A Phelps type result for spaces with asymmetric norms

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

Abstract


If \((X,||\cdot|)\) is a linear space with asymmetric norm and \(Y\) is a \(X\), for every \(f\in Y_{+}^{\ast}\) (the cone of linear bounded functional on \(Y\)) there exists functional \(F\in Y_{+}^{\ast}\) extending \(f\) and preserving the asymmetric norm of \(f\).The problem of uniqueness of the extension in terms of uniqueness of elements of best of \(F\in X_{+}^{\ast}\) by elements of \(Y_{+}^{\perp}=\{G\in X_{+}^{\ast}:\left. G\right \vert _{Y}-0,F\geq G\}\), is discussed.

Authors

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

Keywords

asymmetric norm; extension and approximation.

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C. Mustăţa, A Phelps type result for spaces with asymmetric norms, Bul. Şt. Univ. Baia Mare, Seria B, Fascicola matematică-informatică, 18 (2002) no. 2, 275-280.

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[4] Garcia -Raffi, L.M.; Romaguera S., and Sanchez Pérez E.A., Extension of Asymmetric Norms to Linear Spaces, Rend. Istit. Mat. Trieste XXXIII, 113-125 (2001)
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[8] Mustata, C., Extensions of Semi-Lipschitz functions on quasi-Metric spaces, Rev. Anal. Numér. Théor. Approx.. 30 (2001) No.l, 61-67
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2002-Mustata-A Phelps type result-BulBaiaMare

A PHELPS TYPE THEOREM FOR SPACES WITH ASYMMETRIC NORMS

Costică MUSTĂTA

Abstract

If ( X , ‖ ⋅ ∣ X , ‖ ⋅ ∣ X,||*∣X, \| \cdot \midX,‖⋅∣ ) is a linear space with asymmetric norm and Y Y YYY is a subspace of X X XXX, for every f ∈ Y + ∗ f ∈ Y + ∗ f inY_(+)^(**)f \in Y_{+}^{*}f∈Y+∗ (the cone of linear bounded functional on Y Y YYY ) there exists at most one functional F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗ extending f f fff and preserving the asymmetric norm of f f fff. The problem of uniqueness of the extension in terms of uniqueness of elements of best approximation of F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗ by elements of Y + ⊥ = { G ∈ X + ∗ : G | Y = 0 , F ≥ G } Y + ⊥ = G ∈ X + ∗ : G Y = 0 , F ≥ G Y_(+)^(_|_)={G inX_(+)^(**):G|_(Y)=0,F >= G}Y_{+}^{\perp}=\left\{G \in X_{+}^{*}:\left.G\right|_{Y}=0, F \geq G\right\}Y+⊥={G∈X+∗:G|Y=0,F≥G} is discussed.

