Best approximation in spaces with asymmetric norm

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

Abstract


In this paper we shall present some results on spaces with asymmetric seminorms, with emphasis on best approximation problems in such spaces.

Authors

Stefan Cobzas
Babes-Bolyai University, Cluj-Napoca, Romania

Costică Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy,  Romania

Keywords

Spaces with asymmetric norm; best approximation; Hahn-Banach theorem; characterization of best approximation.

Paper coordinates

S. Cobzas, C. Mustăţa, Best approximation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx. 35 (2006) no. 1, 17-31.

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Journal

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/2006-vol35-no1-art4

Print ISSN

2501-059X

Online ISSN

2457-6794

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2006-Mustata-Best approximation in spaces with asymmetric norm-Jnaat

BEST APPROXIMATION IN SPACES WITH ASYMMETRIC NORM

S. COBZAŞ* and C. MUSTĂŢA † † ^(†){ }^{\dagger}†

Abstract

In this paper we shall present some results on spaces with asymmetric seminorms, with emphasis on best approximation problems in such spaces.

MSC 2000. 41A65.
Keywords. Spaces with asymmetric norm, best approximation, Hahn-Banach theorem, characterization of best approximation.

1. INTRODUCTION

Let X X XXX be a real vector space. An asymmetric seminorm on X X XXX is a positive sublinear functional p : X → [ 0 , ∞ ) p : X → [ 0 , ∞ ) p:X rarr[0,oo)p: X \rightarrow[0, \infty)p:X→[0,∞), i.e. p p ppp satisfies the conditions:
(AN1) p ( x ) ≥ 0 p ( x ) ≥ 0 p(x) >= 0p(x) \geq 0p(x)≥0;
(AN2) 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);
(AN3) 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 and t ≥ 0 t ≥ 0 t >= 0t \geq 0t≥0.
The function 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 positive sublinear functional on X X XXX, called the conjugate of p p ppp, and
(1.1) p s ( x ) = max { p ( x ) , p ( − x ) } , x ∈ X (1.1) p s ( x ) = max { p ( x ) , p ( − x ) } , x ∈ X {:(1.1)p^(s)(x)=max{p(x)","p(-x)}","x in X:}\begin{equation*} p^{s}(x)=\max \{p(x), p(-x)\}, x \in X \tag{1.1} \end{equation*}(1.1)ps(x)=max{p(x),p(−x)},x∈X
is a seminorm on X X XXX and the inequalities
(1.2) | p ( x ) − p ( y ) | ≤ p s ( x − y ) and | p ¯ ( x ) − p ¯ ( y ) | ≤ p s ( x − y ) (1.2) | p ( x ) − p ( y ) | ≤ p s ( x − y )  and  | p ¯ ( x ) − p ¯ ( y ) | ≤ p s ( x − y ) {:(1.2)|p(x)-p(y)| <= p^(s)(x-y)quad" and "quad| bar(p)(x)- bar(p)(y)| <= p^(s)(x-y):}\begin{equation*} |p(x)-p(y)| \leq p^{s}(x-y) \quad \text { and } \quad|\bar{p}(x)-\bar{p}(y)| \leq p^{s}(x-y) \tag{1.2} \end{equation*}(1.2)|p(x)−p(y)|≤ps(x−y) and |p¯(x)−p¯(y)|≤ps(x−y)
hold for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X. If the seminorm p s p s p^(s)p^{s}ps is a norm on X X XXX then we say that p p ppp is an asymmetric norm on X X XXX. This means that, beside (AN1)-(AN3), it satisfies also the condition
(AN4) 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 pair ( X , p X , p X,pX, pX,p ), where X X XXX is a linear space and p p ppp is an asymmetric seminorm on X X XXX is called a space with asymmetric seminorm, respectively a space with asymmetric norm, if p p ppp is an asymmetric norm.
The function ρ : X × X → [ 0 ; ∞ ) ρ : X × X → [ 0 ; ∞ ) rho:X xx X rarr[0;oo)\rho: X \times X \rightarrow[0 ; \infty)ρ:X×X→[0;∞) defined by ρ ( x , y ) = p ( y − x ) , x , y ∈ X ρ ( x , y ) = p ( y − x ) , x , y ∈ X rho(x,y)=p(y-x),x,y in X\rho(x, y)=p(y-x), x, y \in Xρ(x,y)=p(y−x),x,y∈X, is an asymmetric semimetric on X X XXX. Denote by
B p ′ ( x , r ) = { x ′ ∈ X : p ( x ′ − x ) < r } B p ′ ( x , r ) = x ′ ∈ X : p x ′ − x < r B_(p)^(')(x,r)={x^(')in X:p(x^(')-x) < r}B_{p}^{\prime}(x, r)=\left\{x^{\prime} \in X: p\left(x^{\prime}-x\right)<r\right\}Bp′(x,r)={x′∈X:p(x′−x)<r} and B p ( x , r ) = { x ′ ∈ X : p ( x ′ − x ) ≤ r } B p ( x , r ) = x ′ ∈ X : p x ′ − x ≤ r B_(p)(x,r)={x^(')in X:p(x^(')-x) <= r}B_{p}(x, r)=\left\{x^{\prime} \in X: p\left(x^{\prime}-x\right) \leq r\right\}Bp(x,r)={x′∈X:p(x′−x)≤r}, the open, respectively closed, ball in X X XXX of center x x xxx and radius r > 0 r > 0 r > 0r>0r>0. Denoting by
B p ′ = B p ′ ( 0 , 1 ) and B p = B p ( 0 , 1 ) , B p ′ = B p ′ ( 0 , 1 )  and  B p = B p ( 0 , 1 ) , B_(p)^(')=B_(p)^(')(0,1)" and "B_(p)=B_(p)(0,1),B_{p}^{\prime}=B_{p}^{\prime}(0,1) \text { and } B_{p}=B_{p}(0,1),Bp′=Bp′(0,1) and Bp=Bp(0,1),
the corresponding unit balls then
B p ′ ( x , r ) = x + r B p ′ and B p ( x , r ) = x + r B p B p ′ ( x , r ) = x + r B p ′  and  B p ( x , r ) = x + r B p B_(p)^(')(x,r)=x+rB_(p)^(')" and "B_(p)(x,r)=x+rB_(p)B_{p}^{\prime}(x, r)=x+r B_{p}^{\prime} \text { and } B_{p}(x, r)=x+r B_{p}Bp′(x,r)=x+rBp′ and Bp(x,r)=x+rBp
The unit balls B p ′ B p ′ B_(p)^(')B_{p}^{\prime}Bp′ and B p B p B_(p)B_{p}Bp are convex absorbing subsets of the space X X XXX and p p ppp agrees with the Minkowski functional associated to any of them. Recall that for an absorbing subset C C CCC of X X XXX the Minkowski functional p C : X → [ 0 ; ∞ ) p C : X → [ 0 ; ∞ ) p_(C):X rarr[0;oo)p_{C}: X \rightarrow[0 ; \infty)pC:X→[0;∞) is defined by
p C ( x ) = inf { t > 0 : x ∈ t C } p C ( x ) = inf { t > 0 : x ∈ t C } p_(C)(x)=i n f{t > 0:x in tC}p_{C}(x)=\inf \{t>0: x \in t C\}pC(x)=inf{t>0:x∈tC}
If C C CCC is absorbing and convex, then p C p C p_(C)p_{C}pC is a positive sublinear functional, and
{ x ∈ X : p C ( x ) < 1 } ⊂ C ⊂ { x ∈ X : p C ( x ) ≤ 1 } x ∈ X : p C ( x ) < 1 ⊂ C ⊂ x ∈ X : p C ( x ) ≤ 1 {x in X:p_(C)(x) < 1}sub C sub{x in X:p_(C)(x) <= 1}\left\{x \in X: p_{C}(x)<1\right\} \subset C \subset\left\{x \in X: p_{C}(x) \leq 1\right\}{x∈X:pC(x)<1}⊂C⊂{x∈X:pC(x)≤1}
An asymmetric seminorm p p ppp generates a topology τ p τ p tau_(p)\tau_{p}τp on X X XXX, having as basis of neighborhoods of a point x ∈ X x ∈ X x in Xx \in Xx∈X the family { B p ′ ( x , r ) : r > 0 } B p ′ ( x , r ) : r > 0 {B_(p)^(')(x,r):r > 0}\left\{B_{p}^{\prime}(x, r): r>0\right\}{Bp′(x,r):r>0} of open p p ppp-balls. The family { B p ( x , r ) : r > 0 } B p ( x , r ) : r > 0 {B_(p)(x,r):r > 0}\left\{B_{p}(x, r): r>0\right\}{Bp(x,r):r>0} of closed p p ppp-balls is also a neighborhood basis at x x xxx for τ p τ p tau_(p)\tau_{p}τp.
The topology τ p τ p tau_(p)\tau_{p}τp is translation invariant, i.e. the addition + : X × X → X + : X × X → X +:X xx X rarr X+: X \times X \rightarrow X+:X×X→X is continuous, but the multiplication by scalars ⋅ : R × X → X ⋅ : R × X → X *:Rxx X rarr X\cdot: \mathbb{R} \times X \rightarrow X⋅:R×X→X need not be continuous, as it is shown by some examples (see [7]).
The ball B p ′ ( x , r ) B p ′ ( x , r ) B_(p)^(')(x,r)B_{p}^{\prime}(x, r)Bp′(x,r) is τ p τ p tau_(p)\tau_{p}τp-open but the ball B p ( x , r ) B p ( x , r ) B_(p)(x,r)B_{p}(x, r)Bp(x,r) need not to be τ p τ p tau_(p)\tau_{p}τp-closed, as can be seen from the following typical example.
Example 1.1. Consider on R R R\mathbb{R}R the asymmetric seminorm u ( α ) = max { α , 0 } u ( α ) = max { α , 0 } u(alpha)=max{alpha,0}u(\alpha)=\max \{\alpha, 0\}u(α)=max{α,0}, α ∈ R α ∈ R alpha inR\alpha \in \mathbb{R}α∈R, and denote by R u R u R_(u)\mathbb{R}_{u}Ru the space R R R\mathbb{R}R equipped with the topology τ u τ u tau_(u)\tau_{u}τu generated by u u uuu. The conjugate seminorm is u ¯ ( α ) = − min { α , 0 } u ¯ ( α ) = − min { α , 0 } bar(u)(alpha)=-min{alpha,0}\bar{u}(\alpha)=-\min \{\alpha, 0\}u¯(α)=−min{α,0}, and u s ( α ) = max { u ( α ) , u ¯ ( α ) } = | α | u s ( α ) = max { u ( α ) , u ¯ ( α ) } = | α | u^(s)(alpha)=max{u(alpha), bar(u)(alpha)}=|alpha|u^{s}(\alpha)= \max \{u(\alpha), \bar{u}(\alpha)\}=|\alpha|us(α)=max{u(α),u¯(α)}=|α|. The topology τ u τ u tau_(u)\tau_{u}τu, called the upper topology of R R R\mathbb{R}R, is generated by the intervals of the form ( − ∞ ; a ) , a ∈ R ( − ∞ ; a ) , a ∈ R (-oo;a),a inR(-\infty ; a), a \in \mathbb{R}(−∞;a),a∈R, and the family { ( − ∞ ; α + ϵ ) : ϵ > 0 } { ( − ∞ ; α + ϵ ) : ϵ > 0 } {(-oo;alpha+epsilon):epsilon > 0}\{(-\infty ; \alpha+\epsilon): \epsilon>0\}{(−∞;α+ϵ):ϵ>0} is a neighborhood basis of every point α ∈ R α ∈ R alpha inR\alpha \in \mathbb{R}α∈R. The set ( − ∞ ; 1 ) = B u ′ ( 0 , 1 ) ( − ∞ ; 1 ) = B u ′ ( 0 , 1 ) (-oo;1)=B_(u)^(')(0,1)(-\infty ; 1)=B_{u}^{\prime}(0,1)(−∞;1)=Bu′(0,1) is τ u τ u tau_(u)\tau_{u}τu-open, and the ball B u ( 0 , 1 ) = ( − ∞ ; 1 ] B u ( 0 , 1 ) = ( − ∞ ; 1 ] B_(u)(0,1)=(-oo;1]B_{u}(0,1)=(-\infty ; 1]Bu(0,1)=(−∞;1] is not τ u τ u tau_(u)\tau_{u}τu-closed because R ∖ B u ( 0 , 1 ) = ( 1 ; ∞ ) R ∖ B u ( 0 , 1 ) = ( 1 ; ∞ ) R\\B_(u)(0,1)=(1;oo)\mathbb{R} \backslash B_{u}(0,1)=(1 ; \infty)R∖Bu(0,1)=(1;∞) is not τ u τ u tau_(u)\tau_{u}τu-open.
The topology τ p τ p tau_(p)\tau_{p}τp could not be Hausdorff even if p p ppp is an asymmetric norm on X X XXX. A necessary and sufficient condition in order that τ p τ p tau_(p)\tau_{p}τp be Hausdorff was given in [22]. Putting
(1.3) p ~ ( x ) = inf { p ( x ′ ) + p ( x ′ − x ) : x ′ ∈ X } , x ∈ X , (1.3) p ~ ( x ) = inf p x ′ + p x ′ − x : x ′ ∈ X , x ∈ X , {:(1.3) tilde(p)(x)=i n f{p(x^('))+p(x^(')-x):x^(')in X}","x in X",":}\begin{equation*} \tilde{p}(x)=\inf \left\{p\left(x^{\prime}\right)+p\left(x^{\prime}-x\right): x^{\prime} \in X\right\}, x \in X, \tag{1.3} \end{equation*}(1.3)p~(x)=inf{p(x′)+p(x′−x):x′∈X},x∈X,
it follows that p ~ p ~ tilde(p)\tilde{p}p~ is the greatest (symmetric) seminorm majorized by p p ppp and the topology τ p τ p tau_(p)\tau_{p}τp is Hausdorff if and only if p ~ ( x ) > 0 p ~ ( x ) > 0 tilde(p)(x) > 0\tilde{p}(x)>0p~(x)>0 for every x ≠ 0 x ≠ 0 x!=0x \neq 0x≠0. Changing x x xxx
