Characterization of ε-nearest points in space with asymmetric seminorm

Costica Mustata
(original), Approximation Theory, asymmetric norm, best approximation, paper

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

Costica Mustata
“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.

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Revue d’Analyse Numer.Theor. Approx.

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Publishing House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no2-art11

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2501-059X

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2457-6794

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2004-Mustata-Characterization of ε-nearest points-Jnaat

CHARACTERIZATION OF ε ε epsi\varepsilonε-NEAREST POINTS IN SPACES WITH ASYMMETRIC SEMINORM*

COSTICĂ MUSTĂŢA † † ^(†){ }^{\dagger}†Dedicated to professor Elena Popoviciu on the occasion of her 80th anniversary.

Abstract

In this note we are concerned with the characterization of the elements of ε ε epsi\varepsilonε-best approximation ( ε ε epsi\varepsilonε-nearest points) in a subspace Y Y YYY of space X X XXX with asymmetric seminorm. For this we use functionals in the asymmetric dual X b X b X^(b)X^{b}Xb defined and studied in some recent papers 1, 3, 5.

MSC 2000. 41A65.
Keywords. Asymmetric seminormed spaces, ε ε epsi\varepsilonε-nearest points, characterization.

1. INTRODUCTION

Let X X XXX be a real linear space. A functional p : X → [ 0 , ∞ ) p : X → [ 0 , ∞ ) p:X rarr[0,oo)p: X \rightarrow[0, \infty)p:X→[0,∞) with the properties:
(1) p ( x ) ≥ 0 p ( x ) ≥ 0 p(x) >= 0p(x) \geq 0p(x)≥0, for all x ∈ X x ∈ X x in Xx \in Xx∈X,
(2) p ( t x ) = t p ( x ) p ( t x ) = t p ( x ) p(tx)=tp(x)p(t x)=t p(x)p(tx)=tp(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X and t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0,
(3) p ( x + y ) ≤ p ( x ) + p ( y ) p ( x + y ) ≤ p ( x ) + p ( y ) p(x+y) <= p(x)+p(y)p(x+y) \leq p(x)+p(y)p(x+y)≤p(x)+p(y), for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X,
is called asymmetric seminorm on X X XXX, and the pair ( X , p X , p X,pX, pX,p ) is called a space with asymmetric seminorm.
The functional p ¯ : X → [ 0 , ∞ ) p ¯ : X → [ 0 , ∞ ) bar(p):X rarr[0,oo)\bar{p}: X \rightarrow[0, \infty)p¯:X→[0,∞), defined by p ¯ ( x ) = p ( − x ) , x ∈ X p ¯ ( x ) = p ( − x ) , x ∈ X bar(p)(x)=p(-x),x in X\bar{p}(x)=p(-x), x \in Xp¯(x)=p(−x),x∈X is another asymmetric seminorm on X X XXX, called the conjugate of p p ppp.