MSC: 41A65, 41A52, 46A22
Keywords: asymmetric norm, extension and approximation

1. Asymmetric norms

Let X X XXX be a real linear space and ‖ ⋅ ∣: X → [ 0 , ∞ ) ‖ ⋅ ∣: X → [ 0 , ∞ ) ||*∣:X rarr[0,oo)\| \cdot \mid: X \rightarrow[0, \infty)‖⋅∣:X→[0,∞) a function with the following properties:
  1. ‖ x ∣> 0 ‖ x ∣> 0 ||x∣>0\| x \mid>0‖x∣>0 for all x ≠ θ ; 2 ) ‖ λ x | = λ ‖ x | x ≠ θ ; 2 ) ‖ λ x | = λ ‖ x | x!=theta;2)||lambda x|=lambda||x|x \neq \theta ; 2)\|\lambda x|=\lambda \| x|x≠θ;2)‖λx|=λ‖x| for all λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0 and all x ∈ X ; 3 ) ‖ x + y ∣≤ | | x | + | | y | x ∈ X ; 3 ) ‖ x + y ∣≤ | | x | + | | y | x in X;3)||x+y∣≤||x|+||y|x \in X ; 3) \| x+y \mid \leq ||x|+||y|x∈X;3)‖x+y∣≤||x|+||y| for all x , y , ∈ X x , y , ∈ X x,y,in Xx, y, \in Xx,y,∈X. Then the function ‖ ⋅ | ‖ ⋅ | ||*|\| \cdot|‖⋅| is called an asymmetric norm on X X XXX and the pair ( X , ‖ ⋅ ‖ X , ‖ ⋅ ‖ X,||*||X,\|\cdot\|X,‖⋅‖ ) is called a space with asymmetric norm (see [5]). In such a space, in general ‖ − x | ≠ ‖ x | ‖ − x | ≠ ‖ x | ||-x|!=||x|\|-x|\neq \| x|‖−x|≠‖x|.
Example ([1]) Consider the real linear space
C 0 ( [ 0 , 1 ] , 1 , 0 ) = { x : [ 0 , 1 ] → R , x is continuous and ∫ 0 1 x ( t ) d t = 0 } C 0 ( [ 0 , 1 ] , 1 , 0 ) = x : [ 0 , 1 ] → R , x  is continuous and  ∫ 0 1   x ( t ) d t = 0 C_(0)([0,1],1,0)={x:[0,1]rarrR,x" is continuous and "int_(0)^(1)x(t)dt=0}C_{0}([0,1], 1,0)=\left\{x:[0,1] \rightarrow \mathbb{R}, x \text { is continuous and } \int_{0}^{1} x(t) d t=0\right\}C0([0,1],1,0)={x:[0,1]→R,x is continuous and ∫01x(t)dt=0}
The function ‖ ⋅ | : C 0 ( [ 0 , 1 ] , 1 , 0 ) → [ 0 , ∞ ) , ‖ x | = max { x ( t ) : t ∈ [ 0 , 1 ] } ⋅ : C 0 ( [ 0 , 1 ] , 1 , 0 ) → [ 0 , ∞ ) , ‖ x = max { x ( t ) : t ∈ [ 0 , 1 ] } ||*|:C_(0)([0,1],1,0)rarr[0,oo),||x|=max{x(t):t in[0,1]}:}\left\|\cdot\left|: C_{0}([0,1], 1,0) \rightarrow[0, \infty), \| x\right|=\max \{x(t): t \in[0,1]\}\right.‖⋅|:C0([0,1],1,0)→[0,∞),‖x|=max{x(t):t∈[0,1]} satisfies the properties 1) - 3) of asymmetric norm. The functions x α ( t ) = α ( t − 1 2 ) , α ∈ R x α ( t ) = α t − 1 2 , α ∈ R x_(alpha)(t)=alpha(t-(1)/(2)),alpha inRx_{\alpha}(t)=\alpha\left(t-\frac{1}{2}\right), \alpha \in \mathbb{R}xα(t)=α(t−12),α∈R are in C 0 ( [ 0 , 1 ] , 1 , 0 ) C 0 ( [ 0 , 1 ] , 1 , 0 ) C_(0)([0,1],1,0)C_{0}([0,1], 1,0)C0([0,1],1,0) and ‖ x α | = | α | 2 = ‖ − x α | x α = | α | 2 = − x α {:||x_(alpha)|_(=)(|alpha|)/(2)=||-x_(alpha)|\left.