to − x − x -x-x−x and taking x ′ = 0 x ′ = 0 x^(')=0x^{\prime}=0x′=0 it follows that, in this case, p ( x ) > 0 p ( x ) > 0 p(x) > 0p(x)>0p(x)>0 for every x ≠ 0 x ≠ 0 x!=0x \neq 0x≠0, but this condition is not sufficient for τ p τ p tau_(p)\tau_{p}τp to be Hausdorff, see [22].
Spaces with asymmetric seminorms were investigated in a series of papers, emphasizing similarities with seminormed spaces as well as differences, see [1, 2, 3, 7, 17, 18, 19, 21, 22, and the references quoted therein. Among the differences we mention the fact that the dual of a space with asymmetric seminorm is not a linear space but merely a convex cone in the algebraic dual X # X # X^(#)X^{\#}X# of X X XXX. This is due to the fact that the continuity of a linear functional φ φ varphi\varphiφ on ( X , p X , p X,pX, pX,p ) does not imply the continuity of − φ − φ -varphi-\varphi−φ. For instance, φ ( u ) = u φ ( u ) = u varphi(u)=u\varphi(u)=uφ(u)=u is continuous on ( R , u R , u R,u\mathbb{R}, uR,u ) but ψ ( u ) = − u ψ ( u ) = − u psi(u)=-u\psi(u)=-uψ(u)=−u is not continuous. For an other example see [7]. The study of spaces with asymmetric norm was motivated and stimulated also by their applications in the complexity of algorithms, see [20, 40].
Some continuity properties of linear functionals in the symmetric case have their analogs in the asymmetric one.
Proposition 1.2. [21] Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm and φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R a linear functional. Then the following are equivalent.
(1) φ φ varphi\varphiφ is τ p − τ u τ p − τ u tau_(p)-tau_(u)\tau_{p}-\tau_{u}τp−τu-continuous at 0 ∈ X 0 ∈ X 0in X0 \in X0∈X.
(2) φ φ varphi\varphiφ is τ p − τ u τ p − τ u tau_(p)-tau_(u)\tau_{p}-\tau_{u}τp−τu-continuous on X X XXX.
(3) There exists L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 such that
(1.4) ∀ x ∈ X , φ ( x ) ≤ L p ( x ) . (1.4) ∀ x ∈ X , φ ( x ) ≤ L p ( x ) . {:(1.4)AA x in X","quad varphi(x) <= Lp(x).:}\begin{equation*} \forall x \in X, \quad \varphi(x) \leq L p(x) . \tag{1.4} \end{equation*}(1.4)∀x∈X,φ(x)≤Lp(x).
(4) φ φ varphi\varphiφ is upper semi-continuous from ( X , τ p ) X , τ p (X,tau_(p))\left(X, \tau_{p}\right)(X,τp) to ( R , | | ) ( R , | | ) (R,||)(\mathbb{R},| |)(R,||).
A linear functional satisfying (1.4) is called semi-Lipschitz (or p-bounded) and L L LLL a semi-Lipschitz constant. Denote by X p b X p b X_(p)^(b)X_{p}^{b}Xpb the set of all bounded linear functionals on the space with asymmetric seminorm ( X , p X , p X,pX, pX,p ). As we did mention, X p b X p b X_(p)^(b)X_{p}^{b}Xpb is a convex cone in X # X # X^(#)X^{\#}X#.
One can define a norm ‖ | p ‖ p |||_(p)\|\left.\right|_{p}‖|p on X p b X p b X_(p)^(b)X_{p}^{b}Xpb by
(1.5) ‖ φ | p = sup { φ ( x ) : x ∈ B p } , φ ∈ X p b (1.5) ‖ φ p = sup φ ( x ) : x ∈ B p , φ ∈ X p b {:(1.5)|| varphi|_(p)=s u p{varphi(x):x inB_(p)}","varphi inX_(p)^(b):}\begin{equation*} \|\left.\varphi\right|_{p}=\sup \left\{\varphi(x): x \in B_{p}\right\}, \varphi \in X_{p}^{b} \tag{1.5} \end{equation*}(1.5)‖φ|p=sup{φ(x):x∈Bp},φ∈Xpb
Some useful properties of this norm, whose proofs can be found in [9, 12], are collected in the following proposition. We agree to call a linear functional φ φ varphi\varphiφ on ( X , p X , p X,pX, pX,p ), ( p , p ¯ p , p ¯ p, bar(p)p, \bar{p}p,p¯ )-bounded if it is both p p ppp - and p ¯ p ¯ bar(p)\bar{p}p¯-bounded, where p ¯ p ¯ bar(p)\bar{p}p¯ is the seminorm conjugate to p p ppp.
Proposition 1.3. If φ φ varphi\varphiφ is a bounded linear functional on a space with asymmetric seminorm ( X , p X , p X,pX, pX,p ), p ≠ 0 p ≠ 0 p!=0p \neq 0p≠0, then the following assertions hold.
(1) ‖ φ | p ‖ φ p || varphi|_(p)\|\left.\varphi\right|_{p}‖φ|p is the smallest of the numbers L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0 for which the inequality (1.4) holds.
(2) We have
(1.6) ‖ φ | p = sup { φ ( x ) / p ( x ) : x ∈ X , p ( x ) > 0 } (1.7) = sup { φ ( x ) : x ∈ X , p ( x ) < 1 } (1.8) = sup { φ ( x ) : x ∈ X , p ( x ) = 1 } . (1.6) ‖ φ p = sup { φ ( x ) / p ( x ) : x ∈ X , p ( x ) > 0 } (1.7) = sup { φ ( x ) : x ∈ X , p ( x ) < 1 } (1.8) = sup { φ ( x ) : x ∈ X , p ( x ) = 1 } . {:[(1.6)|| varphi|_(p)=s u p{varphi(x)//p(x):x in X","p(x) > 0}],[(1.7)=s u p{varphi(x):x in X","p(x) < 1}],[(1.8)=s u p{varphi(x):x in X","p(x)=1}.]:}\begin{align*} \|\left.\varphi\right|_{p} & =\sup \{\varphi(x) / p(x): x \in X, p(x)>0\} \tag{1.6}\\ & =\sup \{\varphi(x): x \in X, p(x)<1\} \tag{1.7}\\ & =\sup \{\varphi(x): x \in X, p(x)=1\} . \tag{1.8} \end{align*}(1.6)‖φ|p=sup{φ(x)/p(x):x∈X,p(x)>0}(1.7)=sup{φ(x):x∈X,p(x)<1}(1.8)=sup{φ(x):x∈X,p(x)=1}.
(3) If φ ≠ 0 φ ≠ 0 varphi!=0\varphi \neq 0φ≠0, then ‖ φ | p > 0 ‖ φ p > 0 || varphi|_(p) > 0\|\left.\varphi\right|_{p}>0‖φ|p>0. Also, if φ ≠ 0 φ ≠ 0 varphi!=0\varphi \neq 0φ≠0 and φ ( x 0 ) = ‖ φ | p φ x 0 = ‖ φ p varphi(x_(0))=|| varphi|_(p)\varphi\left(x_{0}\right)=\|\left.\varphi\right|_{p}φ(x0)=‖φ|p for some x 0 ∈ B p x 0 ∈ B p x_(0)inB_(p)x_{0} \in B_{p}x0∈Bp, then p ( x 0 ) = 1 p x 0 = 1 p(x_(0))=1p\left(x_{0}\right)=1p(x0)=1.
(4) If φ φ varphi\varphiφ is ( p , p ¯ p , p ¯ p, bar(p)p, \bar{p}p,p¯ )-bounded, then
φ ( r B p ′ ) = ( − r ‖ φ | p ¯ , r ‖ φ | p ) and φ ( r B p ¯ ′ ) = ( − r ‖ φ | p , r ‖ φ | p ¯ ) where B p ′ = { x ∈ X : p ( x ) < 1 } , B p ¯ ′ = { x ∈ X : p ¯ ( x ) < 1 } and r > 0 . φ r B p ′ = − r φ p ¯ , r φ p  and  φ r B p ¯ ′ = − r φ p , r φ p ¯  where  B p ′ = { x ∈ X : p ( x ) < 1 } , B p ¯ ′ = { x ∈ X : p ¯ ( x ) < 1 }  and  r > 0 . {:[varphi(rB_(p)^('))=(-r|| varphi|_( bar(p)),r||varphi|_(p))quad" and "quad varphi(rB_( bar(p))^('))=(-r|| varphi|_(p),r||varphi|_( bar(p)))],[" where "B_(p)^(')={x in X:p(x) < 1}","B_( bar(p))^(')={x in X: bar(p)(x) < 1}" and "r > 0.]:}\begin{gathered} \varphi\left(r B_{p}^{\prime}\right)=\left(-\left.r\left\|\left.\varphi\right|_{\bar{p}}, r\right\| \varphi\right|_{p}\right) \quad \text { and } \quad \varphi\left(r B_{\bar{p}}^{\prime}\right)=\left(-\left.r\left\|\left.\varphi\right|_{p}, r\right\| \varphi\right|_{\bar{p}}\right) \\ \text { where } B_{p}^{\prime}=\{x \in X: p(x)<1\}, B_{\bar{p}}^{\prime}=\{x \in X: \bar{p}(x)<1\} \text { and } r>0 . \end{gathered}φ(rBp′)=(−r‖φ|p¯,r‖φ|p) and φ(rBp¯′)=(−r‖φ|p,r‖φ|p¯) where Bp′={x∈X:p(x)<1},Bp¯′={x∈X:p¯(x)<1} and r>0.
(5) If φ φ varphi\varphiφ is p p ppp-bounded but not p ¯ p ¯ bar(p)\bar{p}p¯-bounded, then
φ ( r B p ′ ) = ( − ∞ , r ‖ φ | p ) φ r B p ′ = − ∞ , r ‖ φ p varphi(rB_(p)^('))=(-oo,r|| varphi|_(p))\varphi\left(r B_{p}^{\prime}\right)=\left(-\infty, r \|\left.\varphi\right|_{p}\right)φ(rBp′)=(−∞,r‖φ|p)
Remark 1.4. A linear functional φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R is ( p , p ¯ ) ( p , p ¯ ) (p, bar(p))(p, \bar{p})(p,p¯)-bounded if and only if
(1.9) ∀ x ∈ X , | φ ( x ) | ≤ L p ( x ) (1.9) ∀ x ∈ X , | φ ( x ) | ≤ L p ( x ) {:(1.9)AA x in X","quad|varphi(x)| <= Lp(x):}\begin{equation*} \forall x \in X, \quad|\varphi(x)| \leq L p(x) \tag{1.9} \end{equation*}(1.9)∀x∈X,|φ(x)|≤Lp(x)
for some L ≥ 0 L ≥ 0 L >= 0L \geq 0L≥0.
Indeed, if L 1 , L 2 ≥ 0 L 1 , L 2 ≥ 0 L_(1),L_(2) >= 0L_{1}, L_{2} \geq 0L1,L2≥0 are such that
φ ( x ) ≤ L 1 p ( x ) and φ ( x ) ≤ L 2 p ( − x ) , φ ( x ) ≤ L 1 p ( x )  and  φ ( x ) ≤ L 2 p ( − x ) , varphi(x) <= L_(1)p(x)quad" and "quad varphi(x) <= L_(2)p(-x),\varphi(x) \leq L_{1} p(x) \quad \text { and } \quad \varphi(x) \leq L_{2} p(-x),φ(x)≤L1p(x) and φ(x)≤L2p(−x),
for all x ∈ X x ∈ X x in Xx \in Xx∈X, then − φ ( x ) = φ ( − x ) ≤ L 2 p ( x ) , x ∈ X − φ ( x ) = φ ( − x ) ≤ L 2 p ( x ) , x ∈ X -varphi(x)=varphi(-x) <= L_(2)p(x),x in X-\varphi(x)=\varphi(-x) \leq L_{2} p(x), x \in X−φ(x)=φ(−x)≤L2p(x),x∈X, so (1.9) holds with L = max { L 1 , L 2 } L = max L 1 , L 2 L=max{L_(1),L_(2)}L=\max \left\{L_{1}, L_{2}\right\}L=max{L1,L2}.
Denote by X p ¯ b X p ¯ b X_( bar(p))^(b)X_{\bar{p}}^{b}Xp¯b the dual cone to ( X , p ¯ X , p ¯ X, bar(p)X, \bar{p}X,p¯ ) and let X ∗ X ∗ X^(**)X^{*}X∗ be the conjugate of the seminormed space ( X , p s ) X , p s (X,p^(s))\left(X, p^{s}\right)(X,ps), where p s p s p^(s)p^{s}ps is the symmetric seminorm associated to p p ppp and p ¯ p ¯ bar(p)\bar{p}p¯ (see (1.1)).
Since
φ ( x ) ≤ L p ( x ) ≤ L p s ( x ) , x ∈ X , φ ( x ) ≤ L p ( x ) ≤ L p s ( x ) , x ∈ X , varphi(x) <= Lp(x) <= Lp^(s)(x),x in X,\varphi(x) \leq L p(x) \leq L p^{s}(x), x \in X,φ(x)≤Lp(x)≤Lps(x),x∈X,
implies | φ ( x ) | ≤ L p ( x ) , x ∈ X | φ ( x ) | ≤ L p ( x ) , x ∈ X |varphi(x)| <= Lp(x),x in X|\varphi(x)| \leq L p(x), x \in X|φ(x)|≤Lp(x),x∈X, it follows that X p b X p b X_(p)^(b)X_{p}^{b}Xpb is contained in the dual X ∗ X ∗ X^(**)X^{*}X∗ of ( X , p s X , p s X,p^(s)X, p^{s}X,ps ). Similarly, X p ¯ b X p ¯ b X_( bar(p))^(b)X_{\bar{p}}^{b}Xp¯b is contained in X ∗ X ∗ X^(**)X^{*}X∗ too.
For x ∗ ∈ X ∗ x ∗ ∈ X ∗ x^(**)inX^(**)x^{*} \in X^{*}x∗∈X∗ put
‖ x ∗ ‖ = sup { x ∗ ( x ) : x ∈ X , p s ( x ) ≤ 1 } . x ∗ = sup x ∗ ( x ) : x ∈ X , p s ( x ) ≤ 1 . ||x^(**)||=s u p{x^(**)(x):x in X,p^(s)(x) <= 1}.\left\|x^{*}\right\|=\sup \left\{x^{*}(x): x \in X, p^{s}(x) \leq 1\right\} .‖x∗‖=sup{x∗(x):x∈X,ps(x)≤1}.
Then ‖ ‖ ‖ ||||:}\left\|\|\right.‖‖ is a norm on X ∗ X ∗ X^(**)X^{*}X∗ and X ∗ X ∗ X^(**)X^{*}X∗ is complete with respect to this norm, i.e. is a Banach space (even if p s p s p^(s)p^{s}ps is not a norm, see [11]).
Proposition 1.5. Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm.
(1) The cones X p b X p b X_(p)^(b)X_{p}^{b}Xpb and X p ¯ b X p ¯ b X_( bar(p))^(b)X_{\bar{p}}^{b}Xp¯b are contained in X ∗ X ∗ X^(**)X^{*}X∗ and