The functional p s : X → [ 0 , ∞ ) p s : X → [ 0 , ∞ ) p^(s):X rarr[0,oo)p^{s}: X \rightarrow[0, \infty)ps:X→[0,∞), defined by
p s ( x ) = max { p ( x ) , p ( − x ) } , x ∈ X p s ( x ) = max { p ( x ) , p ( − x ) } , x ∈ X p^(s)(x)=max{p(x),p(-x)},x in Xp^{s}(x)=\max \{p(x), p(-x)\}, x \in Xps(x)=max{p(x),p(−x)},x∈X
is a seminorm on X X XXX. If p s p s p^(s)p^{s}ps satisfies the axioms of a norm, then p p ppp is called an asymmetric norm on X X XXX. It follows that p p ppp satisfies the properties (1), (2), (3), and
(4) p ( x ) = 0 p ( x ) = 0 p(x)=0p(x)=0p(x)=0 and p ( − x ) = 0 p ( − x ) = 0 p(-x)=0p(-x)=0p(−x)=0 imply x = 0 x = 0 x=0x=0x=0.
The asymmetric seminorm p p ppp on X X XXX generates a topology τ p τ p tau_(p)\tau_{p}τp on X X XXX, having as a basis of neighborhoods of a point x ∈ X x ∈ X x in Xx \in Xx∈X the open p p ppp-balls
B p ′ ( x , r ) = { x ′ ∈ X : p ( x ′ − x ) < r } , r > 0 . B p ′ ( x , r ) = x ′ ∈ X : p x ′ − x < r , r > 0 . 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 .Bp′(x,r)={x′∈X:p(x′−x)<r},r>0.
The family of closed p p ppp-balls
B p ( x , r ) = { x ′ ∈ X : p ( x ′ − x ) ≤ r } , r > 0 B p ( x , r ) = x ′ ∈ X : p x ′ − x ≤ r , r > 0 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>0Bp(x,r)={x′∈X:p(x′−x)≤r},r>0
generates the same topology. This topology τ p τ p tau_(p)\tau_{p}τp could not be Hausdorff (see [5]), and could not be linear (the multiplication by scalars is not continuous in general, see [1].
Let R R R\mathbb{R}R be the set of real numbers and u : R → [ 0 , ∞ ) , u ( a ) = max { a , 0 } u : R → [ 0 , ∞ ) , u ( a ) = max { a , 0 } u:Rrarr[0,oo),u(a)=max{a,0}u: \mathbb{R} \rightarrow[0, \infty), u(a)=\max \{a, 0\}u:R→[0,∞),u(a)=max{a,0}, a ∈ R a ∈ R a inRa \in \mathbb{R}a∈R. Then the function u u uuu is an asymmetric seminorm on R R R\mathbb{R}R and, for a ∈ R a ∈ R a inRa \in \mathbb{R}a∈R, the intervals ( − ∞ , a + ε ) , ε > 0 ( − ∞ , a + ε ) , ε > 0 (-oo,a+epsi),epsi > 0(-\infty, a+\varepsilon), \varepsilon>0(−∞,a+ε),ε>0, form a basis of neighborhoods of a ∈ R a ∈ R a inRa \in \mathbb{R}a∈R in the topology τ u τ u tau_(u)\tau_{u}τu. The conjugate asymmetric seminorm of u u uuu is u ¯ : R → [ 0 , ∞ ) u ¯ : R → [ 0 , ∞ ) bar(u):Rrarr[0,oo)\bar{u}: \mathbb{R} \rightarrow[0, \infty)u¯:R→[0,∞), u ¯ ( a ) = u ( − a ) , a ∈ R u ¯ ( a ) = u ( − a ) , a ∈ R bar(u)(a)=u(-a),a inR\bar{u}(a)=u(-a), a \in \mathbb{R}u¯(a)=u(−a),a∈R, and u s ( a ) = max { u ( a ) , u ( − a ) } = | a | u s ( a ) = max { u ( a ) , u ( − a ) } = | a | u^(s)(a)=max{u(a),u(-a)}=|a|u^{s}(a)=\max \{u(a), u(-a)\}=|a|us(a)=max{u(a),u(−a)}=|a| is a norm on R R R\mathbb{R}R. Consequently, u u uuu is an asymmetric norm on R R R\mathbb{R}R.