\left\|\left.x_{\alpha}\right|_{=} \frac{|\alpha|}{2}=\right\|-x_{\alpha} \right\rvert\,‖xα|=|α|2=‖−xα|, but the functions y n ( t ) = 1 − n t n − 1 , n > 2 ( n ∈ N ) y n ( t ) = 1 − n t n − 1 , n > 2 ( n ∈ N ) y_(n)(t)=1-nt^(n-1),n > 2(n inN)y_{n}(t)=1-n t^{n-1}, n> 2(n \in \mathbb{N})yn(t)=1−ntn−1,n>2(n∈N), which also belong to C ( [ 0 , 1 ] , 1 , 0 ) C ( [ 0 , 1 ] , 1 , 0 ) C([0,1],1,0)C([0,1], 1,0)C([0,1],1,0) satisfy ‖ y n | = 1 ‖ y n = 1 ||y_(n)|^(=1)\|\left. y_{n}\right|^{=1}‖yn|=1 and ‖ − y n ∣= n − 1 > 1 ‖ − y n ∣= n − 1 > 1 ||-y_(n)∣=n-1 > 1\|-y_{n} \mid=n-1>1‖−yn∣=n−1>1, i.e. | | y n | ≠ | | − y n | y n ≠ − y n ||y_(n)|!=||-y_(n)|:}\left|\left|y_{n}\right| \neq\left|\left|-y_{n}\right|\right.\right.||yn|≠||−yn|.
By definition, the balls B ( x , r ) = { y ∈ X : ‖ y − x ∣< r } x ∈ X B ( x , r ) = { y ∈ X : ‖ y − x ∣< r } x ∈ X B(x,r)={y in X:||y-x∣<r}quad x in XB(x, r)=\{y \in X: \| y-x \mid<r\} \quad x \in XB(x,r)={y∈X:‖y−x∣<r}x∈X and r > 0 r > 0 r > 0r>0r>0 form a base of the topology of the space ( X , ‖ ⋅ ∣ ) ( X , ‖ ⋅ ∣ ) (X,||*∣)(X, \| \cdot \mid)(X,‖⋅∣). The space ( X , ‖ ⋅ ∣ X , ‖ ⋅ ∣ X,||*∣X, \| \cdot \midX,‖⋅∣ ) equipped with this topology need not be a topological linear space, since the multiplication by scalars is not continuous. In the preceding example, for x = 0 x = 0 x=0x=0x=0 and λ = − 1 , ( − 1 ) 0 = 0 λ = − 1 , ( − 1 ) 0 = 0 lambda=-1,(-1)0=0\lambda=-1,(-1) 0=0λ=−1,(−1)0=0 and for all r > 0 , − B ( 0 , r ) ⊈ B ( 0 , 1 ) r > 0 , − B ( 0 , r ) ⊈ B ( 0 , 1 ) r > 0,-B(0,r)⊈B(0,1)r>0,-B(0, r) \nsubseteq B(0,1)r>0,−B(0,r)⊈B(0,1) i.e. the multiplication by scalars is not continuous.
For each asymmetric norm ‖ ⋅ ∣ ‖ ⋅ ∣ ||*∣\| \cdot \mid‖⋅∣ on X X XXX one defines ‖ x ‖ = max { ‖ x | , ‖ − x | } ‖ x ‖ = max { ‖ x | , ‖ − x | } ||x||=max{||x|,||-x|}\|x\|=\max \{\|x|, \|-x|\}‖x‖=max{‖x|,‖−x|}. Then ‖ x ∣≤ ‖ x ‖ , x ∈ X ‖ x ∣≤ ‖ x ‖ , x ∈ X ||x∣≤||x||,x in X\| x \mid \leq \|x\|, x \in X‖x∣≤‖x‖,x∈X. If there exists c > 0 c > 0 c > 0c>0c>0 such that ‖ x ‖ ≤ c ‖ x ∣ ‖ x ‖ ≤ c ‖ x ∣ ||x|| <= c||x∣\|x\| \leq c \| x \mid‖x‖≤c‖x∣, i.e. the norm ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖ and asymmetric