‖ φ | p = ‖ φ ‖ , φ ∈ X p b and ‖ ψ | p ¯ = ‖ ψ ‖ , ψ ∈ X p ¯ b φ p = φ ‖ , φ ∈ X p b  and  ψ p ¯ = ψ ‖ , ψ ∈ X p ¯ b || varphi|_(p)=||varphi||,varphi inX_(p)^(b)quad" and "quad|| psi|_( bar(p))=||psi||,psi inX_( bar(p))^(b)\left\|\left.\varphi\right|_{p}=\right\| \varphi \|, \varphi \in X_{p}^{b} \quad \text { and } \quad\left\|\left.\psi\right|_{\bar{p}}=\right\| \psi \|, \psi \in X_{\bar{p}}^{b}‖φ|p=‖φ‖,φ∈Xpb and ‖ψ|p¯=‖ψ‖,ψ∈Xp¯b
(2) We have ‖ φ | p = ‖ − φ | p ¯ φ p = − φ p ¯ || varphi|_(p)=||- varphi|_( bar(p))\left\|\left.\varphi\right|_{p}=\right\|-\left.\varphi\right|_{\bar{p}}‖φ|p=‖−φ|p¯, so that
φ ∈ X p b and ‖ φ | p ≤ r ⟺ − φ ∈ X p ¯ b and ‖ − φ | p ¯ ≤ r . φ ∈ X p b  and  ‖ φ p ≤ r ⟺ − φ ∈ X p ¯ b  and  ‖ − φ p ¯ ≤ r .  varphi inX_(p)^(b)" and "|| varphi|_(p) <= r Longleftrightarrow-varphi inX_( bar(p))^(b)" and "||- varphi|_( bar(p)) <= r". "\varphi \in X_{p}^{b} \text { and } \|\left.\varphi\right|_{p} \leq r \Longleftrightarrow-\varphi \in X_{\bar{p}}^{b} \text { and } \|-\left.\varphi\right|_{\bar{p}} \leq r \text {. }φ∈Xpb and ‖φ|p≤r⟺−φ∈Xp¯b and ‖−φ|p¯≤r. 
The properties of the dual space X p b X p b X_(p)^(b)X_{p}^{b}Xpb were investigated in [21] where, among other things, the analog of the weak* topology of X X XXX was defined. This is denoted by w b w b w^(b)w^{b}wb and has a neighborhood basis at a point φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb, the family
V x 1 , … , x n ; ϵ ( φ ) = { ψ ∈ X p b : ψ ( x k ) − φ ( x k ) < ϵ , k = 1 , … , n } V x 1 , … , x n ; ϵ ( φ ) = ψ ∈ X p b : ψ x k − φ x k < ϵ , k = 1 , … , n V_(x_(1),dots,x_(n);epsilon)(varphi)={psi inX_(p)^(b):psi(x_(k))-varphi(x_(k)) < epsilon,k=1,dots,n}V_{x_{1}, \ldots, x_{n} ; \epsilon}(\varphi)=\left\{\psi \in X_{p}^{b}: \psi\left(x_{k}\right)-\varphi\left(x_{k}\right)<\epsilon, k=1, \ldots, n\right\}Vx1,…,xn;ϵ(φ)={ψ∈Xpb:ψ(xk)−φ(xk)<ϵ,k=1,…,n}
for n ∈ N , x 1 , … , x n ∈ X n ∈ N , x 1 , … , x n ∈ X n inN,x_(1),dots,x_(n)in Xn \in \mathbb{N}, x_{1}, \ldots, x_{n} \in Xn∈N,x1,…,xn∈X and ϵ > 0 ϵ > 0 epsilon > 0\epsilon>0ϵ>0. The w b w b w^(b)w^{b}wb-convergence of a net ( φ i : i ∈ I φ i : i ∈ I varphi_(i):i in I\varphi_{i}: i \in Iφi:i∈I ) in X p b X p b X_(p)^(b)X_{p}^{b}Xpb to φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb can be characterized in the following way
φ i → w b φ ⟺ ∀ x ∈ X , φ i ( x ) → φ ( x ) in ( R , u ) . φ i → w b φ ⟺ ∀ x ∈ X , φ i ( x ) → φ ( x )  in  ( R , u ) .  varphi_(i)rarr"w^(b)"varphi Longleftrightarrow AA x in X,varphi_(i)(x)rarr varphi(x)" in "(R,u)". "\varphi_{i} \xrightarrow{w^{b}} \varphi \Longleftrightarrow \forall x \in X, \varphi_{i}(x) \rightarrow \varphi(x) \text { in }(\mathbb{R}, u) \text {. }φi→wbφ⟺∀x∈X,φi(x)→φ(x) in (R,u). 
It was shown that w b w b w^(b)w^{b}wb is the restriction of the topology w ∗ = σ ( X ∗ , X ) w ∗ = σ X ∗ , X w^(**)=sigma(X^(**),X)w^{*}=\sigma\left(X^{*}, X\right)w∗=σ(X∗,X) on X ∗ X ∗ X^(**)X^{*}X∗ to X p b X p b X_(p)^(b)X_{p}^{b}Xpb (see [21]). This study was continued in [9] where separation theorems for convex sets and a Krein-Milman type theorem were proved. In 10 asymmetric locally convex spaces were introduced and their basic properties were studied.
Another direction of investigation is that of best approximation in spaces with asymmetric seminorm. Due to the asymmetry of the seminorm we have two distances. For a nonempty subset Y Y YYY of a space with asymmetric seminorm ( X , p X , p X,pX, pX,p ) and x ∈ X x ∈ X x in Xx \in Xx∈X put
(1.10) d p ( x , Y ) = inf { p ( y − x ) : y ∈ Y } (1.10) d p ( x , Y ) = inf { p ( y − x ) : y ∈ Y } {:(1.10)d_(p)(x","Y)=i n f{p(y-x):y in Y}:}\begin{equation*} d_{p}(x, Y)=\inf \{p(y-x): y \in Y\} \tag{1.10} \end{equation*}(1.10)dp(x,Y)=inf{p(y−x):y∈Y}
and
(1.11) d p ( Y , x ) = inf { p ( x − y ) : y ∈ Y } . (1.11) d p ( Y , x ) = inf { p ( x − y ) : y ∈ Y } . {:(1.11)d_(p)(Y","x)=i n f{p(x-y):y in Y}.:}\begin{equation*} d_{p}(Y, x)=\inf \{p(x-y): y \in Y\} . \tag{1.11} \end{equation*}(1.11)dp(Y,x)=inf{p(x−y):y∈Y}.
Note that d p ( Y , x ) = d p ¯ ( x , Y ) d p ( Y , x ) = d p ¯ ( x , Y ) d_(p)(Y,x)=d_( bar(p))(x,Y)d_{p}(Y, x)=d_{\bar{p}}(x, Y)dp(Y,x)=dp¯(x,Y).
Duality formulae and characterization results for best approximation in spaces with asymmetric norm were obtained in [5, 6, 9, 12, 34, 35, The papers [32, 33, 39] are concerned with best approximation in spaces of semiLipschitz functions defined on asymmetric metric spaces (called quasi-metric spaces) in connection with the extension properties of these functions. In the papers [13, 24, 25, 36, supposing that p p ppp is the Minkowski functional p C p C p_(C)p_{C}pC of a bounded convex body C C CCC in a normed space ( X , ‖ ‖ X , ‖ ‖ X,||||X,\| \|X,‖‖ ), some generic existence results for best approximation with respect to the asymmetric norm p C p C p_(C)p_{C}pC were proved, extending similar results from the normed case. As in the symmetric case, the geometric properties of the body C C CCC (or, equivalently, of the functional p C p C p_(C)p_{C}pC ) are essential. A study of the moduli of convexity and smoothness corresponding to p C p C p_(C)p_{C}pC is done in [43].
Best approximation with respect to some asymmetric norms in concrete function spaces of continuous or of integrable functions, called sign-sensitive approximation, was also studied in a series of papers, see [14, 15, 16, 41, the references quoted therein, and the monograph by Krein and Nudelman [23, Ch. 9, §5]).
The present paper, which can be viewed as a sequel to 12 and [9], is concerned mainly with characterizations of the elements of best approximation in a subspace Y Y YYY of a space with asymmetric norm ( X , p X , p X,pX, pX,p ) and duality results for best approximation. As in the case of (symmetric) normed spaces the characterizations will be done in terms of some linear bounded functionals vanishing on Y Y YYY. The duality results will involve the annihilator in X p b X p b X_(p)^(b)X_{p}^{b}Xpb of the subspace Y Y YYY. For this reason we start by recalling some extension results for bounded linear functionals on spaces with asymmetric seminorm. For proofs, all resorting to the classical Hahn-Banach extension theorem, see [9, 12].
Theorem 1.6. Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm and Y Y YYY a linear subspace of X X XXX. If φ 0 : Y → R φ 0 : Y → R varphi_(0):Y rarrR\varphi_{0}: Y \rightarrow \mathbb{R}φ0:Y→R is a linear p p ppp-bounded functional on Y Y YYY then there exists a p p ppp-bounded linear functional φ φ varphi\varphiφ defined on the whole X X XXX such that
φ | Y = φ 0 and ‖ φ | p = ‖ φ 0 | p . φ Y = φ 0  and  φ p = φ 0 p . varphi|_(Y)=varphi_(0)quad" and " quad|| varphi|_(p)=||varphi_(0)|_(p).\left.\varphi\right|_{Y}=\varphi_{0} \quad \text { and }\left.\quad\left\|\left.\varphi\right|_{p}=\right\| \varphi_{0}\right|_{p} .φ|Y=φ0 and ‖φ|p=‖φ0|p.
We agree to call a functional φ φ varphi\varphiφ satisfying the conclusions of the above theorem a norm preserving extension of φ 0 φ 0 varphi_(0)\varphi_{0}φ0.
Based on this extension result one can prove the following existence result.
Proposition 1.7. Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm and x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X such that p ( x 0 ) > 0 p x 0 > 0 p(x_(0)) > 0p\left(x_{0}\right)>0p(x0)>0. Then there exists a p p ppp-bounded linear functional φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R such that
‖ φ | p = 1 and φ ( x 0 ) = p ( x 0 ) . ‖ φ p = 1  and  φ x 0 = p x 0 . || varphi|_(p)=1quad" and "quad varphi(x_(0))=p(x_(0)).\|\left.\varphi\right|_{p}=1 \quad \text { and } \quad \varphi\left(x_{0}\right)=p\left(x_{0}\right) .‖φ|p=1 and φ(x0)=p(x0).
In its turn, this proposition has the following corollary.
Corollary 1.8. If p ( x 0 ) > 0 p x 0 > 0 p(x_(0)) > 0p\left(x_{0}\right)>0p(x0)>0 then
p ( x 0 ) = sup { φ ( x 0 ) : φ ∈ X p b , ‖ φ | p ≤ 1 } . p x 0 = sup φ x 0 : φ ∈ X p b , ‖ φ p ≤ 1 . p(x_(0))=s u p{varphi(x_(0)):varphi inX_(p)^(b),|| varphi|_(p) <= 1}.p\left(x_{0}\right)=\sup \left\{\varphi\left(x_{0}\right): \varphi \in X_{p}^{b}, \|\left.\varphi\right|_{p} \leq 1\right\} .p(x0)=sup{φ(x0):φ∈Xpb,‖φ|p≤1}.
Moreover, there exists φ 0 ∈ X p b , ‖ φ 0 | p = 1 φ 0 ∈ X p b , ‖ φ 0 p = 1 varphi_(0)inX_(p)^(b),||varphi_(0)|_(p)=1\varphi_{0} \in X_{p}^{b}, \|\left.\varphi_{0}\right|_{p}=1φ0∈Xpb,‖φ0|p=1, such that φ 0 ( x 0 ) = p ( x 0 ) φ 0 x 0 = p x 0 varphi_(0)(x_(0))=p(x_(0))\varphi_{0}\left(x_{0}\right)=p\left(x_{0}\right)φ0(x0)=p(x0).
The following proposition is the asymmetric analog of a well known result of Hahn.
Proposition 1.9. ([12]) Let Y Y YYY be a subspace of a space with asymmetric seminorm ( X , p X , p X,pX, pX,p ) and x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X.
(1) If 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, then there exists φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb such that
(i) φ | Y = 0 φ Y = 0 varphi|_(Y)=0\left.\varphi\right|_{Y}=0φ|Y=0,
(ii) ‖ φ | p = 1 , ‖ φ p = 1 , || varphi|_(p)=1,quad\|\left.\varphi\right|_{p}=1, \quad‖φ|p=1, and
(iii) φ ( − x 0 ) = d φ − x 0 = d varphi(-x_(0))=d\varphi\left(-x_{0}\right)=dφ(−x0)=d.
(2) If d ¯ := d p ( Y , x 0 ) > 0 d ¯ := d p Y , x 0 > 0 bar(d):=d_(p)(Y,x_(0)) > 0\bar{d}:=d_{p}\left(Y, x_{0}\right)>0d¯:=dp(Y,x0)>0, then there exists ψ ∈ X p b ψ ∈ X p b psi inX_(p)^(b)\psi \in X_{p}^{b}ψ∈Xpb such that
(j) ψ | Y = 0 ψ Y = 0 psi|_(Y)=0\left.\psi\right|_{Y}=0ψ|Y=0,
( jj ) ‖ ψ | p = 1 , ( jj ) ‖ ψ p = 1 , (jj)|| psi|_(p)=1,quad(\mathrm{jj}) \|\left.\psi\right|_{p}=1, \quad(jj)‖ψ|p=1, and
( jjj ) ψ ( x 0 ) = d ¯ ( jjj ) ψ x 0 = d ¯ (jjj)psi(x_(0))= bar(d)(\mathrm{jjj}) \psi\left(x_{0}\right)=\bar{d}(jjj)ψ(x0)=d¯.