Let φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R be a linear functional. The continuity of φ φ varphi\varphiφ with respect to the topologies τ p τ p tau_(p)\tau_{p}τp and τ u τ u tau_(u)\tau_{u}τu is called ( p , u p , u p,up, up,u )-continuity, and it is equivalent to the upper semicontinuity of φ φ varphi\varphiφ as a functional from ( X , τ p ) X , τ p (X,tau_(p))\left(X, \tau_{p}\right)(X,τp) to ( R , | ⋅ | ) ( R , | ⋅ | ) (R,|*|)(\mathbb{R},|\cdot|)(R,|⋅|).
The linear functional φ : ( X , τ p ) → ( R , u ) φ : X , τ p → ( R , u ) varphi:(X,tau_(p))rarr(R,u)\varphi:\left(X, \tau_{p}\right) \rightarrow(\mathbb{R}, u)φ:(X,τp)→(R,u) is ( p , u ) ( p , u ) (p,u)(p, u)(p,u)-continuous if and only if it is p p ppp-bounded, i.e. there exists L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 such that
φ ( x ) ≤ L p ( x ) , for all x ∈ X . φ ( x ) ≤ L p ( x ) ,  for all  x ∈ X . varphi(x) <= Lp(x),quad" for all "x in X.\varphi(x) \leq L p(x), \quad \text { for all } x \in X .φ(x)≤Lp(x), for all x∈X.
The set of all ( p , u p , u p,up, up,u )-continuous functionals is denoted by X p b X p b X_(p)^(b)X_{p}^{b}Xpb. With respect to pointwise addition and multiplication by real scalars, the set X p b X p b X_(p)^(b)X_{p}^{b}Xpb is a cone, i.e. λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0 and φ , ψ ∈ X p b φ , ψ ∈ X p b varphi,psi inX_(p)^(b)\varphi, \psi \in X_{p}^{b}φ,ψ∈Xpb imply φ + ψ ∈ X b φ + ψ ∈ X b varphi+psi inX^(b)\varphi+\psi \in X^{b}φ+ψ∈Xb and λ φ ∈ X p b λ φ ∈ X p b lambda varphi inX_(p)^(b)\lambda \varphi \in X_{p}^{b}λφ∈Xpb.
The functional ‖ . ∣: X p b → [ 0 , ∞ ) ‖ . ∣: X p b → [ 0 , ∞ ) ||.∣:X_(p)^(b)rarr[0,oo)\| . \mid: X_{p}^{b} \rightarrow[0, \infty)‖.∣:Xpb→[0,∞) defined by
‖ φ | p = sup { φ ( x ) : x ∈ X , p ( x ) ≤ 1 } , φ ∈ X p b ‖ φ p = sup { φ ( x ) : x ∈ X , p ( x ) ≤ 1 } , φ ∈ X p b || 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}‖φ|p=sup{φ(x):x∈X,p(x)≤1},φ∈Xpb
satisfies the properties of an asymmetric seminorm, and the pair ( X p b , ‖ ⋅ | p X p b , ‖ ⋅ p X_(p)^(b),||*|_(p)X_{p}^{b}, \|\left.\cdot\right|_{p}Xpb,‖⋅|p ) is called the asymmetric dual of the asymmetric seminormed space ( X , p X , p X,pX, 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 X^(b)X^{b}Xb and ‖ φ ∣ ‖ φ ∣ ||varphi∣\| \varphi \mid‖φ∣ instead of X p b X p b X_(p)^(b)X_{p}^{b}Xpb and ‖ φ | p ‖ φ p || varphi|_(p)\|\left.\varphi\right|_{p}‖φ|p, respectively.