norm ‖ ⋅ ∣ ‖ ⋅ ∣ ||*∣\| \cdot \mid‖⋅∣ are equivalent, then ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) is a topological linear space. Such a situation occurs when dim X < ∞ dim X < ∞ dim X < oo\operatorname{dim} X<\inftydimX<∞. In this case all the norms and asymmetric norms are equivalent ([5], I.2.1. pp.21-23). If ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖ and ‖ ⋅ ∣ ‖ ⋅ ∣ ||*∣\| \cdot \mid‖⋅∣ are equivalent then ‖ ⋅ ∣ ‖ ⋅ ∣ ||*∣\| \cdot \mid‖⋅∣ is continuous on X X XXX.
An example of an asymmetric norm on the normed space ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) is given by ‖ x ∣= ‖ x ‖ + φ ( x ) , x ∈ X ‖ x ∣= ‖ x ‖ + φ ( x ) , x ∈ X ||x∣=||x||+varphi(x),x in X\| x \mid= \|x\|+\varphi(x), x \in X‖x∣=‖x‖+φ(x),x∈X where φ ∈ X ∗ , φ ≠ 0 φ ∈ X ∗ , φ ≠ 0 varphi inX^(**),varphi!=0\varphi \in X^{*}, \varphi \neq 0φ∈X∗,φ≠0, (a linear and continuous functional on X X XXX ).
2. Linear and bounded functional on a linear space with asymmetric norm. Let ( X , ‖ ‖ ) ( X , ‖ ‖ ) (X,||||)(X,\| \|)(X,‖‖) be a space with asymmetric norm and f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R a linear functional. The linear functional f f fff is called bounded if
(1) ‖ f | := sup { f ( x ) ‖ x ∣ < ∞ : x ≠ 0 } < ∞ . (1) ‖ f := sup f ( x ) ‖ x ∣ < ∞ : x ≠ 0 < ∞ . {:(1)||f|:=s u p{(f(x))/(||x∣) < oo:x!=0} < oo.:}:}\begin{equation*} \| f \left\lvert\,:=\sup \left\{\frac{f(x)}{\| x \mid}<\infty: x \neq 0\right\}<\infty .\right. \tag{1} \end{equation*}(1)‖f|:=sup{f(x)‖x∣<∞:x≠0}<∞.
(see [5], Ch.9, Sec.5, p.483). If f f fff is a linear and bounded functional, then
(2) f ( x ) ≤ | | f | ⋅ | | x | , x ∈ X , (2) f ( x ) ≤ | | f | ⋅ | | x | , x ∈ X , {:(2)f(x) <= ||f|*||x|","x in X",":}\begin{equation*} f(x) \leq||f| \cdot||x|, x \in X, \tag{2} \end{equation*}(2)f(x)≤||f|⋅||x|,x∈X,
and, changing x x xxx with − x − x -x-x−x, one obtains − f ( x ) = f ( − x ) ≤ ‖ f | ⋅ ‖ − x | − f ( x ) = f ( − x ) ≤ ‖ f | ⋅ ‖ − x | -f(x)=f(-x) <= ||f|*||-x|-f(x)=f(-x) \leq\|f|\cdot \|-x|−f(x)=f(−x)≤‖f|⋅‖−x|. Consequently
− ‖ f | ⋅ ‖ − x | ≤ f ( x ) ≤ ‖ f | ⋅ ‖ x | , x ∈ X − ‖ f | ⋅ ‖ − x | ≤ f ( x ) ≤ ‖ f | ⋅ ‖ x | , x ∈ X -||f|*||-x| <= f(x) <= ||f|*||x|,x in X-\|f|\cdot\|-x|\leq f(x) \leq\|f|\cdot \| x|, x \in X−‖f|⋅‖−x|≤f(x)≤‖f|⋅‖x|,x∈X