2. BEST APPROXIMATION IN SPACES WITH ASYMMETRIC SEMINORM

Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm, p ¯ p ¯ bar(p)\bar{p}p¯ the seminorm conjugate to p p ppp and Y Y YYY a nonempty subset of X X XXX. The distances d p ( x , Y ) d p ( x , Y ) d_(p)(x,Y)d_{p}(x, Y)dp(x,Y) and d p ( Y , x ) d p ( Y , x ) d_(p)(Y,x)d_{p}(Y, x)dp(Y,x) from an element x ∈ X x ∈ X x in Xx \in Xx∈X to Y Y YYY are defined by the formulae (1.10) and (1.11). An element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y such that p ( y 0 − x ) = d p ( x , Y ) p y 0 − x = d p ( x , Y ) p(y_(0)-x)=d_(p)(x,Y)p\left(y_{0}-x\right)=d_{p}(x, Y)p(y0−x)=dp(x,Y) will be called a p p ppp-nearest point to x x xxx in Y Y YYY, and an element y 1 ∈ Y y 1 ∈ Y y_(1)in Yy_{1} \in Yy1∈Y such that p ( x − y 1 ) = p ¯ ( y 1 − x ) = d p ¯ ( x , Y ) p x − y 1 = p ¯ y 1 − x = d p ¯ ( x , Y ) p(x-y_(1))= bar(p)(y_(1)-x)=d_( bar(p))(x,Y)p\left(x-y_{1}\right)=\bar{p}\left(y_{1}-x\right)=d_{\bar{p}}(x, Y)p(x−y1)=p¯(y1−x)=dp¯(x,Y) is called a p ¯ p ¯ bar(p)\bar{p}p¯-nearest point to x x xxx in Y Y YYY.
Denote by
P Y ( x ) = { y ∈ Y : p ( y − x ) = d p ( x , Y ) } , and (2.1) P ¯ Y ( x ) = { y ∈ Y : p ( x − y ) = d p ( Y , x ) } , P Y ( x ) = y ∈ Y : p ( y − x ) = d p ( x , Y ) ,  and  (2.1) P ¯ Y ( x ) = y ∈ Y : p ( x − y ) = d p ( Y , x ) , {:[P_(Y)(x)={y in Y:p(y-x)=d_(p)(x,Y)}","" and "],[(2.1) bar(P)_(Y)(x)={y in Y:p(x-y)=d_(p)(Y,x)}","]:}\begin{align*} & P_{Y}(x)=\left\{y \in Y: p(y-x)=d_{p}(x, Y)\right\}, \text { and } \\ & \bar{P}_{Y}(x)=\left\{y \in Y: p(x-y)=d_{p}(Y, x)\right\}, \tag{2.1} \end{align*}PY(x)={y∈Y:p(y−x)=dp(x,Y)}, and (2.1)P¯Y(x)={y∈Y:p(x−y)=dp(Y,x)},
the possibly empty sets of p p ppp-nearest points, respectively p ¯ p ¯ bar(p)\bar{p}p¯-nearest points, to x x xxx in Y Y YYY. The set Y Y YYY is called p p ppp-proximinal, p p ppp-semi-Chebyshev, p p ppp-Chebyshev if
for every x ∈ X x ∈ X x in Xx \in Xx∈X the set P Y ( x ) P Y ( x ) P_(Y)(x)P_{Y}(x)PY(x) is nonempty, contains at most one element, contains exactly one element, respectively. Similar definitions are given in the case of p ¯ p ¯ bar(p)\bar{p}p¯-nearest points. A semi-Chebyshev set is called also a uniqueness set.
For a nonempty subset Y Y YYY of a space with asymmetric seminorm ( X , p X , p X,pX, pX,p ), denote by Y p ⊥ Y p ⊥ Y_(p)^(_|_)Y_{p}^{\perp}Yp⊥ the annihilator of Y Y YYY in X p b X p b X_(p)^(b)X_{p}^{b}Xpb, i.e.
Y p ⊥ = { φ ∈ X p b : φ | Y = 0 } . Y p ⊥ = φ ∈ X p b : φ Y = 0 . Y_(p)^(_|_)={varphi inX_(p)^(b): varphi|_(Y)=0}.Y_{p}^{\perp}=\left\{\varphi \in X_{p}^{b}:\left.\varphi\right|_{Y}=0\right\} .Yp⊥={φ∈Xpb:φ|Y=0}.
We start by a characterization of nearest points given in [12] we shall need in the sequel.
Proposition 2.1 ( [ 12 ] [ 12 ] [12][12][12] ) Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm, Y Y YYY a subspace of X X XXX and x 0 x 0 x_(0)x_{0}x0 a point in X X XXX.
(1) Suppose 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. An element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is a p p ppp-nearest point to x 0 x 0 x_(0)x_{0}x0 in Y Y YYY if and only if there exists a bounded linear functional φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R such that
(i) φ | Y = 0 φ Y = 0 varphi|_(Y)=0\left.\varphi\right|_{Y}=0φ|Y=0,
(ii) ‖ 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 ) φ − x 0 = p y 0 − x 0 varphi(-x_(0))=p(y_(0)-x_(0))\varphi\left(-x_{0}\right)=p\left(y_{0}-x_{0}\right)φ(−x0)=p(y0−x0).
(2) Suppose that d ¯ := d p ( Y , x 0 ) > 0 d ¯ := d p Y , x 0 > 0 bar(d):=d_(p)(Y,x_(0)) > 0\bar{d}:=d_{p}\left(Y, x_{0}\right)>0d¯:=dp(Y,x0)>0. An element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y is a p ¯ p ¯ bar(p)\bar{p}p¯-nearest point to x 0 x 0 x_(0)x_{0}x0 in Y Y YYY if and only if there exists a bounded linear functional ψ : X → R ψ : X → R psi:X rarrR\psi: X \rightarrow \mathbb{R}ψ:X→R such that
(j) ψ | Y = 0 ψ Y = 0 psi|_(Y)=0\left.\psi\right|_{Y}=0ψ|Y=0,
( jj ) ‖ ψ | p = 1 ( jj ) ‖ ψ p = 1 (jj)|| psi|_(p)=1(\mathrm{jj}) \|\left.\psi\right|_{p}=1(jj)‖ψ|p=1,
(jjj) ψ ( x 0 ) = p ( x 0 − y 0 ) ψ x 0 = p x 0 − y 0 psi(x_(0))=p(x_(0)-y_(0))\psi\left(x_{0}\right)=p\left(x_{0}-y_{0}\right)ψ(x0)=p(x0−y0).
From this theorem one can obtain characterizations of sets of nearest points.
Corollary 2.2. Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm, Y Y YYY a subspace of X , x ∈ X X , x ∈ X X,x in XX, x \in XX,x∈X, and Z Z ZZZ a nonempty subset of Y Y YYY.
(1) If 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 then Z ⊂ P Y ( x ) Z ⊂ P Y ( x ) Z subP_(Y)(x)Z \subset P_{Y}(x)Z⊂PY(x) if and only if there exists a functional φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb such that
(i) φ | Y = 0 φ Y = 0 varphi|_(Y)=0\left.\varphi\right|_{Y}=0φ|Y=0,
(ii) ‖ φ | p = 1 ‖ φ p = 1 || varphi|_(p)=1\|\left.\varphi\right|_{p}=1‖φ|p=1,
(iii) ∀ y ∈ Z , φ ( − x 0 ) = p ( y − x 0 ) ∀ y ∈ Z , φ − x 0 = p y − x 0 AA y in Z,varphi(-x_(0))=p(y-x_(0))\forall y \in Z, \varphi\left(-x_{0}\right)=p\left(y-x_{0}\right)∀y∈Z,φ(−x0)=p(y−x0).
(2) If d ¯ = d p ( Y , x 0 ) > 0 d ¯ = d p Y , x 0 > 0 bar(d)=d_(p)(Y,x_(0)) > 0\bar{d}=d_{p}\left(Y, x_{0}\right)>0d¯=dp(Y,x0)>0 then Z ⊂ P ¯ Y ( x ) Z ⊂ P ¯ Y ( x ) Z sub bar(P)_(Y)(x)Z \subset \bar{P}_{Y}(x)Z⊂P¯Y(x) if and only if there exists a functional ψ ∈ X p b ψ ∈ X p b psi inX_(p)^(b)\psi \in X_{p}^{b}ψ∈Xpb such that
(j) ψ | Y = 0 ψ Y = 0 psi|_(Y)=0\left.\psi\right|_{Y}=0ψ|Y=0,
( jj ) ‖ ψ | p = 1 ( jj ) ‖ ψ p = 1 (jj)|| psi|_(p)=1(\mathrm{jj}) \|\left.\psi\right|_{p}=1(jj)‖ψ|p=1,
( jjj ) ∀ y ∈ Z , ψ ( x 0 ) = p ( x 0 − y ) ( jjj ) ∀ y ∈ Z , ψ x 0 = p x 0 − y (jjj)AA y in Z,psi(x_(0))=p(x_(0)-y)(\mathrm{jjj}) \forall y \in Z, \psi\left(x_{0}\right)=p\left(x_{0}-y\right)(jjj)∀y∈Z,ψ(x0)=p(x0−y).
In the next proposition we extend to the asymmetric case some characterization results for semi-Chebyshev subspaces (see 42, Chapter I, Theorem 3.2]).
Theorem 2.3. Let Y Y YYY be a subspace of a space with asymmetric norm ( X , p X , p X,pX, pX,p ) such that p ( x ) > 0 p ( x ) > 0 p(x) > 0p(x)>0p(x)>0 for every x ≠ 0 x ≠ 0 x!=0x \neq 0x≠0. Then the following assertions are equivalent.
(1) Y Y YYY is a p p ppp-semi-Chebyshev subspace of X X XXX.
(2) There are no φ ∈ Y p ⊥ φ ∈ Y p ⊥ varphi inY_(p)^(_|_)\varphi \in Y_{p}^{\perp}φ∈Yp⊥ and x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X with x 1 − x 2 ∈ Y ∖ { 0 } x 1 − x 2 ∈ Y ∖ { 0 } x_(1)-x_(2)in Y\\{0}x_{1}-x_{2} \in Y \backslash\{0\}x1−x2∈Y∖{0}, such that
(i) ‖ φ | p = 1 ‖ φ p = 1 || varphi|_(p)=1quad\|\left.\varphi\right|_{p}=1 \quad‖φ|p=1 and (ii) φ ( − x i ) = p ( − x i ) , i = 1 , 2 φ − x i = p − x i , i = 1 , 2 varphi(-x_(i))=p(-x_(i)),i=1,2\varphi\left(-x_{i}\right)=p\left(-x_{i}\right), i=1,2φ(−xi)=p(−xi),i=1,2.
(3) There are no ψ ∈ Y p ⊥ , x ∈ X ψ ∈ Y p ⊥ , x ∈ X psi inY_(p)^(_|_),x in X\psi \in Y_{p}^{\perp}, x \in Xψ∈Yp⊥,x∈X, and y 0 ∈ Y ∖ { 0 } y 0 ∈ Y ∖ { 0 } y_(0)in Y\\{0}y_{0} \in Y \backslash\{0\}y0∈Y∖{0} such that
(j) ‖ ψ | p = 1 and (jj) ψ ( − x ) = p ( − x ) = p ( y 0 − x ) .  (j)  ‖ ψ p = 1  and   (jj)  ψ ( − x ) = p ( − x ) = p y 0 − x .  " (j) "|| psi|_(p)=1quad" and "quad" (jj) "psi(-x)=p(-x)=p(y_(0)-x)". "\text { (j) } \|\left.\psi\right|_{p}=1 \quad \text { and } \quad \text { (jj) } \psi(-x)=p(-x)=p\left(y_{0}-x\right) \text {. } (j) ‖ψ|p=1 and  (jj) ψ(−x)=p(−x)=p(y0−x). 
Proof. (1) ⇒ ⇒ =>\Rightarrow⇒ (2) Suppose that (2) does not hold. Let φ ∈ Y p ⊥ φ ∈ Y p ⊥ varphi inY_(p)^(_|_)\varphi \in Y_{p}^{\perp}φ∈Yp⊥ and x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X with x 1 − x 2 ∈ Y ∖ { 0 } x 1 − x 2 ∈ Y ∖ { 0 } x_(1)-x_(2)in Y\\{0}x_{1}-x_{2} \in Y \backslash\{0\}x1−x2∈Y∖{0}, such that the conditions (i) and (ii) of the assertion (2) are satisfied, and put y 0 = x 1 − x 2 y 0 = x 1 − x 2 y_(0)=x_(1)-x_(2)y_{0}=x_{1}-x_{2}y0=x1−x2. Then
φ ( − x 2 ) = p ( − x 2 ) ⟺ φ ( y 0 − x 1 ) = p ( y 0 − x 1 ) , φ − x 2 = p − x 2 ⟺ φ y 0 − x 1 = p y 0 − x 1 , varphi(-x_(2))=p(-x_(2))Longleftrightarrow varphi(y_(0)-x_(1))=p(y_(0)-x_(1)),\varphi\left(-x_{2}\right)=p\left(-x_{2}\right) \Longleftrightarrow \varphi\left(y_{0}-x_{1}\right)=p\left(y_{0}-x_{1}\right),φ(−x2)=p(−x2)⟺φ(y0−x1)=p(y0−x1),
