Abstract
In this note we are concerned with the characterization of the elements of \(\varepsilon\)-best approximation (\varepsilon\)-nearest points) in a subspace \(Y\) of space \(X\) with asymmetric seminorm. For this we use functionals in the asymmetric dual \(X^{b}\) defined and studied in some recent papers [1], [3], [5].
Authors
“Tiberiu Popovicu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Asymmetric seminormed spaces; ε-nearest points; characterization.
Paper coordinates
C. Mustăţa, Characterization of ε-nearest points in space with asymmetric seminorm, Rev. Anal. Numer. Theor. Approx. 33 (2004) no. 2, 203-208, https://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no2-art11
About this paper
Journal
Revue d’Analyse Numer.Theor. Approx.
Publisher Name
Publishing House of the Romanian Academy
Print ISSN
2501-059X
Online ISSN
2457-6794
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Paper (preprint) in HTML form
CHARACTERIZATION OF epsi\varepsilon-NEAREST POINTS IN SPACES WITH ASYMMETRIC SEMINORM*
Abstract
In this note we are concerned with the characterization of the elements of epsi\varepsilon-best approximation ( epsi\varepsilon-nearest points) in a subspace YY of space XX with asymmetric seminorm. For this we use functionals in the asymmetric dual X^(b)X^{b} defined and studied in some recent papers 1, 3, 5.
MSC 2000. 41A65.
Keywords. Asymmetric seminormed spaces, epsi\varepsilon-nearest points, characterization.
Keywords. Asymmetric seminormed spaces, epsi\varepsilon-nearest points, characterization.
1. INTRODUCTION
Let XX be a real linear space. A functional p:X rarr[0,oo)p: X \rightarrow[0, \infty) with the properties:
(1) p(x) >= 0p(x) \geq 0, for all x in Xx \in X,
(2) p(tx)=tp(x)p(t x)=t p(x), for all x in Xx \in X and t >= 0t \geq 0,
(3) p(x+y) <= p(x)+p(y)p(x+y) \leq p(x)+p(y), for all x,y in Xx, y \in X,
is called asymmetric seminorm on XX, and the pair ( X,pX, p ) is called a space with asymmetric seminorm.
(1) p(x) >= 0p(x) \geq 0, for all x in Xx \in X,
(2) p(tx)=tp(x)p(t x)=t p(x), for all x in Xx \in X and t >= 0t \geq 0,
(3) p(x+y) <= p(x)+p(y)p(x+y) \leq p(x)+p(y), for all x,y in Xx, y \in X,
is called asymmetric seminorm on XX, and the pair ( X,pX, p ) is called a space with asymmetric seminorm.
The functional bar(p):X rarr[0,oo)\bar{p}: X \rightarrow[0, \infty), defined by bar(p)(x)=p(-x),x in X\bar{p}(x)=p(-x), x \in X is another asymmetric seminorm on XX, called the conjugate of pp.
The functional p^(s):X rarr[0,oo)p^{s}: X \rightarrow[0, \infty), defined by
p^(s)(x)=max{p(x),p(-x)},x in Xp^{s}(x)=\max \{p(x), p(-x)\}, x \in X
is a seminorm on XX. If p^(s)p^{s} satisfies the axioms of a norm, then pp is called an asymmetric norm on XX. It follows that pp satisfies the properties (1), (2), (3), and
(4) p(x)=0p(x)=0 and p(-x)=0p(-x)=0 imply x=0x=0.
(4) p(x)=0p(x)=0 and p(-x)=0p(-x)=0 imply x=0x=0.
The asymmetric seminorm pp on XX generates a topology tau_(p)\tau_{p} on XX, having as a basis of neighborhoods of a point x in Xx \in X the open pp-balls
B_(p)^(')(x,r)={x^(')in X:p(x^(')-x) < r},r > 0.B_{p}^{\prime}(x, r)=\left\{x^{\prime} \in X: p\left(x^{\prime}-x\right)<r\right\}, r>0 .
The family of closed pp-balls
B_(p)(x,r)={x^(')in X:p(x^(')-x) <= r},r > 0B_{p}(x, r)=\left\{x^{\prime} \in X: p\left(x^{\prime}-x\right) \leq r\right\}, r>0
generates the same topology. This topology tau_(p)\tau_{p} could not be Hausdorff (see [5]), and could not be linear (the multiplication by scalars is not continuous in general, see [1].