and in general, ‖ − f | ≠ ‖ f | − f | ≠ ‖ f | ||-f|!=||f|:}\left\|-f|\neq \| f|\right.‖−f|≠‖f|. Denote by X # X # X^(#)X^{\#}X# the algebraic dual of the linear space X X XXX and by X + ∗ X + ∗ X_(+)^(**)X_{+}^{*}X+∗ the set of all linear and bounded functional on the space X X XXX with an asymmetric norm |||.
For f , g ∈ X + ∗ f , g ∈ X + ∗ f,g inX_(+)^(**)f, g \in X_{+}^{*}f,g∈X+∗ one obtains f + g ∈ X + ∗ f + g ∈ X + ∗ f+g inX_(+)^(**)f+g \in X_{+}^{*}f+g∈X+∗ and λ f ∈ X + ∗ ( λ ≥ 0 ) ( λ f + μ g ∈ X + ∗ λ f ∈ X + ∗ ( λ ≥ 0 ) λ f + μ g ∈ X + ∗ lambda f inX_(+)^(**)(lambda >= 0)(lambda f+mu g inX_(+)^(**):}\lambda f \in X_{+}^{*}(\lambda \geq 0)\left(\lambda f+\mu g \in X_{+}^{*}\right.λf∈X+∗(λ≥0)(λf+μg∈X+∗, for all f , g , ∈ X + ∗ f , g , ∈ X + ∗ f,g,inX_(+)^(**)f, g, \in X_{+}^{*}f,g,∈X+∗ and all λ , μ ≥ 0 λ , μ ≥ 0 lambda,mu >= 0\lambda, \mu \geq 0λ,μ≥0 ). Consequently X + ∗ X + ∗ X_(+)^(**)X_{+}^{*}X+∗ is a convex cone in X # X # X^(#)X^{\#}X#.
The functional ‖ ⋅ ∣: X + ∗ → [ 0 , ∞ ) ‖ ⋅ ∣: X + ∗ → [ 0 , ∞ ) ||*∣:X_(+)^(**)rarr[0,oo)\| \cdot \mid: X_{+}^{*} \rightarrow[0, \infty)‖⋅∣:X+∗→[0,∞) defined by formula (1) satisfies the axioms 1) - 3) of an asymmetric norm. Indeed, if f ≠ 0 f ≠ 0 f!=0f \neq 0f≠0 then there exists x ∈ X , x ≠ 0 x ∈ X , x ≠ 0 x in X,x!=0x \in X, x \neq 0x∈X,x≠0 such that f ( x ) > 0 f ( x ) > 0 f(x) > 0f(x)>0f(x)>0 or f ( − x ) > 0 f ( − x ) > 0 f(-x) > 0f(-x)>0f(−x)>0. It follows that ‖ f ‖ = sup ( f ( x ) / ‖ x ‖ ) > 0 ‖ f ‖ = sup ( f ( x ) / ‖ x ‖ ) > 0 ||f||=s u p(f(x)//||x||) > 0\|f\|=\sup (f(x) /\|x\|)>0‖f‖=sup(f(x)/‖x‖)>0. If λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0 then ‖ λ f | = λ ‖ f | ‖ λ f | = λ ‖ f | ||lambda f|=lambda||f|\|\lambda f|=\lambda \| f|‖λf|=λ‖f| and | | f + g | ≤ | | f | + | | g | | | f + g | ≤ | | f | + | | g | ||f+g| <= ||f|+||g|||f+g| \leq||f|+||g|||f+g|≤||f|+||g| are evidently fullfielled.
Finally, observe that the function d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty)d:X×X→[0,∞) defined by
(3) d ( x , y ) = ‖ x − y ∣ , x , y ∈ X (3) d ( x , y ) = ‖ x − y ∣ , x , y ∈ X {:(3)d(x","y)=||x-y∣","quad x","y in X:}\begin{equation*} d(x, y)=\| x-y \mid, \quad x, y \in X \tag{3} \end{equation*}(3)d(x,y)=‖x−y∣,x,y∈X
where X X XXX is a space with asymmetric norm ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖, is a quasi-metric on X X XXX, i.e. d d ddd satisfies the conditions:
a) d ( x , y ) = 0 ⟺ x = y d ( x , y ) = 0 ⟺ x = y d(x,y)=0Longleftrightarrow x=yd(x, y)=0 \Longleftrightarrow x=yd(x,y)=0⟺x=y;
b) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) , x , y , z ∈ X ( d ( x , y ) ≤ d ( x , z ) + d ( z , y ) , x , y , z ∈ X ( d(x,y) <= d(x,z)+d(z,y),quad x,y,z in X(d(x, y) \leq d(x, z)+d(z, y), \quad x, y, z \in X(d(x,y)≤d(x,z)+d(z,y),x,y,z∈X( see [6] ) ) )))
For f ∈ X + ∗ f ∈ X + ∗ f inX_(+)^(**)f \in X_{+}^{*}f∈X+∗ and all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X, we have f ( x − y ) ≤ ‖ f | ⋅ ‖ x − y | f ( x − y ) ≤ ‖ f | ⋅ ‖ x − y | f(x-y) <= ||f|*||x-y|f(x-y) \leq\|f|\cdot \| x-y|f(x−y)≤‖f|⋅‖x−y|, so that
(4) f ( x ) − f ( y ) ≤ ‖ f | ⋅ ‖ x − y | , x , y ∈ X . (4) f ( x ) − f ( y ) ≤ ‖ f | ⋅ ‖ x − y | , x , y ∈ X . {:(4)f(x)-f(y) <= ||f|*||x-y|","quad x","y in X.:}\begin{equation*} f(x)-f(y) \leq\|f|\cdot \| x-y|, \quad x, y \in X . \tag{4} \end{equation*}(4)f(x)−f(y)≤‖f|⋅‖x−y|,x,y∈X.
The last inequality means that every bounded linear functional f f fff on ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) is semiLipschitz (see [11]) i.e. X + ∗ ⊂ S Lip 0 X X + ∗ ⊂ S Lip 0 X X_(+)^(**)sub SLip_(0)XX_{+}^{*} \subset S \operatorname{Lip}_{0} XX+∗⊂SLip0X where
S Lip 0 X = { f : X → R , f ( 0 ) = 0 , sup ( f ( x ) − f ( y ) ) ∨ 0 ‖ x − y ‖ < ∞ } S Lip 0 X = f : X → R , f ( 0 ) = 0 , sup ( f ( x ) − f ( y ) ) ∨ 0 ‖ x − y ‖ < ∞ SLip_(0)X={f:X rarrR,f(0)=0,s u p((f(x)-f(y))vv0)/(||x-y||) < oo}S \operatorname{Lip}_{0} X=\left\{f: X \rightarrow \mathbb{R}, f(0)=0, \sup \frac{(f(x)-f(y)) \vee 0}{\|x-y\|}<\infty\right\}SLip0X={f:X→R,f(0)=0,sup(f(x)−f(y))∨0‖x−y‖<∞}
is the semi-linear space of semi-Lipschitz real functions defined on ( X , ‖ ⋅ ‖ ) ( X , ‖ ⋅ ‖ ) (X,||*||)(X,\|\cdot\|)(X,‖⋅‖) (see [11]). Because X + ∗ ⊂ S Lip 0 X X + ∗ ⊂ S Lip 0 X X_(+)^(**)sub SLip_(0)XX_{+}^{*} \subset S \operatorname{Lip}_{0} XX+∗⊂SLip0X, for every f ∈ X + ∗ f ∈ X + ∗ f inX_(+)^(**)f \in X_{+}^{*}f∈X+∗, we have
(5) sup x ≠ 0 f ( x ) ∨ 0 ‖ x ∣ = sup x ≠ 0 f ( x ) ‖ x ‖ and sup x − y ≠ 0 f ( x ) − f ( y ) ‖ x − y ‖ = ‖ f | (5) sup x ≠ 0   f ( x ) ∨ 0 ‖ x ∣ = sup x ≠ 0   f ( x ) ‖ x ‖  and  sup x − y ≠ 0   f ( x ) − f ( y ) ‖ x − y ‖ = ‖ f {:(5)s u p_(x!=0)(f(x)vv0)/(||x∣)=s u p_(x!=0)(f(x))/(||x||)quad" and "{: quads u p_(x-y!=0)(f(x)-f(y))/(||x-y||)=||f|:}\begin{equation*} \sup _{x \neq 0} \frac{f(x) \vee 0}{\| x \mid}=\sup _{x \neq 0} \frac{f(x)}{\|x\|} \quad \text { and } \left.\quad \sup _{x-y \neq 0} \frac{f(x)-f(y)}{\|x-y\|}=\| f \right\rvert\, \tag{5} \end{equation*}(5)supx≠0f(x)∨0‖x∣=supx≠0f(x)‖x‖ and supx−y≠0f(x)−f(y)‖x−y‖=‖f|