and
φ ( − x 1 ) = p ( − x 1 ) ⟺ φ ( 0 − x 1 ) = p ( 0 − x 1 ) . φ − x 1 = p − x 1 ⟺ φ 0 − x 1 = p 0 − x 1 . varphi(-x_(1))=p(-x_(1))Longleftrightarrow varphi(0-x_(1))=p(0-x_(1)).\varphi\left(-x_{1}\right)=p\left(-x_{1}\right) \Longleftrightarrow \varphi\left(0-x_{1}\right)=p\left(0-x_{1}\right) .φ(−x1)=p(−x1)⟺φ(0−x1)=p(0−x1).
By Proposition 2.1, it follows that 0 and y 0 y 0 y_(0)y_{0}y0 are p p ppp-nearest points to x 1 x 1 x_(1)x_{1}x1 in Y Y YYY.
( 2 ) ⇒ ( 3 ) ( 2 ) ⇒ ( 3 ) (2)=>(3)(2) \Rightarrow(3)(2)⇒(3) Suppose that (3) does not hold. Then there exist ψ ∈ Y p ⊥ , x ∈ X ψ ∈ Y p ⊥ , x ∈ X psi inY_(p)^(_|_),x in X\psi \in Y_{p}^{\perp}, x \in Xψ∈Yp⊥,x∈X, and y 0 ∈ Y ∖ { 0 } y 0 ∈ Y ∖ { 0 } y_(0)in Y\\{0}y_{0} \in Y \backslash\{0\}y0∈Y∖{0} such that the conditions ( j ) and ( jj ) of the assertion (3) are fulfilled. It follows that the conditions (i) and (ii) of the assertion (2) are satisfied by φ = ψ , x 1 = x φ = ψ , x 1 = x varphi=psi,x_(1)=x\varphi=\psi, x_{1}=xφ=ψ,x1=x and x 2 = y 0 − x x 2 = y 0 − x x_(2)=y_(0)-xx_{2}=y_{0}-xx2=y0−x, i.e. (2) does not hold.
( 3 ) ⇒ ( 1 ) ( 3 ) ⇒ ( 1 ) (3)=>(1)(3) \Rightarrow(1)(3)⇒(1) Supposing that (1) does not hold, there exist z ∈ X ∖ Y z ∈ X ∖ Y z in X\\Yz \in X \backslash Yz∈X∖Y and y 1 , y 2 ∈ Y , y 1 ≠ y 2 y 1 , y 2 ∈ Y , y 1 ≠ y 2 y_(1),y_(2)in Y,y_(1)!=y_(2)y_{1}, y_{2} \in Y, y_{1} \neq y_{2}y1,y2∈Y,y1≠y2, such that
p ( y 1 − z ) = p ( y 2 − z ) = d p ( z , Y ) p y 1 − z = p y 2 − z = d p ( z , Y ) p(y_(1)-z)=p(y_(2)-z)=d_(p)(z,Y)p\left(y_{1}-z\right)=p\left(y_{2}-z\right)=d_{p}(z, Y)p(y1−z)=p(y2−z)=dp(z,Y)
If d p ( z , Y ) = 0 d p ( z , Y ) = 0 d_(p)(z,Y)=0d_{p}(z, Y)=0dp(z,Y)=0, then y 1 = y 2 = z y 1 = y 2 = z y_(1)=y_(2)=zy_{1}=y_{2}=zy1=y2=z, a contradiction which shows that d p ( z , Y ) > 0 d p ( z , Y ) > 0 d_(p)(z,Y) > 0d_{p}(z, Y)>0dp(z,Y)>0.
If x := z − y 1 x := z − y 1 x:=z-y_(1)x:=z-y_{1}x:=z−y1, then
d p ( z − y 1 , Y ) = inf { p ( y + y 1 − z ) : y ∈ Y } = inf { p ( y ′ − z ) : y ′ ∈ Y } = d p ( z , Y ) = p ( y 1 − z ) = p ( y 2 − z ) d p z − y 1 , Y = inf p y + y 1 − z : y ∈ Y = inf p y ′ − z : y ′ ∈ Y = d p ( z , Y ) = p y 1 − z = p y 2 − z {:[d_(p)(z-y_(1),Y)=i n f{p(y+y_(1)-z):y in Y}],[=i n f{p(y^(')-z):y^(')in Y}],[=d_(p)(z","Y)],[=p(y_(1)-z)],[=p(y_(2)-z)]:}\begin{aligned} d_{p}\left(z-y_{1}, Y\right) & =\inf \left\{p\left(y+y_{1}-z\right): y \in Y\right\} \\ & =\inf \left\{p\left(y^{\prime}-z\right): y^{\prime} \in Y\right\} \\ & =d_{p}(z, Y) \\ & =p\left(y_{1}-z\right) \\ & =p\left(y_{2}-z\right) \end{aligned}dp(z−y1,Y)=inf{p(y+y1−z):y∈Y}=inf{p(y′−z):y′∈Y}=dp(z,Y)=p(y1−z)=p(y2−z)
By Proposition 2.1, there exists ψ ∈ Y p ⊥ , ‖ ψ | p = 1 ψ ∈ Y p ⊥ , ‖ ψ p = 1 psi inY_(p)^(_|_),|| psi|_(p)=1\psi \in Y_{p}^{\perp}, \|\left.\psi\right|_{p}=1ψ∈Yp⊥,‖ψ|p=1, such that
ψ ( y 1 − z ) = p ( y 1 − z ) = p ( y 2 − z ) ψ y 1 − z = p y 1 − z = p y 2 − z psi(y_(1)-z)=p(y_(1)-z)=p(y_(2)-z)\psi\left(y_{1}-z\right)=p\left(y_{1}-z\right)=p\left(y_{2}-z\right)ψ(y1−z)=p(y1−z)=p(y2−z)
or, denoting y 0 := y 2 − y 1 y 0 := y 2 − y 1 y_(0):=y_(2)-y_(1)y_{0}:=y_{2}-y_{1}y0:=y2−y1, this is equivalent to
ψ ( − x ) = p ( − x ) = p ( y 0 − x ) ψ ( − x ) = p ( − x ) = p y 0 − x psi(-x)=p(-x)=p(y_(0)-x)\psi(-x)=p(-x)=p\left(y_{0}-x\right)ψ(−x)=p(−x)=p(y0−x)
showing that (3) does not hold.
Remark 2.4. Obviously that a similar characterization result holds for p ¯ p ¯ bar(p)\bar{p}p¯ -semi-Chebyshev subspaces.
Using Corollary 2.2, one can extend Theorem 2.3 to obtain characterizations of pseudo-Chebyshev subspaces, a notion introduced by Mohebi [28] in the case of normed spaces. Concerning other weaker notions of Chebyshev spaces - quasi-Chebyshev subspaces, weak-Chebyshev subspaces, as well as for their behaviour in concrete function spaces, see the papers [26, 27, 29, 31. For a subset Z Z ZZZ of a vector space X X XXX denote by aff( Z Z ZZZ ) the affine hull of the set Z Z ZZZ, i.e.
aff ( Z ) = { x ∈ X : ∃ n ∈ N , ∃ z 1 , … , z n ∈ Z , ∃ a 1 , … , a n ∈ R , a 1 + … + a n = aff ( Z ) = x ∈ X : ∃ n ∈ N , ∃ z 1 , … , z n ∈ Z , ∃ a 1 , … , a n ∈ R , a 1 + … + a n = aff(Z)={x in X:EE n inN,EEz_(1),dots,z_(n)in Z,EEa_(1),dots,a_(n)inR,a_(1)+dots+a_(n)=:}\operatorname{aff}(Z)=\left\{x \in X: \exists n \in \mathbb{N}, \exists z_{1}, \ldots, z_{n} \in Z, \exists a_{1}, \ldots, a_{n} \in \mathbb{R}, a_{1}+\ldots+a_{n}=\right.aff(Z)={x∈X:∃n∈N,∃z1,…,zn∈Z,∃a1,…,an∈R,a1+…+an= 1 such that x = a 1 z 1 + … + a n z n x = a 1 z 1 + … + a n z n x=a_(1)z_(1)+dots+a_(n)z_(n)x=a_{1} z_{1}+\ldots+a_{n} z_{n}x=a1z1+…+anzn }. There exists a unique subspace Y Y YYY of X X XXX such that aff ( Z ) = z + Y aff ( Z ) = z + Y aff(Z)=z+Y\operatorname{aff}(Z)=z+Yaff(Z)=z+Y, for an arbitrary z ∈ Z z ∈ Z z in Zz \in Zz∈Z. By definition, the affine dimension of the set Z Z ZZZ is the dimension of this subspace Y Y YYY of X X XXX.
A subspace Y Y YYY of a space with asymmetric norm ( X , p X , p X,pX, pX,p ) is called p p ppp-pseudoChebyshev if it is p p ppp-proximinal and the set P Y ( x ) P Y ( x ) P_(Y)(x)P_{Y}(x)PY(x) has finite affine dimension for every x ∈ X x ∈ X x in Xx \in Xx∈X.
The following theorem extends a result proved by Mohebi [28] in normed spaces.
Theorem 2.5. Let Y Y YYY be a subspace of an asymmetric normed space ( X , p X , p X,pX, pX,p ) such that p ( x ) > 0 p ( x ) > 0 p(x) > 0p(x)>0p(x)>0 for every x ≠ 0 x ≠ 0 x!=0x \neq 0x≠0. The following assertions are equivalent.
(1) The subspace Y Y YYY is p-pseudo-Chebyshev.
(2) There do not exist φ ∈ Y p ⊥ , x 0 ∈ X φ ∈ Y p ⊥ , x 0 ∈ X varphi inY_(p)^(_|_),x_(0)in X\varphi \in Y_{p}^{\perp}, x_{0} \in Xφ∈Yp⊥,x0∈X, and infinitely many linearly independent elements x n ∈ X , n ∈ N x n ∈ X , n ∈ N x_(n)in X,n inNx_{n} \in X, n \in \mathbb{N}xn∈X,n∈N, with x 0 − x n ∈ Y , n ∈ N x 0 − x n ∈ Y , n ∈ N x_(0)-x_(n)in Y,n inNx_{0}-x_{n} \in Y, n \in \mathbb{N}x0−xn∈Y,n∈N, such that
(i) ‖ φ | p = 1 ‖ φ p = 1 || varphi|_(p)=1\|\left.\varphi\right|_{p}=1‖φ|p=1 and
(ii) φ ( − x n ) = p ( − x n ) , n = 0 , 1 , … φ − x n = p − x n , n = 0 , 1 , … varphi(-x_(n))=p(-x_(n)),n=0,1,dots\varphi\left(-x_{n}\right)=p\left(-x_{n}\right), n=0,1, \ldotsφ(−xn)=p(−xn),n=0,1,….
(3) There do not exist ψ ∈ Y p ⊥ , x 0 ∈ X ψ ∈ Y p ⊥ , x 0 ∈ X psi inY_(p)^(_|_),x_(0)in X\psi \in Y_{p}^{\perp}, x_{0} \in Xψ∈Yp⊥,x0∈X, and infinitely many linearly independent elements y n ∈ Y , n ∈ N y n ∈ Y , n ∈ N y_(n)in Y,n inNy_{n} \in Y, n \in \mathbb{N}yn∈Y,n∈N, such that
(j) ‖ ψ | p = 1 ‖ ψ p = 1 || psi|_(p)=1\|\left.\psi\right|_{p}=1‖ψ|p=1 and
(jj) ψ ( − x 0 ) = p ( − x 0 ) = p ( y n − x 0 ) , n = 1 , 2 , … ψ − x 0 = p − x 0 = p y n − x 0 , n = 1 , 2 , … psi(-x_(0))=p(-x_(0))=p(y_(n)-x_(0)),n=1,2,dots\psi\left(-x_{0}\right)=p\left(-x_{0}\right)=p\left(y_{n}-x_{0}\right), n=1,2, \ldotsψ(−x0)=p(−x0)=p(yn−x0),n=1,2,….
Proof. (1) ⇒ ⇒ =>\Rightarrow⇒ (2) Suppose that (2) does not hold. Then there exist φ ∈ Y p ⊥ , x 0 ∈ X φ ∈ Y p ⊥ , x 0 ∈ X varphi inY_(p)^(_|_),x_(0)in X\varphi \in Y_{p}^{\perp}, x_{0} \in Xφ∈Yp⊥,x0∈X, and infinitely many linearly independent elements x n ∈ X x n ∈ X x_(n)in Xx_{n} \in Xxn∈X, with x 0 − x n ∈ Y , n ∈ N x 0 − x n ∈ Y , n ∈ N x_(0)-x_(n)in Y,n inNx_{0}-x_{n} \in Y, n \in \mathbb{N}x0−xn∈Y,n∈N, satisfying the conditions (i) and (ii). The elements y n := x 0 − x n , n ∈ N y n := x 0 − x n , n ∈ N y_(n):=x_(0)-x_(n),n inNy_{n}:=x_{0}-x_{n}, n \in \mathbb{N}yn:=x0−xn,n∈N, all belong to Y Y YYY, are linearly independent, and
φ ( y n − x 0 ) = φ ( − x n ) = p ( − x n ) = p ( y n − x 0 ) , φ y n − x 0 = φ − x n = p − x n = p y n − x 0 , varphi(y_(n)-x_(0))=varphi(-x_(n))=p(-x_(n))=p(y_(n)-x_(0)),\varphi\left(y_{n}-x_{0}\right)=\varphi\left(-x_{n}\right)=p\left(-x_{n}\right)=p\left(y_{n}-x_{0}\right),φ(yn−x0)=φ(−xn)=p(−xn)=p(yn−x0),
so that, by Corollary 2.2, they are all contained in P Y ( x 0 ) P Y x 0 P_(Y)(x_(0))P_{Y}\left(x_{0}\right)PY(x0), showing that Y Y YYY is not p p ppp-pseudo-Chebyshev.