Let R\mathbb{R} be the set of real numbers and u:Rrarr[0,oo),u(a)=max{a,0}u: \mathbb{R} \rightarrow[0, \infty), u(a)=\max \{a, 0\}, a inRa \in \mathbb{R}. Then the function uu is an asymmetric seminorm on R\mathbb{R} and, for a inRa \in \mathbb{R}, the intervals (-oo,a+epsi),epsi > 0(-\infty, a+\varepsilon), \varepsilon>0, form a basis of neighborhoods of a inRa \in \mathbb{R} in the topology tau_(u)\tau_{u}. The conjugate asymmetric seminorm of uu is bar(u):Rrarr[0,oo)\bar{u}: \mathbb{R} \rightarrow[0, \infty), bar(u)(a)=u(-a),a inR\bar{u}(a)=u(-a), a \in \mathbb{R}, and u^(s)(a)=max{u(a),u(-a)}=|a|u^{s}(a)=\max \{u(a), u(-a)\}=|a| is a norm on R\mathbb{R}. Consequently, uu is an asymmetric norm on R\mathbb{R}.
Let varphi:X rarrR\varphi: X \rightarrow \mathbb{R} be a linear functional. The continuity of varphi\varphi with respect to the topologies tau_(p)\tau_{p} and tau_(u)\tau_{u} is called ( p,up, u )-continuity, and it is equivalent to the upper semicontinuity of varphi\varphi as a functional from (X,tau_(p))\left(X, \tau_{p}\right) to (R,|*|)(\mathbb{R},|\cdot|).
The linear functional varphi:(X,tau_(p))rarr(R,u)\varphi:\left(X, \tau_{p}\right) \rightarrow(\mathbb{R}, u) is (p,u)(p, u)-continuous if and only if it is pp-bounded, i.e. there exists L >= 0L \geq 0 such that
varphi(x) <= Lp(x),quad" for all "x in X.\varphi(x) \leq L p(x), \quad \text { for all } x \in X .
The set of all ( p,up, u )-continuous functionals is denoted by X_(p)^(b)X_{p}^{b}. With respect to pointwise addition and multiplication by real scalars, the set X_(p)^(b)X_{p}^{b} is a cone, i.e. lambda >= 0\lambda \geq 0 and varphi,psi inX_(p)^(b)\varphi, \psi \in X_{p}^{b} imply varphi+psi inX^(b)\varphi+\psi \in X^{b} and lambda varphi inX_(p)^(b)\lambda \varphi \in X_{p}^{b}.
The functional ||.∣:X_(p)^(b)rarr[0,oo)\| . \mid: X_{p}^{b} \rightarrow[0, \infty) defined by
|| varphi|_(p)=s u p{varphi(x):x in X,p(x) <= 1},quad varphi inX_(p)^(b)\|\left.\varphi\right|_{p}=\sup \{\varphi(x): x \in X, p(x) \leq 1\}, \quad \varphi \in X_{p}^{b}
satisfies the properties of an asymmetric seminorm, and the pair ( X_(p)^(b),||*|_(p)X_{p}^{b}, \|\left.\cdot\right|_{p} ) is called the asymmetric dual of the asymmetric seminormed space ( X,pX, p ) (see [5]). Some properties of this dual are presented in [1], [3, [5]. If there is no danger of confusion we shall use the notation X^(b)X^{b} and ||varphi∣\| \varphi \mid instead of X_(p)^(b)X_{p}^{b} and || varphi|_(p)\|\left.\varphi\right|_{p}, respectively.
Let ( X,pX, p ) be an asymmetric seminormed space and YY a subspace of XX. Let Y^(b)Y^{b} be the asymmetric dual of (Y,p)(Y, p).
The following result is the analog of a well known extension result for linear functionals in normed spaces.
Theorem 1 (Hahn-Banach). Let ( Y,pY, p ) be a subspace of asymmetric seminormed space ( X,pX, p ). Then for every varphi_(0)inY^(b)\varphi_{0} \in Y^{b} there exists varphi inX^(b)\varphi \in X^{b} such that
{:[ varphi|_(Y)=varphi_(0)],[||varphi∣=||varphi_(0)∣.]:}\begin{aligned}
\left.\varphi\right|_{Y} & =\varphi_{0} \\
\| \varphi \mid & =\| \varphi_{0} \mid .