Let ( X , p X , p X,pX, pX,p ) be an asymmetric seminormed space and Y Y YYY a subspace of X X XXX. Let Y b Y b Y^(b)Y^{b}Yb be the asymmetric dual of ( Y , p ) ( Y , p ) (Y,p)(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 , p Y , p Y,pY, pY,p ) be a subspace of asymmetric seminormed space ( X , p X , p X,pX, pX,p ). Then for every φ 0 ∈ Y b φ 0 ∈ Y b varphi_(0)inY^(b)\varphi_{0} \in Y^{b}φ0∈Yb there exists φ ∈ X b φ ∈ X b varphi inX^(b)\varphi \in X^{b}φ∈Xb such that
φ | Y = φ 0 ‖ φ ∣ = ‖ φ 0 ∣ . φ Y = φ 0 ‖ φ ∣ = ‖ φ 0 ∣ . {:[ varphi|_(Y)=varphi_(0)],[||varphi∣=||varphi_(0)∣.]:}\begin{aligned} \left.\varphi\right|_{Y} & =\varphi_{0} \\ \| \varphi \mid & =\| \varphi_{0} \mid . \end{aligned}φ|Y=φ0‖φ∣=‖φ0∣.
Proof. We consider the functional q : X → [ 0 , ∞ ) , q ( x ) = ‖ φ 0 ∣ ⋅ p ( x ) q : X → [ 0 , ∞ ) , q ( x ) = ‖ φ 0 ∣ ⋅ p ( x ) q:X rarr[0,oo),q(x)=||varphi_(0)∣*p(x)q: X \rightarrow[0, \infty), q(x)=\| \varphi_{0} \mid \cdot p(x)q:X→[0,∞),q(x)=‖φ0∣⋅p(x), x ∈ X x ∈ X x in Xx \in Xx∈X. Obviously q q qqq is subadditive and positive homogeneous and for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
φ 0 ( y ) ≤ ‖ φ 0 ∣ ⋅ p ( y ) = q ( y ) φ 0 ( y ) ≤ ‖ φ 0 ∣ ⋅ p ( y ) = q ( y ) varphi_(0)(y) <= ||varphi_(0)∣*p(y)=q(y)\varphi_{0}(y) \leq \| \varphi_{0} \mid \cdot p(y)=q(y)φ0(y)≤‖φ0∣⋅p(y)=q(y)
i.e. φ 0 φ 0 varphi_(0)\varphi_{0}φ0 is majorized by q q qqq on Y Y YYY.
By Hahn-Banach extension theorem it results that there exists the linear functional φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R with properties:
φ | Y = φ 0 and φ ( x ) ≤ ‖ φ 0 ∣ ⋅ p ( x ) , for every x ∈ X . φ Y = φ 0  and  φ ( x ) ≤ ‖ φ 0 ∣ ⋅ p ( x ) ,  for every  x ∈ X . {:[ 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}φ|Y=φ0 and φ(x)≤‖φ0∣⋅p(x), for every x∈X.
It follows ‖ φ | ≤ ‖ φ 0 | φ ≤ ‖ φ 0 ||varphi| <= ||varphi_(0)|:}\left\|\varphi\left|\leq \| \varphi_{0}\right|\right.‖φ|≤‖φ0|, and, because
‖ φ ∣ = sup { φ ( x ) : x ∈ X , p ( x ) ≤ 1 } ≥ sup { φ ( y ) : y ∈ Y , p ( y ) ≤ 1 } = sup { φ 0 ( y ) : y ∈ Y , p ( y ) ≤ 1 } = ‖ φ 0 ∣ ‖ φ ∣ = sup { φ ( x ) : x ∈ X , p ( x ) ≤ 1 } ≥ sup { φ ( y ) : y ∈ Y , p ( y ) ≤ 1 } = sup φ 0 ( y ) : y ∈ Y , p ( y ) ≤ 1 = ‖ φ 0 ∣ {:[||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}‖φ∣=sup{φ(x):x∈X,p(x)≤1}≥sup{φ(y):y∈Y,p(y)≤1}=sup{φ0(y):y∈Y,p(y)≤1}=‖φ0∣
we have ‖ φ | = ‖ φ 0 | φ = ‖ φ 0 ||varphi|=||varphi_(0)|:}\left\|\varphi\left|=\| \varphi_{0}\right|\right.‖φ|=‖φ0|.