(2) ⇒ ⇒ =>\Rightarrow⇒ (3) Suppose again that (3) does not hold, and let ψ ∈ Y p ⊥ , x 0 ∈ X ψ ∈ Y p ⊥ , x 0 ∈ X psi inY_(p)^(_|_),x_(0)in X\psi \in Y_{p}^{\perp}, x_{0} \in Xψ∈Yp⊥,x0∈X, and the linearly independent elements { y n : n = 1 , 2 , … } ⊂ Y y n : n = 1 , 2 , … ⊂ Y {y_(n):n=1,2,dots}sub Y\left\{y_{n}: n=1,2, \ldots\right\} \subset Y{yn:n=1,2,…}⊂Y fulfilling the conditions ( j ) and ( jj ).
Then x n := x 0 − y n , n = 1 , 2 , … x n := x 0 − y n , n = 1 , 2 , … x_(n):=x_(0)-y_(n),n=1,2,dotsx_{n}:=x_{0}-y_{n}, n=1,2, \ldotsxn:=x0−yn,n=1,2,…, are linearly independent elements in X X XXX and
ψ ( − x n ) = ψ ( y n − x 0 ) = p ( − x n ) , n = 0 , 1 , 2 , … , ψ − x n = ψ y n − x 0 = p − x n , n = 0 , 1 , 2 , … , psi(-x_(n))=psi(y_(n)-x_(0))=p(-x_(n)),n=0,1,2,dots,\psi\left(-x_{n}\right)=\psi\left(y_{n}-x_{0}\right)=p\left(-x_{n}\right), n=0,1,2, \ldots,ψ(−xn)=ψ(yn−x0)=p(−xn),n=0,1,2,…,
showing that (2) does not hold.
(3) ⇒ ⇒ =>\Rightarrow⇒ (1) Supposing that (1) does not hold, there exist an element z ∈ X z ∈ X z in Xz \in Xz∈X and an infinite set { y n : n = 1 , 2 , … } y n : n = 1 , 2 , … {y_(n):n=1,2,dots}\left\{y_{n}: n=1,2, \ldots\right\}{yn:n=1,2,…} of linearly independent elements contained in P Y ( z ) P Y ( z ) P_(Y)(z)P_{Y}(z)PY(z).
By Corollary 2.2, there exists φ ∈ Y p ⊥ , ‖ φ | p = 1 φ ∈ Y p ⊥ , ‖ φ p = 1 varphi inY_(p)^(_|_),|| varphi|_(p)=1\varphi \in Y_{p}^{\perp}, \|\left.\varphi\right|_{p}=1φ∈Yp⊥,‖φ|p=1, such that
φ ( y n − z ) = d p ( z , Y ) = p ( y n − z ) , n = 1 , 2 , … φ y n − z = d p ( z , Y ) = p y n − z , n = 1 , 2 , … varphi(y_(n)-z)=d_(p)(z,Y)=p(y_(n)-z),n=1,2,dots\varphi\left(y_{n}-z\right)=d_{p}(z, Y)=p\left(y_{n}-z\right), n=1,2, \ldotsφ(yn−z)=dp(z,Y)=p(yn−z),n=1,2,…
Putting x := z − y 1 x := z − y 1 x:=z-y_(1)x:=z-y_{1}x:=z−y1 we have
d p ( x , Y ) = inf { p ( y + y 1 − z ) : y ∈ Y } = inf { p ( y ′ − z ) : y ′ ∈ Y } = = d p ( z , Y ) = p ( y n − z ) = p ( y n − y 1 − x ) , n = 2 , 3 , … , d p ( x , Y ) = inf p y + y 1 − z : y ∈ Y = inf p y ′ − z : y ′ ∈ Y = = d p ( z , Y ) = p y n − z = p y n − y 1 − x , n = 2 , 3 , … , {:[d_(p)(x","Y)=i n f{p(y+y_(1)-z):y in Y}=i n f{p(y^(')-z):y^(')in Y}=],[=d_(p)(z","Y)=p(y_(n)-z)=p(y_(n)-y_(1)-x)","n=2","3","dots","]:}\begin{aligned} d_{p}(x, Y) & =\inf \left\{p\left(y+y_{1}-z\right): y \in Y\right\}=\inf \left\{p\left(y^{\prime}-z\right): y^{\prime} \in Y\right\}= \\ & =d_{p}(z, Y)=p\left(y_{n}-z\right)=p\left(y_{n}-y_{1}-x\right), n=2,3, \ldots, \end{aligned}dp(x,Y)=inf{p(y+y1−z):y∈Y}=inf{p(y′−z):y′∈Y}==dp(z,Y)=p(yn−z)=p(yn−y1−x),n=2,3,…,
showing that { y n − y 1 : n = 2 , 3 , … } ⊂ P Y ( x ) y n − y 1 : n = 2 , 3 , … ⊂ P Y ( x ) {y_(n)-y_(1):n=2,3,dots}subP_(Y)(x)\left\{y_{n}-y_{1}: n=2,3, \ldots\right\} \subset P_{Y}(x){yn−y1:n=2,3,…}⊂PY(x). By Corollary 2.2, there exists ψ ∈ Y p ⊥ ψ ∈ Y p ⊥ psi inY_(p)^(_|_)\psi \in Y_{p}^{\perp}ψ∈Yp⊥ with ‖ ψ | p = 1 ‖ ψ p = 1 || psi|_(p)=1\|\left.\psi\right|_{p}=1‖ψ|p=1 such that
ψ ( y n − y y − x 0 ) = p ( y n − y 1 − x 0 ) , n = 2 , 3 , … , ψ y n − y y − x 0 = p y n − y 1 − x 0 , n = 2 , 3 , … , psi(y_(n)-y_(y)-x_(0))=p(y_(n)-y_(1)-x_(0)),n=2,3,dots,\psi\left(y_{n}-y_{y}-x_{0}\right)=p\left(y_{n}-y_{1}-x_{0}\right), n=2,3, \ldots,ψ(yn−yy−x0)=p(yn−y1−x0),n=2,3,…,
showing that (3) does not hold.
Phelps [37] emphasized for the first time some close connections existing between the approximation properties of the annihilator Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ of a subspace Y Y YYY of a normed space X X XXX and the extension properties of the subspace Y Y YYY. A presentation of various situations in which such a connection occurs is done in [8]. The case of spaces with asymmetric norms was considered in 34, 35.
Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm and Y Y YYY a subspace of X X XXX. For a p p ppp-bounded linear functional φ : Y → R φ : Y → R varphi:Y rarrR\varphi: Y \rightarrow \mathbb{R}φ:Y→R denote by
E p ( φ ) = { ψ ∈ X p b : ψ | Y = φ , ‖ ψ | p = ‖ φ | p } E p ( φ ) = ψ ∈ X p b : ψ Y = φ , ψ p = φ p E_(p)(varphi)={psi inX_(p)^(b): psi|_(Y)=varphi,|| psi|_(p)=||varphi|_(p)}E_{p}(\varphi)=\left\{\psi \in X_{p}^{b}:\left.\psi\right|_{Y}=\varphi,\left.\left\|\left.\psi\right|_{p}=\right\| \varphi\right|_{p}\right\}Ep(φ)={ψ∈Xpb:ψ|Y=φ,‖ψ|p=‖φ|p}
the set of all norm-preserving extensions of the functional φ φ varphi\varphiφ. By the HahnBanach theorem (Theorem 1.6) the set E p ( φ ) E p ( φ ) E_(p)(varphi)E_{p}(\varphi)Ep(φ) is always nonempty.
For φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb consider the following minimization problem
(2.2) γ ( φ , Y p ⊥ ) := inf { ‖ φ + ψ | p : ψ ∈ Y p ⊥ } . (2.2) γ φ , Y p ⊥ := inf ‖ φ + ψ p : ψ ∈ Y p ⊥ . {:(2.2)gamma(varphi,Y_(p)^(_|_)):=i n f{||varphi+ psi|_(p):psi inY_(p)^(_|_)}.:}\begin{equation*} \gamma\left(\varphi, Y_{p}^{\perp}\right):=\inf \left\{\| \varphi+\left.\psi\right|_{p}: \psi \in Y_{p}^{\perp}\right\} . \tag{2.2} \end{equation*}(2.2)γ(φ,Yp⊥):=inf{‖φ+ψ|p:ψ∈Yp⊥}.
A solution to this problem is an element ψ 0 ∈ Y p ⊥ ψ 0 ∈ Y p ⊥ psi_(0)inY_(p)^(_|_)\psi_{0} \in Y_{p}^{\perp}ψ0∈Yp⊥ such that ‖ φ + ψ 0 | p = γ ( φ , Y p ⊥ ) ‖ φ + ψ 0 p = γ φ , Y p ⊥ ||varphi+psi_(0)|_(p)=gamma(varphi,Y_(p)^(_|_))\| \varphi+\left.\psi_{0}\right|_{p}= \gamma\left(\varphi, Y_{p}^{\perp}\right)‖φ+ψ0|p=γ(φ,Yp⊥). Denote by Π Y p ⊥ ( φ ) Π Y p ⊥ ( φ ) Pi_(Y_(p)^(_|_))(varphi)\Pi_{Y_{p}^{\perp}}(\varphi)ΠYp⊥(φ) the set of all these solutions.
Theorem 2.6. If the linear functional φ : X → R φ : X → R varphi:X rarrR\varphi: X \rightarrow \mathbb{R}φ:X→R is ( p , p ¯ ) ( p , p ¯ ) (p, bar(p))(p, \bar{p})(p,p¯)-bounded then the minimization problem (2.2) has a solution and the following formulae hold
(2.3) γ ( φ , Y p ⊥ ) = ‖ φ | Y | p and Π Y p ⊥ ( φ ) = E p ( φ | Y ) − φ (2.3) γ φ , Y p ⊥ = ‖ φ Y p  and  Π Y p ⊥ ( φ ) = E p φ Y − φ {:(2.3)gamma(varphi,Y_(p)^(_|_))=|| varphi|_(Y)|_(p)quad" and "quadPi_(Y_(p)^(_|_))(varphi)=E_(p)( varphi|_(Y))-varphi:}\begin{equation*} \gamma\left(\varphi, Y_{p}^{\perp}\right)=\|\left.\left.\varphi\right|_{Y}\right|_{p} \quad \text { and } \quad \Pi_{Y_{p}^{\perp}}(\varphi)=E_{p}\left(\left.\varphi\right|_{Y}\right)-\varphi \tag{2.3} \end{equation*}(2.3)γ(φ,Yp⊥)=‖φ|Y|p and ΠYp⊥(φ)=Ep(φ|Y)−φ
Proof. Let φ ∈ X p b ∩ X p ¯ b φ ∈ X p b ∩ X p ¯ b varphi inX_(p)^(b)nnX_( bar(p))^(b)\varphi \in X_{p}^{b} \cap X_{\bar{p}}^{b}φ∈Xpb∩Xp¯b and ψ ∈ Y p ⊥ ψ ∈ Y p ⊥ psi inY_(p)^(_|_)\psi \in Y_{p}^{\perp}ψ∈Yp⊥. Then
‖ φ + ψ | p ≥ ‖ ( φ + ψ ) | Y | p = ‖ φ | Y | p φ + ψ p ≥ ( φ + ψ ) Y p = ‖ φ Y p ||varphi+ psi|_(p) >= ||(varphi+psi)|_(Y)|_(p)=|| varphi|_(Y)|_(p)\left.\left.\left\|\varphi+\left.\psi\right|_{p} \geq\right\|(\varphi+\psi)\right|_{Y}\right|_{p}=\|\left.\left.\varphi\right|_{Y}\right|_{p}‖φ+ψ|p≥‖(φ+ψ)|Y|p=‖φ|Y|p
implying γ ( φ , Y p ⊥ ) ≥ ‖ φ | Y | p γ φ , Y p ⊥ ≥ ‖ φ Y p gamma(varphi,Y_(p)^(_|_)) >= || varphi|_(Y)|_(p)\gamma\left(\varphi, Y_{p}^{\perp}\right) \geq \|\left.\left.\varphi\right|_{Y}\right|_{p}γ(φ,Yp⊥)≥‖φ|Y|p.
If Φ ∈ E p ( φ | Y ) Φ ∈ E p φ Y Phi inE_(p)( varphi|_(Y))\Phi \in E_{p}\left(\left.\varphi\right|_{Y}\right)Φ∈Ep(φ|Y) then, because φ φ varphi\varphiφ is ( p , p ¯ ) ( p , p ¯ ) (p, bar(p))(p, \bar{p})(p,p¯)-bounded, − φ ∈ X p b − φ ∈ X p b -varphi inX_(p)^(b)-\varphi \in X_{p}^{b}−φ∈Xpb (see Proposition 1.5. ψ := Φ − φ ∈ Y p ⊥ ψ := Φ − φ ∈ Y p ⊥ psi:=Phi-varphi inY_(p)^(_|_)\psi:=\Phi-\varphi \in Y_{p}^{\perp}ψ:=Φ−φ∈Yp⊥, and γ ( φ , Y p ⊥ ) ≤ ‖ φ + ψ | p = ‖ Φ | p γ φ , Y p ⊥ ≤ φ + ψ p = Φ p gamma(varphi,Y_(p)^(_|_)) <= ||varphi+ psi|_(p)=||Phi|_(p)\gamma\left(\varphi, Y_{p}^{\perp}\right) \leq\left.\left\|\varphi+\left.\psi\right|_{p}=\right\| \Phi\right|_{p}γ(φ,Yp⊥)≤‖φ+ψ|p=‖Φ|p. Therefore γ ( φ , Y p ⊥ ) = ‖ φ | Y | p γ φ , Y p ⊥ = ‖ φ Y p gamma(varphi,Y_(p)^(_|_))=|| varphi|_(Y)|_(p)\gamma\left(\varphi, Y_{p}^{\perp}\right)=\|\left.\left.\varphi\right|_{Y}\right|_{p}γ(φ,Yp⊥)=‖φ|Y|p and
E p ( φ | Y ) − φ ⊂ Π Y p ⊥ ( φ ) . E p φ Y − φ ⊂ Π Y p ⊥ ( φ ) . E_(p)( varphi|_(Y))-varphi subPi_(Y_(p)^(_|_))(varphi).E_{p}\left(\left.\varphi\right|_{Y}\right)-\varphi \subset \Pi_{Y_{p}^{\perp}}(\varphi) .Ep(φ|Y)−φ⊂ΠYp⊥(φ).