\end{aligned}
Proof. We consider the functional q:X rarr[0,oo),q(x)=||varphi_(0)∣*p(x)q: X \rightarrow[0, \infty), q(x)=\| \varphi_{0} \mid \cdot p(x), x in Xx \in X. Obviously qq is subadditive and positive homogeneous and for every y in Yy \in Y we have
varphi_(0)(y) <= ||varphi_(0)∣*p(y)=q(y)\varphi_{0}(y) \leq \| \varphi_{0} \mid \cdot p(y)=q(y)
i.e. varphi_(0)\varphi_{0} is majorized by qq on YY.
By Hahn-Banach extension theorem it results that there exists the linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} with properties:
By Hahn-Banach extension theorem it results that there exists the linear functional varphi:X rarrR\varphi: X \rightarrow \mathbb{R} with properties:
{:[ varphi|_(Y)=varphi_(0)quad" and "],[varphi(x) <= ||varphi_(0)∣*p(x)","" for every "x in X.]:}\begin{aligned}
\left.\varphi\right|_{Y} & =\varphi_{0} \quad \text { and } \\
\varphi(x) & \leq \| \varphi_{0} \mid \cdot p(x), \text { for every } x \in X .
\end{aligned}
It follows ||varphi| <= ||varphi_(0)|:}\left\|\varphi\left|\leq \| \varphi_{0}\right|\right., and, because
{:[||varphi∣=s u p{varphi(x):x in X","p(x) <= 1}],[ >= s u p{varphi(y):y in Y","p(y) <= 1}],[=s u p{varphi_(0)(y):y in Y,p(y) <= 1}],[=||varphi_(0)∣]:}\begin{aligned}
\| \varphi \mid & =\sup \{\varphi(x): x \in X, p(x) \leq 1\} \\
& \geq \sup \{\varphi(y): y \in Y, p(y) \leq 1\} \\
& =\sup \left\{\varphi_{0}(y): y \in Y, p(y) \leq 1\right\} \\
& =\| \varphi_{0} \mid
\end{aligned}
we have ||varphi|=||varphi_(0)|:}\left\|\varphi\left|=\| \varphi_{0}\right|\right..
2. THE epsi\varepsilon-BEST APPROXIMATION IN ( X,pX, p )
Let YY be a nonvoid subset of the asymmetric seminormed space ( X,pX, p ).
The problem of best approximation of the element x in Xx \in X by elements in YY is: find an element y_(0)in Yy_{0} \in Y such that
The problem of best approximation of the element x in Xx \in X by elements in YY is: find an element y_(0)in Yy_{0} \in Y such that
{:(1)d_(p)(x","Y):=i n f{p(y-x):y in Y}=p(y_(0)-x):}\begin{equation*}
d_{p}(x, Y):=\inf \{p(y-x): y \in Y\}=p\left(y_{0}-x\right) \tag{1}
\end{equation*}
Let epsi > 0\varepsilon>0. The problem of epsi\varepsilon-best approximation of x in Xx \in X by elements in YY is: find y_(0)in Yy_{0} \in Y such that
{:(2)p(y_(0)-x) <= d_(p)(x","Y)+epsi:}\begin{equation*}
p\left(y_{0}-x\right) \leq d_{p}(x, Y)+\varepsilon \tag{2}
\end{equation*}
Obviously, the problem of epsi\varepsilon-best approximation always admits a solution, because for every number n inNn \in \mathbb{N} there exists y_(n)in Yy_{n} \in Y such that p(y_(n)-x) <= d_(p)(x,Y)+(1)/(n)p\left(y_{n}-x\right) \leq d_{p}(x, Y)+\frac{1}{n}, so that p(y_(n)-x) <= d_(p)(x,Y)+epsip\left(y_{n}-x\right) \leq d_{p}(x, Y)+\varepsilon, for n > [(1)/(epsi)]+1n>\left[\frac{1}{\varepsilon}\right]+1.
In the following we denote by
{:(3)P_(Y,epsi)(x)={y in Y:p(y-x) <= d_(p)(x,Y)+epsi}","x in X:}\begin{equation*}
P_{Y, \varepsilon}(x)=\left\{y \in Y: p(y-x) \leq d_{p}(x, Y)+\varepsilon\right\}, x \in X \tag{3}
\end{equation*}
the nonvoid set of the elements of epsi\varepsilon-best approximation for x in Xx \in X in YY.