2. THE ε ε epsi\varepsilonε-BEST APPROXIMATION IN ( X , p X , p X,pX, pX,p )

Let Y Y YYY be a nonvoid subset of the asymmetric seminormed space ( X , p X , p X,pX, pX,p ).
The problem of best approximation of the element x ∈ X x ∈ X x in Xx \in Xx∈X by elements in Y Y YYY is: find an element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that
(1) d p ( x , Y ) := inf { p ( y − x ) : y ∈ Y } = p ( y 0 − x ) (1) d p ( x , Y ) := inf { p ( y − x ) : y ∈ Y } = p y 0 − x {:(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*}(1)dp(x,Y):=inf{p(y−x):y∈Y}=p(y0−x)
Let ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. The problem of ε ε epsi\varepsilonε-best approximation of x ∈ X x ∈ X x in Xx \in Xx∈X by elements in Y Y YYY is: find y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that
(2) p ( y 0 − x ) ≤ d p ( x , Y ) + ε (2) p y 0 − x ≤ d p ( x , Y ) + ε {:(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*}(2)p(y0−x)≤dp(x,Y)+ε
Obviously, the problem of ε ε epsi\varepsilonε-best approximation always admits a solution, because for every number n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N there exists y n ∈ Y y n ∈ Y y_(n)in Yy_{n} \in Yyn∈Y such that p ( y n − x ) ≤ d p ( x , Y ) + 1 n p y n − x ≤ d p ( x , Y ) + 1 n p(y_(n)-x) <= d_(p)(x,Y)+(1)/(n)p\left(y_{n}-x\right) \leq d_{p}(x, Y)+\frac{1}{n}p(yn−x)≤dp(x,Y)+1n, so that p ( y n − x ) ≤ d p ( x , Y ) + ε p y n − x ≤ d p ( x , Y ) + ε p(y_(n)-x) <= d_(p)(x,Y)+epsip\left(y_{n}-x\right) \leq d_{p}(x, Y)+\varepsilonp(yn−x)≤dp(x,Y)+ε, for n > [ 1 ε ] + 1 n > 1 ε + 1 n > [(1)/(epsi)]+1n>\left[\frac{1}{\varepsilon}\right]+1n>[1ε]+1.
In the following we denote by
(3) P Y , ε ( x ) = { y ∈ Y : p ( y − x ) ≤ d p ( x , Y ) + ε } , x ∈ X (3) P Y , ε ( x ) = y ∈ Y : p ( y − x ) ≤ d p ( x , Y ) + ε , x ∈ X {:(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*}(3)PY,ε(x)={y∈Y:p(y−x)≤dp(x,Y)+ε},x∈X
the nonvoid set of the elements of ε ε epsi\varepsilonε-best approximation for x ∈ X x ∈ X x in Xx \in Xx∈X in Y Y YYY.
The paper [3] contains characterizations, in terms of functionals in X b X b X^(b)X^{b}Xb vanishing on Y Y YYY, of the elements of best approximation of x ∈ X x ∈ X x in Xx \in Xx∈X by elements in a subspace Y Y YYY of X X XXX. Let us observe firstly that, one can consider also the problem of ε ε epsi\varepsilonε-best approximation by using the conjugate p ¯ p ¯ bar(p)\bar{p}p¯ of p p ppp. In this case, for
(4) d p ¯ ( x , Y ) = inf { p ( x − y ) : y ∈ Y } (4) d p ¯ ( x , Y ) = inf { p ( x − y ) : y ∈ Y } {:(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*}(4)dp¯(x,Y)=inf{p(x−y):y∈Y}
one looks for y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that
(5) p ( x − y 0 ) ≤ d p ¯ ( x , Y ) + ε . (5) p x − y 0 ≤ d p ¯ ( x , Y ) + ε . {:(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*}(5)p(x−y0)≤dp¯(x,Y)+ε.