i.e. the asymmetric norm of f ∈ X + ∗ f ∈ X + ∗ f inX_(+)^(**)f \in X_{+}^{*}f∈X+∗ is the smallest semi-Lipschitz constant of f f fff.
Let Y Y YYY be a subspace of the linear space X X XXX with asymmetric norm ‖ ⋅ ∣ ‖ ⋅ ∣ ||*∣\| \cdot \mid‖⋅∣ and let f ∈ Y + ∗ f ∈ Y + ∗ f inY_(+)^(**)f \in Y_{+}^{*}f∈Y+∗. Then f ∈ S Lip 0 Y f ∈ S Lip 0 Y f in SLip_(0)Yf \in S \operatorname{Lip}_{0} Yf∈SLip0Y and, by an analogue of an extension theorem of McShane ([7]), there exists at least one function F ∈ S F ∈ S F in SF \in SF∈S Lip 0 X 0 X _(0)X_{0} X0X such that F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f and ‖ F | = ‖ f | ‖ F | = ‖ f | ||F|=||f|\|F|=\| f|‖F|=‖f| (see [8], Th.2). In our case the following result holds:
Theorem 1. ([5]). Let X X XXX be a real linear space with the asymmetric norm ‖ ⋅ ∣ ‖ ⋅ ∣ ||*∣\| \cdot \mid‖⋅∣ and Y Y YYY be a subspace of X X XXX. Then for every f ∈ Y + ∗ f ∈ Y + ∗ f inY_(+)^(**)f \in Y_{+}^{*}f∈Y+∗ there exists F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗ such that
a) F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f,
b) ‖ F | = ‖ f | ‖ F | = ‖ f | ||F|=||f|\|F|=\| f|‖F|=‖f|.
Proof. If f ∈ Y + ∗ f ∈ Y + ∗ f inY_(+)^(**)f \in Y_{+}^{*}f∈Y+∗ let p : X → R p : X → R p:X rarrRp: X \rightarrow \mathbb{R}p:X→R be defined by p ( x ) = ‖ f | ⋅ ‖ x | p ( x ) = ‖ f | ⋅ ‖ x | p(x)=||f|*||x|p(x)=\|f|\cdot \| x|p(x)=‖f|⋅‖x|. Then f ( y ) ≤ ‖ f | ⋅ ‖ y | = p ( y ) , y ≤ Y f ( y ) ≤ ‖ f | ⋅ ‖ y | = p ( y ) , y ≤ Y f(y) <= ||f|*||y|=p(y),y <= Yf(y) \leq\| f|\cdot \| y|= p(y), y \leq Yf(y)≤‖f|⋅‖y|=p(y),y≤Y, and by Hahn-Banach theorem, there exists F ∈ X ∗ F ∈ X ∗ F inX^(**)F \in X^{*}F∈X∗ such that
F | Y = f and F ( x ) ≤ ‖ f | ⋅ ‖ x | , x ∈ X . F Y = f  and  F ( x ) ≤ ‖ f | ⋅ ‖ x | , x ∈ X . F|_(Y)=f" and "F(x) <= ||f|*||x|,quad x in X.\left.F\right|_{Y}=f \text { and } F(x) \leq\|f|\cdot \| x|, \quad x \in X .F|Y=f and F(x)≤‖f|⋅‖x|,x∈X.
Then
F ( x ) ‖ x ‖ ≤ ‖ f | , x ∈ X , x ≠ 0 F ( x ) ‖ x ‖ ≤ ‖ f , x ∈ X , x ≠ 0 {:(F(x))/(||x||) <= ||f|,x in X,x!=0\left.\frac{F(x)}{\|x\|} \leq \| f \right\rvert\,, x \in X, x \neq 0F(x)‖x‖≤‖f|,x∈X,x≠0