The paper [3] contains characterizations, in terms of functionals in X^(b)X^{b} vanishing on YY, of the elements of best approximation of x in Xx \in X by elements in a subspace YY of XX. Let us observe firstly that, one can consider also the problem of epsi\varepsilon-best approximation by using the conjugate bar(p)\bar{p} of pp. In this case, for
The paper [3] contains characterizations, in terms of functionals in X^(b)X^{b} vanishing on YY, of the elements of best approximation of x in Xx \in X by elements in a subspace YY of XX. Let us observe firstly that, one can consider also the problem of epsi\varepsilon-best approximation by using the conjugate bar(p)\bar{p} of pp. In this case, for
{:(4)d_( bar(p))(x","Y)=i n f{p(x-y):y in Y}:}\begin{equation*}
d_{\bar{p}}(x, Y)=\inf \{p(x-y): y \in Y\} \tag{4}
\end{equation*}
one looks for y_(0)in Yy_{0} \in Y such that
{:(5)p(x-y_(0)) <= d_( bar(p))(x","Y)+epsi.:}\begin{equation*}
p\left(x-y_{0}\right) \leq d_{\bar{p}}(x, Y)+\varepsilon . \tag{5}
\end{equation*}
Let us denote by
{:(6) bar(P)_(Y,epsi)(x)={y in Y:( bar(p))(y-x)=p(x-y) <= d_( bar(p))(x,Y)+epsi}:}\begin{equation*}
\bar{P}_{Y, \varepsilon}(x)=\left\{y \in Y: \bar{p}(y-x)=p(x-y) \leq d_{\bar{p}}(x, Y)+\varepsilon\right\} \tag{6}
\end{equation*}
the set of epsi\varepsilon-best approximation of x in Xx \in X with respect to the conjugate asymmetric seminorm bar(p)\bar{p}.
In the following we obtain characterizations of elements of epsi\varepsilon-best approximation of x in Xx \in X by elements of a subspace YY, with respect to the asymmetric seminorms pp and bar(p)\bar{p}.
Results of this type, for elements of best approximations in a normed space XX, using the elements of dual X^(**)X^{*} are obtained in [17] (see also [3, 9, [1], [13], [16], [18]).
Concerning the characterizations of elements of epsi\varepsilon-best approximation in normed space, see papers [14, [15].
Theorem 2. Let ( X,pX, p ) be an asymmetric seminormed space, YY a subspace of XX and x_(0)in X\\Yx_{0} \in X \backslash Y, such that d=d_(p)(x_(0),Y) > 0d=d_{p}\left(x_{0}, Y\right)>0 and bar(d)=d_( bar(p))(x_(0),Y) > 0\bar{d}=d_{\bar{p}}\left(x_{0}, Y\right)>0. Then
(a) An element y_(0)in Yy_{0} \in Y is in P_(Y,epsi)(x_(0))P_{Y, \varepsilon}\left(x_{0}\right) if and only if there exists varphi inX_(p)^(b)\varphi \in X_{p}^{b} with the properties:
(i) varphi(y)=0\varphi(y)=0, for all y in Yy \in Y,
(iii) || varphi|_(p)=1\|\left.\varphi\right|_{p}=1,
(iii) varphi(-x_(0)) >= p(y_(0)-x_(0))-epsi\varphi\left(-x_{0}\right) \geq p\left(y_{0}-x_{0}\right)-\varepsilon.
(b) An element y_(0)y_{0} is in bar(P)_(Y,epsi)(x_(0))\bar{P}_{Y, \varepsilon}\left(x_{0}\right) if and only if there exists psi inX_(p)^(b)\psi \in X_{p}^{b} with the properties:
(j) psi(y)=0\psi(y)=0, for all y in Yy \in Y,
(jj) || psi|_(p)=1\|\left.\psi\right|_{p}=1,
(jjj) psi(x_(0)) >= p(x_(0)-y_(0))-epsi\psi\left(x_{0}\right) \geq p\left(x_{0}-y_{0}\right)-\varepsilon.