Let us denote by
(6) P ¯ Y , ε ( x ) = { y ∈ Y : p ¯ ( y − x ) = p ( x − y ) ≤ d p ¯ ( x , Y ) + ε } (6) P ¯ Y , ε ( x ) = y ∈ Y : p ¯ ( y − x ) = p ( x − y ) ≤ d p ¯ ( x , Y ) + ε {:(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*}(6)P¯Y,ε(x)={y∈Y:p¯(y−x)=p(x−y)≤dp¯(x,Y)+ε}
the set of ε ε epsi\varepsilonε-best approximation of x ∈ X x ∈ X x in Xx \in Xx∈X with respect to the conjugate asymmetric seminorm p ¯ p ¯ bar(p)\bar{p}p¯.
In the following we obtain characterizations of elements of ε ε epsi\varepsilonε-best approximation of x ∈ X x ∈ X x in Xx \in Xx∈X by elements of a subspace Y Y YYY, with respect to the asymmetric seminorms p p ppp and p ¯ p ¯ bar(p)\bar{p}p¯.
Results of this type, for elements of best approximations in a normed space X X XXX, using the elements of dual X ∗ X ∗ X^(**)X^{*}X∗ are obtained in [17] (see also [3, 9, [1], [13], [16], [18]).
Concerning the characterizations of elements of ε ε epsi\varepsilonε-best approximation in normed space, see papers [14, [15].
Theorem 2. Let ( X , p X , p X,pX, pX,p ) be an asymmetric seminormed space, Y Y YYY a subspace of X X XXX and x 0 ∈ X ∖ Y x 0 ∈ X ∖ Y x_(0)in X\\Yx_{0} \in X \backslash Yx0∈X∖Y, such that d = d p ( x 0 , Y ) > 0 d = d p x 0 , Y > 0 d=d_(p)(x_(0),Y) > 0d=d_{p}\left(x_{0}, Y\right)>0d=dp(x0,Y)>0 and d ¯ = d p ¯ ( x 0 , Y ) > 0 d ¯ = d p ¯ x 0 , Y > 0 bar(d)=d_( bar(p))(x_(0),Y) > 0\bar{d}=d_{\bar{p}}\left(x_{0}, Y\right)>0d¯=dp¯(x0,Y)>0. Then
(a) An element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is in P Y , ε ( x 0 ) P Y , ε x 0 P_(Y,epsi)(x_(0))P_{Y, \varepsilon}\left(x_{0}\right)PY,ε(x0) if and only if there exists φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb with the properties:
(i) φ ( y ) = 0 φ ( y ) = 0 varphi(y)=0\varphi(y)=0φ(y)=0, for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y,
(iii) ‖ φ | p = 1 ‖ φ p = 1 || varphi|_(p)=1\|\left.\varphi\right|_{p}=1‖φ|p=1,
(iii) φ ( − x 0 ) ≥ p ( y 0 − x 0 ) − ε φ − x 0 ≥ p y 0 − x 0 − ε varphi(-x_(0)) >= p(y_(0)-x_(0))-epsi\varphi\left(-x_{0}\right) \geq p\left(y_{0}-x_{0}\right)-\varepsilonφ(−x0)≥p(y0−x0)−ε.
(b) An element y 0 y 0 y_(0)y_{0}y0 is in P ¯ Y , ε ( x 0 ) P ¯ Y , ε x 0 bar(P)_(Y,epsi)(x_(0))\bar{P}_{Y, \varepsilon}\left(x_{0}\right)P¯Y,ε(x0) if and only if there exists ψ ∈ X p b ψ ∈ X p b psi inX_(p)^(b)\psi \in X_{p}^{b}ψ∈Xpb with the properties:
(j) ψ ( y ) = 0 ψ ( y ) = 0 psi(y)=0\psi(y)=0ψ(y)=0, for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y,
(jj) ‖ ψ | p = 1 ‖ ψ p = 1 || psi|_(p)=1\|\left.\psi\right|_{p}=1‖ψ|p=1,
(jjj) ψ ( x 0 ) ≥ p ( x 0 − y 0 ) − ε ψ x 0 ≥ p x 0 − y 0 − ε psi(x_(0)) >= p(x_(0)-y_(0))-epsi\psi\left(x_{0}\right) \geq p\left(x_{0}-y_{0}\right)-\varepsilonψ(x0)≥p(x0−y0)−ε.
