

Proposition 4

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 $X.$

Proof â–¼

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

\[ \left\Vert \varphi -h\right\Vert _{\infty }\leq \left\Vert f_{x}-\varphi \right\Vert _{\infty } \]

for all $\varphi \in L^{\infty }\left( \mu ,G\right) .$ Taking $\varphi =f_{g},$ it follows

\[ \left\Vert h-f_{g}\right\Vert _{\infty }\leq \left\Vert f_{x}-f_{g}\right\Vert _{\infty }=\left\Vert x-g\right\Vert \]

for all $g\in G.$ That is, for every $g\in G$

\[ \inf _{E\in \aleph }\sup _{s\in E^{c}}\left\Vert h\left( s\right) -g\right\Vert \leq \left\Vert x-g\right\Vert . \]

Taking $E_{g}\in \aleph $ such that

\[ \left\Vert h-f_{g}\right\Vert _{\infty }=\sup _{s\in E_{g}^{c}}\left\Vert h\left( s\right) -g\right\Vert , \]

it follows that

\[ \left\Vert h\left( s\right) -g\right\Vert \leq \left\Vert x-g\right\Vert \text{ for all }s\in E_{g}^{c}\ . \]

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 $G.$

Proof â–¼

Proposition 5

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 $X.$

Proof â–¼

Let $x\in X$ . Define $f_{x}\in L^{p}\left( \mu ,X\right) $ by

\[ \ f_{x}=\mu (A)^{\frac{1}{p}-1}x\ \chi _{A}. \]

By hypothesis, there exists $h\in L^{p}\left( \mu ,G\right) $ such that

\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}

For $g\in G$ let $\varphi _{g}\in L^{p}\left( \mu ,G\right) $ be given by

\[ \varphi _{g}=\mu (A)^{\frac{1}{p}-1}g\ \chi _{A}. \]

Then

\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}

Now, let

\[ g_{0}={\displaystyle \int \nolimits _{\Omega }} h(s)\chi _{A}(s)d\mu (s)\in G. \]

If $1{\lt}p{\lt}\infty ,$ then

\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*}

which show that $\mu (A)^{-1/p}g_{0}$ is a best coapproximation element to $x$ in $G.$

If $p=1,$ then, instead of Hőlder Inequality, the following inequality ca be used:

\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*}

Remark 6

If by a trivial space one understands that $\mu (\Sigma )\neq \left\{ 0,\infty \right\} ,$ i.e. $\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 $0{\lt}\mu (A){\lt}\infty .$

Corollary 7

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 $X.$

Proof â–¼

The proof follows from proposition 4 and proposition 5.

Proof â–¼

Bibliography

1: C. Franchetti, M. Furi, Some characteristic properties of real Hilbert spaces, Rev. Romaine Math. Pures Appl., 17 (1972), pp. 1045–1048.
2: E. Abu-Sirhan, A remark on best coapproximation in $L^{\infty }(\mu ,X),$ Intern. J. Math. Anal., 13 (2019) no. 9, pp. 449–458. $\includegraphics[scale=0.1]{ext-link.png}$
3: H. Mazaheri, S. Javad Jesmani, Some results on best coapproximation in $L^{1}(\mu ,X)$, Mediterr. J. Math., 4 (2007) no. 4, pp. 497–503. $\includegraphics[scale=0.1]{ext-link.png}$
4: J. Jawdat, Best coapproximation in $L^{\infty }(\mu ,X),$ TWMS J. App. Eng. Math., 8 (2018) no. 2, pp. 448–453.
5: J. Mendoza, Proximinality in $L^{p}(\mu ,X)$, J. Approx. Theory, 93 (1998), pp. 331–343. $\includegraphics[scale=0.1]{ext-link.png}$
6: K. Kuratowiski, C. Ryll-Nardzewski, A general theorem on selector, Bull. Acad. Polonaise Science, Series Math. Astr. Phys., 13 (1965), pp. 379–403.
7: M. R. Haddadi, N. Hejazjpoor, H. Mazaheri, Some result about best coapproximation in $L^{p}(S,X),$ Anal. Theory Appl., 26 (2010) no. 1, pp. 69–75. $\includegraphics[scale=0.1]{ext-link.png}$
8: P.L. Papini, I. Singer, Best coapproximation in normed linear spaces, Mh. Math., 88 (1979), pp. 27–44. $\includegraphics[scale=0.1]{ext-link.png}$
9: R. Khalil, Best approximation in $L^{p}(\mu ,X)$, Math. Proc. Cambridge Philos. Soc., 94 (1983), pp. 277–279. $\includegraphics[scale=0.1]{ext-link.png}$
10: R. Khalil, W. Deeb, Best approximation in $L^{p}(\mu ,X)$, II, J. Approx. Theory, 59 (1989), pp. 296–299. $\includegraphics[scale=0.1]{ext-link.png}$
11: W.A. Light, Proximinality in $L^{p}(\mu ,X)$, Rocky Mountain J. Math., 19 (1989), pp. 251–259. $\includegraphics[scale=0.1]{ext-link.png}$
12: Y. Zhao-Yong, G. Tie-Xin, Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in $L^{p}(\mu ,X)$, J. Approx. Theory, 78 (1994), pp. 314–320. $\includegraphics[scale=0.1]{ext-link.png}$

Pointwise Best Coapproximation in the Space of Bochner Integrable Functions

1 Introduction

2 Best Coapproximation in \(L^{\lowercase {p}}(\mu ,X)\), \(1\leq \lowercase {p}\leq \infty .\)

Bibliography