Proof. Let x_(0)in X\\Yx_{0} \in X \backslash Y and Z=Y+(:x_(0):)Z=Y+\left\langle x_{0}\right\rangle be the direct sum of YY with the space generated by x_(0)x_{0}. Consider the functional varphi_(0):Z rarrR\varphi_{0}: Z \rightarrow \mathbb{R} defined by
(a) An element y_(0)in Yy_{0} \in Y is in P_(Y,epsi)(x_(0))P_{Y, \varepsilon}\left(x_{0}\right) if and only if there exists varphi inX_(p)^(b)\varphi \in X_{p}^{b} with the properties:
(i) varphi(y)=0\varphi(y)=0, for all y in Yy \in Y,
(iii) || varphi|_(p)=1\|\left.\varphi\right|_{p}=1,
(iii) varphi(-x_(0)) >= p(y_(0)-x_(0))-epsi\varphi\left(-x_{0}\right) \geq p\left(y_{0}-x_{0}\right)-\varepsilon.
(b) An element y_(0)y_{0} is in bar(P)_(Y,epsi)(x_(0))\bar{P}_{Y, \varepsilon}\left(x_{0}\right) if and only if there exists psi inX_(p)^(b)\psi \in X_{p}^{b} with the properties:
(j) psi(y)=0\psi(y)=0, for all y in Yy \in Y,
(jj) || psi|_(p)=1\|\left.\psi\right|_{p}=1,
(jjj) psi(x_(0)) >= p(x_(0)-y_(0))-epsi\psi\left(x_{0}\right) \geq p\left(x_{0}-y_{0}\right)-\varepsilon.
Proof. Let x_(0)in X\\Yx_{0} \in X \backslash Y and Z=Y+(:x_(0):)Z=Y+\left\langle x_{0}\right\rangle be the direct sum of YY with the space generated by x_(0)x_{0}. Consider the functional varphi_(0):Z rarrR\varphi_{0}: Z \rightarrow \mathbb{R} defined by
varphi_(0)(z)=varphi(y+lambdax_(0))=-lambda,\varphi_{0}(z)=\varphi\left(y+\lambda x_{0}\right)=-\lambda,
where z in Zz \in Z, and zz is uniquely represented in the form z=y+lambdax_(0)z=y+\lambda x_{0}.
The functional varphi_(0)\varphi_{0} is linear on ZZ.
Observe that varphi_(0)∣Y=0\varphi_{0} \mid Y=0, and for every lambda > 0\lambda>0 we have
The functional varphi_(0)\varphi_{0} is linear on ZZ.
Observe that varphi_(0)∣Y=0\varphi_{0} \mid Y=0, and for every lambda > 0\lambda>0 we have
p(y-lambdax_(0))=lambda p((1)/(lambda)y-x_(0)) >= lambda d=d*varphi_(0)(y-lambdax_(0)).p\left(y-\lambda x_{0}\right)=\lambda p\left(\frac{1}{\lambda} y-x_{0}\right) \geq \lambda d=d \cdot \varphi_{0}\left(y-\lambda x_{0}\right) .
It follows that
varphi_(0)(y-lambdax_(0)) <= (1)/(d)*p(y-lambdax_(0)),\varphi_{0}\left(y-\lambda x_{0}\right) \leq \frac{1}{d} \cdot p\left(y-\lambda x_{0}\right),
for every lambda > 0\lambda>0.
Because the last inequality is also valid if varphi_(0)(y-tx_(0))=t <= 0\varphi_{0}\left(y-t x_{0}\right)=t \leq 0, it follows
Because the last inequality is also valid if varphi_(0)(y-tx_(0))=t <= 0\varphi_{0}\left(y-t x_{0}\right)=t \leq 0, it follows
||varphi_(0)|_(p) <= (1)/(d)," and consequently "varphi_(0)inZ_(p)^(b).\|\left.\varphi_{0}\right|_{p} \leq \frac{1}{d}, \text { and consequently } \varphi_{0} \in Z_{p}^{b} .
Now, let (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1} be a sequence in YY such that p(y_(n)-x_(0))rarr dp\left(y_{n}-x_{0}\right) \rightarrow d, for n rarr oon \rightarrow \infty, and such that p(y_(n)-x_(0)) > 0p\left(y_{n}-x_{0}\right)>0 for every n inNn \in \mathbb{N}. Then
||varphi_(0)|_(p) >= varphi_(0)((y_(n)-x_(0))/(p(y_(n)-x_(0))))=(1)/(p(y_(n)-x_(0)))rarr(1)/(d),\|\left.\varphi_{0}\right|_{p} \geq \varphi_{0}\left(\frac{y_{n}-x_{0}}{p\left(y_{n}-x_{0}\right)}\right)=\frac{1}{p\left(y_{n}-x_{0}\right)} \rightarrow \frac{1}{d},
and, consequently, ||varphi_(0)|_(p)=(1)/(d)\|\left.\varphi_{0}\right|_{p}=\frac{1}{d}.