Proof. Let x 0 ∈ X ∖ Y x 0 ∈ X ∖ Y x_(0)in X\\Yx_{0} \in X \backslash Yx0∈X∖Y and Z = Y + ⟨ x 0 ⟩ Z = Y + x 0 Z=Y+(:x_(0):)Z=Y+\left\langle x_{0}\right\rangleZ=Y+⟨x0⟩ be the direct sum of Y Y YYY with the space generated by x 0 x 0 x_(0)x_{0}x0. Consider the functional φ 0 : Z → R φ 0 : Z → R varphi_(0):Z rarrR\varphi_{0}: Z \rightarrow \mathbb{R}φ0:Z→R defined by
φ 0 ( z ) = φ ( y + λ x 0 ) = − λ , φ 0 ( z ) = φ y + λ x 0 = − λ , varphi_(0)(z)=varphi(y+lambdax_(0))=-lambda,\varphi_{0}(z)=\varphi\left(y+\lambda x_{0}\right)=-\lambda,φ0(z)=φ(y+λx0)=−λ,
where z ∈ Z z ∈ Z z in Zz \in Zz∈Z, and z z zzz is uniquely represented in the form z = y + λ x 0 z = y + λ x 0 z=y+lambdax_(0)z=y+\lambda x_{0}z=y+λx0.
The functional φ 0 φ 0 varphi_(0)\varphi_{0}φ0 is linear on Z Z ZZZ.
Observe that φ 0 ∣ Y = 0 φ 0 ∣ Y = 0 varphi_(0)∣Y=0\varphi_{0} \mid Y=0φ0∣Y=0, and for every λ > 0 λ > 0 lambda > 0\lambda>0λ>0 we have
p ( y − λ x 0 ) = λ p ( 1 λ y − x 0 ) ≥ λ d = d ⋅ φ 0 ( y − λ x 0 ) . p y − λ x 0 = λ p 1 λ y − x 0 ≥ λ d = d ⋅ φ 0 y − λ x 0 . 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) .p(y−λx0)=λp(1λy−x0)≥λd=d⋅φ0(y−λx0).
It follows that
φ 0 ( y − λ x 0 ) ≤ 1 d ⋅ p ( y − λ x 0 ) , φ 0 y − λ x 0 ≤ 1 d ⋅ p y − λ x 0 , 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),φ0(y−λx0)≤1d⋅p(y−λx0),
for every λ > 0 λ > 0 lambda > 0\lambda>0λ>0.
Because the last inequality is also valid if φ 0 ( y − t x 0 ) = t ≤ 0 φ 0 y − t x 0 = t ≤ 0 varphi_(0)(y-tx_(0))=t <= 0\varphi_{0}\left(y-t x_{0}\right)=t \leq 0φ0(y−tx0)=t≤0, it follows
‖ φ 0 | p ≤ 1 d , and consequently φ 0 ∈ Z p b . ‖ φ 0 p ≤ 1 d ,  and consequently  φ 0 ∈ Z p b . ||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} .‖φ0|p≤1d, and consequently φ0∈Zpb.
Now, let ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 be a sequence in Y Y YYY such that p ( y n − x 0 ) → d p y n − x 0 → d p(y_(n)-x_(0))rarr dp\left(y_{n}-x_{0}\right) \rightarrow dp(yn−x0)→d, for n → ∞ n → ∞ n rarr oon \rightarrow \inftyn→∞, and such that p ( y n − x 0 ) > 0 p y n − x 0 > 0 p(y_(n)-x_(0)) > 0p\left(y_{n}-x_{0}\right)>0p(yn−x0)>0 for every n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N. Then
‖ φ 0 | p ≥ φ 0 ( y n − x 0 p ( y n − x 0 ) ) = 1 p ( y n − x 0 ) → 1 d , ‖ φ 0 p ≥ φ 0 y n − x 0 p y n − x 0 = 1 p y n − x 0 → 1 d , ||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},‖φ0|p≥φ0(yn−x0p(yn−x0))=1p(yn−x0)→1d,
and, consequently, ‖ φ 0 | p = 1 d ‖ φ 0 p = 1 d ||varphi_(0)|_(p)=(1)/(d)\|\left.\varphi_{0}\right|_{p}=\frac{1}{d}‖φ0|p=1d.
By Theorem 1, there exists φ 1 ∈ X b φ 1 ∈ X b varphi_(1)inX^(b)\varphi_{1} \in X^{b}φ1∈Xb such that
φ 1 | z = φ 0 , ‖ φ 1 | = ‖ φ 0 | p = 1 d φ 1 z = φ 0 , φ 1 = ‖ φ 0 p = 1 d 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.φ1|z=φ0,‖φ1|=‖φ0|p=1d
Then, the functional φ = d ⋅ φ 1 φ = d ⋅ φ 1 varphi=d*varphi_(1)\varphi=d \cdot \varphi_{1}φ=d⋅φ1 satisfies the properties: φ ∈ X p b , φ ∣ Y = d ⋅ φ 1 | Y = 0 φ ∈ X p b , φ ∣ Y = d ⋅ φ 1 Y = 0 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}=0φ∈Xpb,φ∣Y=d⋅φ1|Y=0,