Conversely, if ψ ∈ Π Y p ⊥ ( φ ) ψ ∈ Π Y p ⊥ ( φ ) psi inPi_(Y_(p)^(_|_))(varphi)\psi \in \Pi_{Y_{p}^{\perp}}(\varphi)ψ∈ΠYp⊥(φ), then Φ := φ + ψ Φ := φ + ψ Phi:=varphi+psi\Phi:=\varphi+\psiΦ:=φ+ψ satisfies Φ | Y = φ | Y Φ Y = φ Y Phi|_(Y)= varphi|_(Y)\left.\Phi\right|_{Y}=\left.\varphi\right|_{Y}Φ|Y=φ|Y and ‖ Φ | p = ‖ φ + ψ | p = γ ( φ , Y p ⊥ ) = ‖ φ | Y | p Φ p = φ + ψ p = γ φ , Y p ⊥ = ‖ φ Y p || Phi|_(p)=||varphi+ psi|_(p)=gamma(varphi,Y_(p)^(_|_))=|| varphi|_(Y)|_(p)\left\|\left.\Phi\right|_{p}=\right\| \varphi+\left.\psi\right|_{p}=\gamma\left(\varphi, Y_{p}^{\perp}\right)=\|\left.\left.\varphi\right|_{Y}\right|_{p}‖Φ|p=‖φ+ψ|p=γ(φ,Yp⊥)=‖φ|Y|p, i.e. Φ ∈ E p ( φ | Y ) Φ ∈ E p φ Y Phi inE_(p)( varphi|_(Y))\Phi \in E_{p}\left(\left.\varphi\right|_{Y}\right)Φ∈Ep(φ|Y) and
φ + Π Y p ⊥ ( φ ) ⊂ E p ( φ | Y ) ⟺ Π Y p ⊥ ( φ ) ⊂ E p ( φ | Y ) − φ . φ + Π Y p ⊥ ( φ ) ⊂ E p φ Y ⟺ Π Y p ⊥ ( φ ) ⊂ E p φ Y − φ . varphi+Pi_(Y_(p)^(_|_))(varphi)subE_(p)( varphi|_(Y))LongleftrightarrowPi_(Y_(p)^(_|_))(varphi)subE_(p)( varphi|_(Y))-varphi.\varphi+\Pi_{Y_{p}^{\perp}}(\varphi) \subset E_{p}\left(\left.\varphi\right|_{Y}\right) \Longleftrightarrow \Pi_{Y_{p}^{\perp}}(\varphi) \subset E_{p}\left(\left.\varphi\right|_{Y}\right)-\varphi .φ+ΠYp⊥(φ)⊂Ep(φ|Y)⟺ΠYp⊥(φ)⊂Ep(φ|Y)−φ.
Denoting by
(2.4) Y ⊥ = { ψ ∈ X ∗ : ψ | Y = 0 } (2.4) Y ⊥ = ψ ∈ X ∗ : ψ Y = 0 {:(2.4)Y^(_|_)={psi inX^(**): psi|_(Y)=0}:}\begin{equation*} Y^{\perp}=\left\{\psi \in X^{*}:\left.\psi\right|_{Y}=0\right\} \tag{2.4} \end{equation*}(2.4)Y⊥={ψ∈X∗:ψ|Y=0}
the annihilator Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ of a subspace Y Y YYY of X X XXX in the symmetric dual X ∗ X ∗ X^(**)X^{*}X∗ of the seminormed space ( X , p s ) ∗ X , p s ∗ (X,p^(s))^(**)\left(X, p^{s}\right)^{*}(X,ps)∗, it follows that Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is a subspace of X ∗ X ∗ X^(**)X^{*}X∗. Consider on X ∗ X ∗ X^(**)X^{*}X∗ the asymmetric extended norm ‖ | p ∗ : X ∗ → [ 0 ; ∞ ] ‖ p ∗ : X ∗ → [ 0 ; ∞ ] |||_(p)^(**):X^(**)rarr[0;oo]\|\left.\right|_{p} ^{*}: X^{*} \rightarrow[0 ; \infty]‖|p∗:X∗→[0;∞] defined by
‖ φ | p ∗ = sup φ ( B p ) . ‖ φ p ∗ = sup φ B p . || varphi|_(p)^(**)=s u p varphi(B_(p)).\|\left.\varphi\right|_{p} ^{*}=\sup \varphi\left(B_{p}\right) .‖φ|p∗=supφ(Bp).
We have for any φ ∈ X ∗ φ ∈ X ∗ varphi inX^(**)\varphi \in X^{*}φ∈X∗
φ ∈ X p b ⟺ ‖ φ | p ∗ < ∞ , φ ∈ X p b ⟺ ‖ φ p ∗ < ∞ , varphi inX_(p)^(b)Longleftrightarrow|| varphi|_(p)^(**) < oo,\varphi \in X_{p}^{b} \Longleftrightarrow \|\left.\varphi\right|_{p} ^{*}<\infty,φ∈Xpb⟺‖φ|p∗<∞,
and ‖ φ | p ∗ = ‖ φ | p φ p ∗ = φ p || varphi|_(p)^(**)=||varphi|_(p)\left.\left\|\left.\varphi\right|_{p} ^{*}=\right\| \varphi\right|_{p}‖φ|p∗=‖φ|p for φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb (see Proposition 1.5).
For φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb consider the distance from φ φ varphi\varphiφ to Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ defined by
d p ( Y ⊥ , φ ) = inf { ‖ φ − ψ | p ∗ : ψ ∈ Y ⊥ } . d p Y ⊥ , φ = inf ‖ φ − ψ p ∗ : ψ ∈ Y ⊥ . d_(p)(Y^(_|_),varphi)=i n f{||varphi- psi|_(p)^(**):psi inY^(_|_)}.d_{p}\left(Y^{\perp}, \varphi\right)=\inf \left\{\| \varphi-\left.\psi\right|_{p} ^{*}: \psi \in Y^{\perp}\right\} .dp(Y⊥,φ)=inf{‖φ−ψ|p∗:ψ∈Y⊥}.
Because ‖ φ − 0 | p ∗ = ‖ φ | p < ∞ φ − 0 p ∗ = φ p < ∞ ||varphi-0|_(p)^(**)=||varphi|_(p) < oo\left.\left\|\varphi-\left.0\right|_{p} ^{*}=\right\| \varphi\right|_{p}<\infty‖φ−0|p∗=‖φ|p<∞ this distance is always finite. Put
P Y ⊥ ( φ ) = { ψ ∈ Y ⊥ : ‖ φ − ψ | p = d p ( Y ⊥ , φ ) } . P Y ⊥ ( φ ) = ψ ∈ Y ⊥ : ‖ φ − ψ p = d p Y ⊥ , φ . P_(Y^(_|_))(varphi)={psi inY^(_|_):||varphi- psi|_(p)=d_(p)(Y^(_|_),varphi)}.P_{Y^{\perp}}(\varphi)=\left\{\psi \in Y^{\perp}: \| \varphi-\left.\psi\right|_{p}=d_{p}\left(Y^{\perp}, \varphi\right)\right\} .PY⊥(φ)={ψ∈Y⊥:‖φ−ψ|p=dp(Y⊥,φ)}.
Theorem 2.7. Every φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb has a p ¯ p ¯ bar(p)\bar{p}p¯-nearest point in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ and the following formulae hold
d p ( Y ⊥ , φ ) = ‖ φ | Y | p and P Y ⊥ ( φ ) = φ − E p ( φ | Y ) . d p Y ⊥ , φ = ‖ φ Y p  and  P Y ⊥ ( φ ) = φ − E p φ Y . d_(p)(Y^(_|_),varphi)=|| varphi|_(Y)|_(p)quad" and "quadP_(Y^(_|_))(varphi)=varphi-E_(p)( varphi|_(Y)).d_{p}\left(Y^{\perp}, \varphi\right)=\|\left.\left.\varphi\right|_{Y}\right|_{p} \quad \text { and } \quad P_{Y^{\perp}}(\varphi)=\varphi-E_{p}\left(\left.\varphi\right|_{Y}\right) .dp(Y⊥,φ)=‖φ|Y|p and PY⊥(φ)=φ−Ep(φ|Y).
Proof. For ψ ∈ Y ⊥ ψ ∈ Y ⊥ psi inY^(_|_)\psi \in Y^{\perp}ψ∈Y⊥ we have
‖ φ − ψ | p ∗ ≥ ‖ ( φ − ψ ) | Y | p ∗ = ‖ φ | Y | p φ − ψ p ∗ ≥ ( φ − ψ ) Y p ∗ = ‖ φ Y p ||varphi- psi|_(p)^(**) >= ||(varphi-psi)|_(Y)|_(p)^(**)=|| varphi|_(Y)|_(p)\left.\left.\left\|\varphi-\left.\psi\right|_{p} ^{*} \geq\right\|(\varphi-\psi)\right|_{Y}\right|_{p} ^{*}=\|\left.\left.\varphi\right|_{Y}\right|_{p}‖φ−ψ|p∗≥‖(φ−ψ)|Y|p∗=‖φ|Y|p
implying d p ( Y ⊥ , φ ) ≥ ‖ φ | Y | p d p Y ⊥ , φ ≥ ‖ φ Y p d_(p)(Y^(_|_),varphi) >= || varphi|_(Y)|_(p)d_{p}\left(Y^{\perp}, \varphi\right) \geq \|\left.\left.\varphi\right|_{Y}\right|_{p}dp(Y⊥,φ)≥‖φ|Y|p. If Φ ∈ E p ( φ | Y ) Φ ∈ E p φ Y Phi inE_(p)( varphi|_(Y))\Phi \in E_{p}\left(\left.\varphi\right|_{Y}\right)Φ∈Ep(φ|Y), then ψ := φ − Φ ∈ Y ⊥ ψ := φ − Φ ∈ Y ⊥ psi:=varphi-Phi inY^(_|_)\psi:=\varphi-\Phi \in Y^{\perp}ψ:=φ−Φ∈Y⊥ and
d p ( Y ⊥ , φ ) ≤ ‖ φ − ψ | p ∗ = ‖ Φ | p = ‖ φ | Y | p . d p Y ⊥ , φ ≤ φ − ψ p ∗ = Φ p = ‖ φ Y p . d_(p)(Y^(_|_),varphi) <= ||varphi- psi|_(p)^(**)=||Phi|_(p)=|| varphi|_(Y)|_(p).d_{p}\left(Y^{\perp}, \varphi\right) \leq\left.\left\|\varphi-\left.\psi\right|_{p} ^{*}=\right\| \Phi\right|_{p}=\|\left.\left.\varphi\right|_{Y}\right|_{p} .dp(Y⊥,φ)≤‖φ−ψ|p∗=‖Φ|p=‖φ|Y|p.
Therefore d p ( Y ⊥ , φ ) = ‖ φ | Y | p d p Y ⊥ , φ = ‖ φ Y p d_(p)(Y^(_|_),varphi)=|| varphi|_(Y)|_(p)d_{p}\left(Y^{\perp}, \varphi\right)=\|\left.\left.\varphi\right|_{Y}\right|_{p}dp(Y⊥,φ)=‖φ|Y|p and φ − E p ( φ | Y ) ⊂ P Y ⊥ ( φ ) φ − E p φ Y ⊂ P Y ⊥ ( φ ) varphi-E_(p)( varphi|_(Y))subP_(Y^(_|_))(varphi)\varphi-E_{p}\left(\left.\varphi\right|_{Y}\right) \subset P_{Y^{\perp}}(\varphi)φ−Ep(φ|Y)⊂PY⊥(φ).
If ψ ∈ P Y ⊥ ( φ ) ψ ∈ P Y ⊥ ( φ ) psi inP_(Y^(_|_))(varphi)\psi \in P_{Y^{\perp}}(\varphi)ψ∈PY⊥(φ) and Φ := φ − ψ Φ := φ − ψ Phi:=varphi-psi\Phi:=\varphi-\psiΦ:=φ−ψ, then Φ | Y = φ | Y Φ Y = φ Y Phi|_(Y)= varphi|_(Y)\left.\Phi\right|_{Y}=\left.\varphi\right|_{Y}Φ|Y=φ|Y and ‖ Φ | p = ‖ φ − ψ | p = d p ( Y ⊥ , φ ) = ‖ φ | Y | p Φ p = φ − ψ p = d p Y ⊥ , φ = ‖ φ Y p || Phi|_(p)=||varphi- psi|_(p)=d_(p)(Y^(_|_),varphi)=|| varphi|_(Y)|_(p)\left\|\left.\Phi\right|_{p}=\right\| \varphi-\left.\psi\right|_{p}= d_{p}\left(Y^{\perp}, \varphi\right)=\|\left.\left.\varphi\right|_{Y}\right|_{p}‖Φ|p=‖φ−ψ|p=dp(Y⊥,φ)=‖φ|Y|p, i.e. Φ ∈ E p ( φ | Y ) Φ ∈ E p φ Y Phi inE_(p)( varphi|_(Y))\Phi \in E_{p}\left(\left.\varphi\right|_{Y}\right)Φ∈Ep(φ|Y), showing that φ − P Y ⊥ ( φ ) ⊂ E p ( φ | Y ) φ − P Y ⊥ ( φ ) ⊂ E p φ Y varphi-P_(Y^(_|_))(varphi)subE_(p)( varphi|_(Y))\varphi-P_{Y^{\perp}}(\varphi) \subset E_{p}\left(\left.\varphi\right|_{Y}\right)φ−PY⊥(φ)⊂Ep(φ|Y), or equivalently, P Y ⊥ ( φ ) ⊂ φ − E p ( φ | Y ) P Y ⊥ ( φ ) ⊂ φ − E p φ Y P_(Y^(_|_))(varphi)sub varphi-E_(p)( varphi|_(Y))P_{Y^{\perp}}(\varphi) \subset \varphi-E_{p}\left(\left.\varphi\right|_{Y}\right)PY⊥(φ)⊂φ−Ep(φ|Y).
From these theorems we obtain some uniqueness conditions for the minimization problems we have considered, in terms of the uniqueness of normpreserving extensions.
Corollary 2.8. Let ( X , p ) ( X , p ) (X,p)(X, p)(X,p) be a space with asymmetric seminorm and Y Y YYY a subspace of X X XXX.
(1) If every f ∈ Y p b f ∈ Y p b f inY_(p)^(b)f \in Y_{p}^{b}f∈Ypb has a unique norm preserving extension F ∈ X p b F ∈ X p b F inX_(p)^(b)F \in X_{p}^{b}F∈Xpb, then the minimization problem (2.2) has a unique solution for every φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb.
(2) Every point φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb has a unique p ¯ p ¯ bar(p)\bar{p}p¯-nearest point in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ if and only if every f ∈ Y p b f ∈ Y p b f inY_(p)^(b)f \in Y_{p}^{b}f∈Ypb has a unique norm-preserving extension F ∈ X p b F ∈ X p b F inX_(p)^(b)F \in X_{p}^{b}F∈Xpb.