By Theorem 1, there exists varphi_(1)inX^(b)\varphi_{1} \in X^{b} such that
By Theorem 1, there exists varphi_(1)inX^(b)\varphi_{1} \in X^{b} such that
varphi_(1)|z=varphi_(0),quad||varphi_(1)|=||varphi_(0)|_(p)=(1)/(d):}\varphi_{1}\left|z=\varphi_{0}, \quad\left\|\varphi_{1}\left|=\| \varphi_{0}\right|_{p}=\frac{1}{d}\right.\right.
Then, the functional varphi=d*varphi_(1)\varphi=d \cdot \varphi_{1} satisfies the properties: varphi inX_(p)^(b),varphi∣Y=d*varphi_(1)|_(Y)=0\varphi \in X_{p}^{b}, \varphi \mid Y= \left.d \cdot \varphi_{1}\right|_{Y}=0,
{:[varphi(-x_(0))=varphi(y_(0))+varphi(-x_(0))],[=varphi(y_(0)-x_(0))],[ >= p(y_(0)-x_(0))],[ >= p(y_(0)-x_(0))-epsi.]:}\begin{aligned}
\varphi\left(-x_{0}\right) & =\varphi\left(y_{0}\right)+\varphi\left(-x_{0}\right) \\
& =\varphi\left(y_{0}-x_{0}\right) \\
& \geq p\left(y_{0}-x_{0}\right) \\
& \geq p\left(y_{0}-x_{0}\right)-\varepsilon .
\end{aligned}
Conversely, if y_(0)in Yy_{0} \in Y and there exists varphi inX_(p)^(b)\varphi \in X_{p}^{b} with the properties (a) (i)-(iii), then for every y in Yy \in Y we have
{:[p(y_(0)-x_(0)) <= varphi(-x_(0))+epsi],[=varphi(y-x_(0))+epsi],[ <= || varphi|_(p)*p(y-x_(0))+epsi],[ <= p(y-x_(0))+epsi]:}\begin{aligned}
p\left(y_{0}-x_{0}\right) & \leq \varphi\left(-x_{0}\right)+\varepsilon \\
& =\varphi\left(y-x_{0}\right)+\varepsilon \\
& \leq \|\left.\varphi\right|_{p} \cdot p\left(y-x_{0}\right)+\varepsilon \\
& \leq p\left(y-x_{0}\right)+\varepsilon
\end{aligned}
Taking the infimum with respect to y in Yy \in Y, one finds
p(y_(0)-x_(0)) <= d_(p)(x_(0),Y)+epsi;p\left(y_{0}-x_{0}\right) \leq d_{p}\left(x_{0}, Y\right)+\varepsilon ;
so that y_(0)inP_(Y,epsi)(x_(0))y_{0} \in P_{Y, \varepsilon}\left(x_{0}\right).
Similarly, defining psi_(0):Z=Y+(:x_(0):)rarrR\psi_{0}: Z=Y+\left\langle x_{0}\right\rangle \rightarrow \mathbb{R} by psi(z)=psi_(0)(y+lambdax_(0))=lambda,y in Y\psi(z)=\psi_{0}\left(y+\lambda x_{0}\right)= \lambda, y \in Y and lambda inR\lambda \in \mathbb{R}, and proceeding in the same way, one obtains the claim (b) of the theorem.
Similarly, defining psi_(0):Z=Y+(:x_(0):)rarrR\psi_{0}: Z=Y+\left\langle x_{0}\right\rangle \rightarrow \mathbb{R} by psi(z)=psi_(0)(y+lambdax_(0))=lambda,y in Y\psi(z)=\psi_{0}\left(y+\lambda x_{0}\right)= \lambda, y \in Y and lambda inR\lambda \in \mathbb{R}, and proceeding in the same way, one obtains the claim (b) of the theorem.