φ ( − x 0 ) = φ ( y 0 ) + φ ( − x 0 ) = φ ( y 0 − x 0 ) ≥ p ( y 0 − x 0 ) ≥ p ( y 0 − x 0 ) − ε . φ − x 0 = φ y 0 + φ − x 0 = φ y 0 − x 0 ≥ p y 0 − x 0 ≥ p y 0 − x 0 − ε . {:[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}φ(−x0)=φ(y0)+φ(−x0)=φ(y0−x0)≥p(y0−x0)≥p(y0−x0)−ε.
Conversely, if y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y and there exists φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb with the properties (a) (i)-(iii), then for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
p ( y 0 − x 0 ) ≤ φ ( − x 0 ) + ε = φ ( y − x 0 ) + ε ≤ ‖ φ | p ⋅ p ( y − x 0 ) + ε ≤ p ( y − x 0 ) + ε p y 0 − x 0 ≤ φ − x 0 + ε = φ y − x 0 + ε ≤ ‖ φ p ⋅ p y − x 0 + ε ≤ p y − x 0 + ε {:[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}p(y0−x0)≤φ(−x0)+ε=φ(y−x0)+ε≤‖φ|p⋅p(y−x0)+ε≤p(y−x0)+ε
Taking the infimum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y, one finds
p ( y 0 − x 0 ) ≤ d p ( x 0 , Y ) + ε ; p y 0 − x 0 ≤ d p x 0 , Y + ε ; 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 ;p(y0−x0)≤dp(x0,Y)+ε;
so that y 0 ∈ P Y , ε ( x 0 ) y 0 ∈ P Y , ε x 0 y_(0)inP_(Y,epsi)(x_(0))y_{0} \in P_{Y, \varepsilon}\left(x_{0}\right)y0∈PY,ε(x0).
Similarly, defining ψ 0 : Z = Y + ⟨ x 0 ⟩ → R ψ 0 : Z = Y + x 0 → R psi_(0):Z=Y+(:x_(0):)rarrR\psi_{0}: Z=Y+\left\langle x_{0}\right\rangle \rightarrow \mathbb{R}ψ0:Z=Y+⟨x0⟩→R by ψ ( z ) = ψ 0 ( y + λ x 0 ) = λ , y ∈ Y ψ ( z ) = ψ 0 y + λ x 0 = λ , y ∈ Y psi(z)=psi_(0)(y+lambdax_(0))=lambda,y in Y\psi(z)=\psi_{0}\left(y+\lambda x_{0}\right)= \lambda, y \in Yψ(z)=ψ0(y+λx0)=λ,y∈Y and λ ∈ R λ ∈ R lambda inR\lambda \in \mathbb{R}λ∈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 ⊂ P Y , ε ( x 0 ) M ⊂ P Y , ε x 0 M subP_(Y,epsi)(x_(0))M \subset P_{Y, \varepsilon}\left(x_{0}\right)M⊂PY,ε(x0), if and only if there exists φ ∈ X b φ ∈ X b varphi inX^(b)\varphi \in X^{b}φ∈Xb verifying (a) (i)-(ii) and the condition:
φ ( − x 0 ) ≥ p ( u − x 0 ) − ε , for all u ∈ M ; φ − x 0 ≥ p u − x 0 − ε ,  for all  u ∈ M ; 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 ;φ(−x0)≥p(u−x0)−ε, for all u∈M;
(b') M ⊂ P Y , ε ( x 0 ) M ⊂ P Y , ε x 0 M subP_(Y,epsi)(x_(0))M \subset P_{Y, \varepsilon}\left(x_{0}\right)M⊂PY,ε(x0) if and only if there exists ψ ∈ X b ψ ∈ X b psi inX^(b)\psi \in X^{b}ψ∈Xb with properties (b) ( j ) − ( j j ) ( j ) − ( j j ) (j)-(jj)(j)-(j j)(j)−(jj), and verifying the condition:
ψ ( x 0 ) ≥ p ( x 0 − u ) − ε , for all u ∈ M . ψ x 0 ≥ p x 0 − u − ε ,  for all  u ∈ M .  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 {. }ψ(x0)≥p(x0−u)−ε, for all u∈M. 

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Received by the editors: June 11, 2003.
  1. ∗ ∗ ^(**){ }^{*}∗ 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.
2004

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