Proof. (1) If every f ∈ Y p b f ∈ Y p b f inY_(p)^(b)f \in Y_{p}^{b}f∈Ypb has a unique norm-preserving extension F ∈ X p b F ∈ X p b F inX_(p)^(b)F \in X_{p}^{b}F∈Xpb, then for every φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb the set Π Y p ⊥ ( φ ) = φ + E p ( φ | Y ) Π Y p ⊥ ( φ ) = φ + E p φ Y Pi_(Y_(p)^(_|_))(varphi)=varphi+E_(p)( varphi|_(Y))\Pi_{Y_{p}^{\perp}}(\varphi)=\varphi+E_{p}\left(\left.\varphi\right|_{Y}\right)ΠYp⊥(φ)=φ+Ep(φ|Y) contains exactly one element.
(2) Similarly, P Y ⊥ ( φ ) = φ − E p ( φ | Y ) P Y ⊥ ( φ ) = φ − E p φ Y P_(Y^(_|_))(varphi)=varphi-E_(p)( varphi|_(Y))P_{Y^{\perp}}(\varphi)=\varphi-E_{p}\left(\left.\varphi\right|_{Y}\right)PY⊥(φ)=φ−Ep(φ|Y) contains exactly one element, provided every f ∈ Y p b f ∈ Y p b f inY_(p)^(b)f \in Y_{p}^{b}f∈Ypb has exactly one norm-preserving extension F ∈ X p b F ∈ X p b F inX_(p)^(b)F \in X_{p}^{b}F∈Xpb.
Conversely, suppose that there exists f ∈ Y p b f ∈ Y p b f inY_(p)^(b)f \in Y_{p}^{b}f∈Ypb having two distinct normpreserving extensions F 1 , F 2 ∈ X p b F 1 , F 2 ∈ X p b F_(1),F_(2)inX_(p)^(b)F_{1}, F_{2} \in X_{p}^{b}F1,F2∈Xpb. Then
P Y ⊥ ( F 1 ) = F 1 − E p ( F 1 | Y ) = F 1 − E p ( f ) ⊃ { 0 , F 1 − F 2 } . P Y ⊥ F 1 = F 1 − E p F 1 Y = F 1 − E p ( f ) ⊃ 0 , F 1 − F 2 . P_(Y^(_|_))(F_(1))=F_(1)-E_(p)(F_(1)|_(Y))=F_(1)-E_(p)(f)sup{0,F_(1)-F_(2)}.P_{Y^{\perp}}\left(F_{1}\right)=F_{1}-E_{p}\left(\left.F_{1}\right|_{Y}\right)=F_{1}-E_{p}(f) \supset\left\{0, F_{1}-F_{2}\right\} .PY⊥(F1)=F1−Ep(F1|Y)=F1−Ep(f)⊃{0,F1−F2}.
REMARK 2.9. We can not prove the reverse implication in the assertion (1) of the above corollary. To do this we would need an extension theorem for ( p , p ¯ p , p ¯ p, bar(p)p, \bar{p}p,p¯ )-bounded linear functionals, preserving both p p ppp - and p ¯ p ¯ bar(p)\bar{p}p¯-norm, and we are not aware of such a result.
Some results connecting the ϵ ϵ epsilon\epsilonϵ-approximations and ϵ ϵ epsilon\epsilonϵ-extensions were obtained by Rezapour [38]. In the next proposition we transpose these results to the asymmetric case.
Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm and Y Y YYY a subspace of X X XXX. For x ∈ X x ∈ X x in Xx \in Xx∈X and ϵ > 0 ϵ > 0 epsilon > 0\epsilon>0ϵ>0 let
P Y ϵ ( x ) = { y ∈ Y : p ( y − x ) ≤ d p ( x , Y ) + ϵ } P Y ϵ ( x ) = y ∈ Y : p ( y − x ) ≤ d p ( x , Y ) + ϵ P_(Y)^(epsilon)(x)={y in Y:p(y-x) <= d_(p)(x,Y)+epsilon}P_{Y}^{\epsilon}(x)=\left\{y \in Y: p(y-x) \leq d_{p}(x, Y)+\epsilon\right\}PYϵ(x)={y∈Y:p(y−x)≤dp(x,Y)+ϵ}
and
P ¯ Y ϵ ( x ) = { y ∈ Y : p ( x − y ) ≤ d p ( Y , x ) + ϵ } P ¯ Y ϵ ( x ) = y ∈ Y : p ( x − y ) ≤ d p ( Y , x ) + ϵ bar(P)_(Y)^(epsilon)(x)={y in Y:p(x-y) <= d_(p)(Y,x)+epsilon}\bar{P}_{Y}^{\epsilon}(x)=\left\{y \in Y: p(x-y) \leq d_{p}(Y, x)+\epsilon\right\}P¯Yϵ(x)={y∈Y:p(x−y)≤dp(Y,x)+ϵ}
denote the nonempty sets of ϵ − p ϵ − p epsilon-p\epsilon-pϵ−p-, respectively ϵ − p ¯ ϵ − p ¯ epsilon- bar(p)\epsilon-\bar{p}ϵ−p¯-nearest points to x x xxx in Y Y YYY. For φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb consider the set of ϵ ϵ epsilon\epsilonϵ-solutions of the minimization problem (2.2)
Π Y p ⊥ ϵ ( φ ) = { ψ ∈ Y p ⊥ : ‖ φ + ψ | p ≤ γ ( φ , Y p ⊥ ) + ϵ } Π Y p ⊥ ϵ ( φ ) = ψ ∈ Y p ⊥ : ‖ φ + ψ p ≤ γ φ , Y p ⊥ + ϵ Pi_(Y_(p)^(_|_))^(epsilon)(varphi)={psi inY_(p)^(_|_):||varphi+ psi|_(p) <= gamma(varphi,Y_(p)^(_|_))+epsilon}\Pi_{Y_{p}^{\perp}}^{\epsilon}(\varphi)=\left\{\psi \in Y_{p}^{\perp}: \| \varphi+\left.\psi\right|_{p} \leq \gamma\left(\varphi, Y_{p}^{\perp}\right)+\epsilon\right\}ΠYp⊥ϵ(φ)={ψ∈Yp⊥:‖φ+ψ|p≤γ(φ,Yp⊥)+ϵ}
and, finally, denote by
E p ϵ ( f ) = { F ∈ X p b : F | Y = f and ‖ F | p ≤ ‖ f | p + ϵ } , E p ϵ ( f ) = F ∈ X p b : F Y = f  and  F p ≤ f p + ϵ , E_(p)^(epsilon)(f)={F inX_(p)^(b):F|_(Y)=f" and "||F|_(p) <= ||f|_(p)+epsilon},E_{p}^{\epsilon}(f)=\left\{F \in X_{p}^{b}:\left.F\right|_{Y}=f \text { and }\left.\left\|\left.F\right|_{p} \leq\right\| f\right|_{p}+\epsilon\right\},Epϵ(f)={F∈Xpb:F|Y=f and ‖F|p≤‖f|p+ϵ},
the set of ϵ ϵ epsilon\epsilonϵ-extensions of a functional f ∈ Y p b f ∈ Y p b f inY_(p)^(b)f \in Y_{p}^{b}f∈Ypb.
These two sets are related in the following way.
Proposition 2.10. Let ( X , p X , p X,pX, pX,p ) be a space with asymmetric seminorm, Y Y YYY a subspace of X X XXX and φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb. Then
Π Y p ⊥ ϵ ( φ ) = E p ϵ ( φ | Y ) − φ . Π Y p ⊥ ϵ ( φ ) = E p ϵ φ Y − φ . Pi_(Y_(p)^(_|_))^(epsilon)(varphi)=E_(p)^(epsilon)( varphi|_(Y))-varphi.\Pi_{Y_{p}^{\perp}}^{\epsilon}(\varphi)=E_{p}^{\epsilon}\left(\left.\varphi\right|_{Y}\right)-\varphi .ΠYp⊥ϵ(φ)=Epϵ(φ|Y)−φ.
Proof. Indeed, by Theorem 2.6,
ψ ∈ Π Y p ⊥ ϵ ( φ ) ⟺ ψ ∈ Y p ⊥ and ‖ φ + ψ | p ≤ γ ( φ , Y p ⊥ ) + ϵ = ‖ φ | Y | p + ϵ ⟺ φ + ψ ∈ E p ϵ ( φ | Y ) ψ ∈ Π Y p ⊥ ϵ ( φ ) ⟺ ψ ∈ Y p ⊥  and  φ + ψ p ≤ γ φ , Y p ⊥ + ϵ = φ Y p + ϵ ⟺ φ + ψ ∈ E p ϵ φ Y {:[psi inPi_(Y_(p)^(_|_))^(epsilon)(varphi) Longleftrightarrow psi inY_(p)^(_|_)" and "||varphi+ psi|_(p) <= gamma(varphi,Y_(p)^(_|_))+epsilon=||varphi|_(Y)|_(p)+epsilon],[ Longleftrightarrow varphi+psi inE_(p)^(epsilon)( varphi|_(Y))]:}\begin{aligned} \psi \in \Pi_{Y_{p}^{\perp}}^{\epsilon}(\varphi) & \Longleftrightarrow \psi \in Y_{p}^{\perp} \text { and }\left.\left.\left\|\varphi+\left.\psi\right|_{p} \leq \gamma\left(\varphi, Y_{p}^{\perp}\right)+\epsilon=\right\| \varphi\right|_{Y}\right|_{p}+\epsilon \\ & \Longleftrightarrow \varphi+\psi \in E_{p}^{\epsilon}\left(\left.\varphi\right|_{Y}\right) \end{aligned}ψ∈ΠYp⊥ϵ(φ)⟺ψ∈Yp⊥ and ‖φ+ψ|p≤γ(φ,Yp⊥)+ϵ=‖φ|Y|p+ϵ⟺φ+ψ∈Epϵ(φ|Y)
Working with the annihilator Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ of the subspace Y Y YYY in the symmetric dual X ∗ = ( X , p s ) ∗ X ∗ = X , p s ∗ X^(**)=(X,p^(s))^(**)X^{*}=\left(X, p^{s}\right)^{*}X∗=(X,ps)∗ given by (2.4) and putting
P ¯ Y ⊥ ϵ ( φ ) = { ψ ∈ Y ⊥ : ‖ φ − ψ | p ≤ d p ( φ , Y ⊥ ) + ϵ } P ¯ Y ⊥ ϵ ( φ ) = ψ ∈ Y ⊥ : ‖ φ − ψ p ≤ d p φ , Y ⊥ + ϵ bar(P)_(Y^(_|_))^(epsilon)(varphi)={psi inY^(_|_):||varphi- psi|_(p) <= d_(p)(varphi,Y^(_|_))+epsilon}\bar{P}_{Y^{\perp}}^{\epsilon}(\varphi)=\left\{\psi \in Y^{\perp}: \| \varphi-\left.\psi\right|_{p} \leq d_{p}\left(\varphi, Y^{\perp}\right)+\epsilon\right\}P¯Y⊥ϵ(φ)={ψ∈Y⊥:‖φ−ψ|p≤dp(φ,Y⊥)+ϵ}
we have
Proposition 2.11. Let Y Y YYY be a subspace of a space with asymmetric seminorm ( X , p ) , ϵ > 0 ( X , p ) , ϵ > 0 (X,p),epsilon > 0(X, p), \epsilon>0(X,p),ϵ>0, and φ ∈ X p b φ ∈ X p b varphi inX_(p)^(b)\varphi \in X_{p}^{b}φ∈Xpb. Then
P ¯ Y ϵ ( φ ) = φ − E p ϵ ( φ | Y ) . P ¯ Y ϵ ( φ ) = φ − E p ϵ φ Y . bar(P)_(Y)^(epsilon)(varphi)=varphi-E_(p)^(epsilon)( varphi|_(Y)).\bar{P}_{Y}^{\epsilon}(\varphi)=\varphi-E_{p}^{\epsilon}\left(\left.\varphi\right|_{Y}\right) .P¯Yϵ(φ)=φ−Epϵ(φ|Y).
Proof. Indeed, by Theorem 2.7,
ψ ∈ P ¯ Y ⊥ ϵ ( φ ) ⟺ ψ ∈ Y ⊥ and ‖ φ − ψ | p ≤ d p ( Y ⊥ , φ ) + ϵ = ‖ φ | Y | p + ϵ ⟺ φ − ψ ∈ E p ϵ ( φ | Y ) . ψ ∈ P ¯ Y ⊥ ϵ ( φ ) ⟺ ψ ∈ Y ⊥  and  φ − ψ p ≤ d p Y ⊥ , φ + ϵ = φ Y p + ϵ ⟺ φ − ψ ∈ E p ϵ φ Y . {:[psi in bar(P)_(Y^(_|_))^(epsilon)(varphi) Longleftrightarrow psi inY^(_|_)" and "||varphi- psi|_(p) <= d_(p)(Y^(_|_),varphi)+epsilon=||varphi|_(Y)|_(p)+epsilon],[ Longleftrightarrow varphi-psi inE_(p)^(epsilon)( varphi|_(Y)).]:}\begin{aligned} \psi \in \bar{P}_{Y^{\perp}}^{\epsilon}(\varphi) & \Longleftrightarrow \psi \in Y^{\perp} \text { and }\left.\left.\left\|\varphi-\left.\psi\right|_{p} \leq d_{p}\left(Y^{\perp}, \varphi\right)+\epsilon=\right\| \varphi\right|_{Y}\right|_{p}+\epsilon \\ & \Longleftrightarrow \varphi-\psi \in E_{p}^{\epsilon}\left(\left.\varphi\right|_{Y}\right) . \end{aligned}ψ∈P¯Y⊥ϵ(φ)⟺ψ∈Y⊥ and ‖φ−ψ|p≤dp(Y⊥,φ)+ϵ=‖φ|Y|p+ϵ⟺φ−ψ∈Epϵ(φ|Y).

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Received by the editors: March 9, 2006.
  1. *"Babeş-Bolyai" University, Faculty of Mathematics and Computer Science, 400084 Cluj-Napoca, Romania, e-mail: scobzas@math.ubbcluj.ro.
    † † ^(†){ }^{\dagger}† "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.
2006

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