Theorem 2 has the following consequence:
Corollary 3. In the hypothesis of Theorem 2 we have:
(a') M subP_(Y,epsi)(x_(0))M \subset P_{Y, \varepsilon}\left(x_{0}\right), if and only if there exists varphi inX^(b)\varphi \in X^{b} verifying (a) (i)-(ii) and the condition:
Corollary 3. In the hypothesis of Theorem 2 we have:
(a') M subP_(Y,epsi)(x_(0))M \subset P_{Y, \varepsilon}\left(x_{0}\right), if and only if there exists varphi inX^(b)\varphi \in X^{b} verifying (a) (i)-(ii) and the condition:
varphi(-x_(0)) >= p(u-x_(0))-epsi,quad" for all "quad u in M;\varphi\left(-x_{0}\right) \geq p\left(u-x_{0}\right)-\varepsilon, \quad \text { for all } \quad u \in M ;
(b') M subP_(Y,epsi)(x_(0))M \subset P_{Y, \varepsilon}\left(x_{0}\right) if and only if there exists psi inX^(b)\psi \in X^{b} with properties (b) (j)-(jj)(j)-(j j), and verifying the condition:
psi(x_(0)) >= p(x_(0)-u)-epsi,quad" for all "quad u in M". "\psi\left(x_{0}\right) \geq p\left(x_{0}-u\right)-\varepsilon, \quad \text { for all } \quad u \in M \text {. }
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[2] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx. Theory, 94, no. 1, pp. 54-72, 1998.
[3] Cobzas, S. and Mustata, C., Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Numér. Théor. Approx., 33, no. 1, pp. 39-50, 2004. 줓
[4] Cobzas, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27, pp. 1-22, 2004.
[5] Garcia-Raffi, L. M., Romaguera S. and Sánchez-Pérez, E. A., The dual space of an asymmetric normed linear space, Quaest. Math., 26, no. 1, pp. 83-96, 2003.
[6] Garcia-Raffi, L.M., Romaguera S. and Sánchez-Pérez, E. A., On Hausdorff asymmetric normed linear spaces, Houston J. Math., 29, no. 3, pp. 717-728, 2003 (electronic).
[7] Krein, M. G. and Nudel'man, A. A., The Markov Moment Problem and Extremum Problems, Nauka, Moscow 1973 (in Russian). English translation: American Mathematical Society, Providence, R.I., 1997.
[8] Li, Chong and Ni, Renxing, Derivatives of generalized distance functions and existence of generalized nearest points, J. Approx. Theory, 115, no. 1, pp. 44-55, 2002.
[9] Mabizela, S., Characterization of best approximation in metric linear spaces, Scientiae Mathematicae Japonica, 57, 2, pp. 233-240, 2003.
[10] Mustăţa, C., On the best approximation in metric spaces, Mathematica - Revue d'Analyse Numérique et de Théorie de l'Approximation, L'Analyse Numérique et la Théorie de l'Approximation, 4, pp. 45-50, 1975. 지
[11] Mustăta, C., On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space, Rev. Anal. Numér. Théor. Approx., 32, no. 2, pp. 187-192, 2003. 준
[12] Ni, Renxing, Existence of generalized nearest points, Taiwanese J. Math., 7, no. 1, pp. 115-128, 2003.
[13] Pantelidis, G., Approximations theorie für metrich linear Räume, Math. Ann., 184, pp. 30-48, 1969.
[14] Rezapour, Sh., epsi\varepsilon-pseudo Chebyshev and epsi\varepsilon-quasi Chebyshev subspaces of Banach spaces, Technical Report, Azarbaidjan University of Tarbiot Moallem, 2003.
[15] Rezapour, Sh., epsi\varepsilon-weakly Chebyshev subspaces of Banach spaces, Analysis in Theory and Applications, 19, no. 2, pp. 130-135, 2003.
[16] Schnatz, K., Nonlinear duality and best approximation in metric linear spaces, J. Approx. Theory, 49, no. 3, pp. 201-21, 1987.
[17] Singer, I., Best Approximation in Normed Linear spaces by Elements of Linear subspaces, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New-York-Berlin, 1970.
[18] Singer, I., Caracterisations des éléments de la meilleure approximation dans un espace de Banach quelconque, Acta Sci. Math., 17, pp. 181-189, 1956.
Received by the editors: June 11, 2003.
- ^(**){ }^{*} This work has been supported by the Romanian Academy under Grant GAR 13/2004.
^(†){ }^{\dagger} "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.