and taking the supremum with respect to x ∈ X x ∈ X x in Xx \in Xx∈X one obtains ‖ F | ≤ ‖ f | ‖ F | ≤ ‖ f | ||F| <= ||f|\|F|\leq \| f|‖F|≤‖f|. On the other hand
‖ F ∣ = sup { F ( x ) ‖ x ∣ , x ∈ X , x ≠ 0 } ≥ sup { F ( y ) ‖ y ∣ , y ∈ Y , y ≠ 0 } = (6) = sup { f ( y ) ‖ y ∣ , y ∈ Y , y ≠ 0 } = ‖ f | ‖ F ∣ = sup F ( x ) ‖ x ∣ , x ∈ X , x ≠ 0 ≥ sup F ( y ) ‖ y ∣ , y ∈ Y , y ≠ 0 = (6) = sup f ( y ) ‖ y ∣ , y ∈ Y , y ≠ 0 = ‖ f {:[||F∣=s u p{(F(x))/(||x∣),x in X,x!=0} >= s u p{(F(y))/(||y∣),y in Y,y!=0}=],[(6){:=s u p{(f(y))/(||y∣),y in Y,y!=0}=||f|]:}\begin{align*} \| F \mid & =\sup \left\{\frac{F(x)}{\| x \mid}, x \in X, x \neq 0\right\} \geq \sup \left\{\frac{F(y)}{\| y \mid}, y \in Y, y \neq 0\right\}= \\ & \left.=\sup \left\{\frac{f(y)}{\| y \mid}, y \in Y, y \neq 0\right\}=\| f \right\rvert\, \tag{6} \end{align*}‖F∣=sup{F(x)‖x∣,x∈X,x≠0}≥sup{F(y)‖y∣,y∈Y,y≠0}=(6)=sup{f(y)‖y∣,y∈Y,y≠0}=‖f|
and, consequently ‖ F | = ‖ f | ‖ F | = ‖ f | ||F|=||f|\|F|=\| f|‖F|=‖f|.
By Theorem 1 it follows that if Y Y YYY is a subspace of ( X , ‖ ⋅ ∣ X , ‖ ⋅ ∣ X,||*∣X, \| \cdot \midX,‖⋅∣ ) then for every f ∈ Y + ∗ f ∈ Y + ∗ f inY_(+)^(**)f \in Y_{+}^{*}f∈Y+∗ the set
(6) E ( f ) = { F ∈ X + ∗ : F | Y = f and ‖ F | = ‖ f | } (6) E ( f ) = F ∈ X + ∗ : F Y = f  and  ‖ F | = ‖ f | } {:(6)E(f)={F inX_(+)^(**):F|_(Y)=f" and "||F|=||f|}:}:}\begin{equation*} \mathcal{E}(f)=\left\{F \in X_{+}^{*}:\left.F\right|_{Y}=f \text { and }\|F|=\| f|\}\right. \tag{6} \end{equation*}(6)E(f)={F∈X+∗:F|Y=f and ‖F|=‖f|}
is nonvoid.
Observe that, for every f ∈ Y + ∗ f ∈ Y + ∗ f inY_(+)^(**)f \in Y_{+}^{*}f∈Y+∗, the set E ( f ) E ( f ) E(f)\mathcal{E}(f)E(f) of all extensions of f f fff, is included in S ( 0 ) := { F ∈ X + ∗ : ‖ F | = ‖ f | } S ( 0 ) := F ∈ X + ∗ : ‖ F | = ‖ f | } S(0):={F inX_(+)^(**):||F|=||f|}:}S(0):=\left\{F \in X_{+}^{*}:\|F|=\| f|\}\right.S(0):={F∈X+∗:‖F|=‖f|} and E ( f ) E ( f ) E(f)\mathcal{E}(f)E(f) is convex.
Indeed, if F 1 , F 2 ∈ E ( f ) F 1 , F 2 ∈ E ( f ) F_(1),F_(2)inE(f)F_{1}, F_{2} \in \mathcal{E}(f)F1,F2∈E(f) and λ ∈ [ 0 , 1 ] λ ∈ [ 0 , 1 ] lambda in[0,1]\lambda \in[0,1]λ∈[0,1] then f = λ F 1 | Y + ( 1 − λ ) F 2 | Y f = λ F 1 Y + ( 1 − λ ) F 2 Y f= lambdaF_(1)|_(Y)+(1-lambda)F_(2)|_(Y)f=\left.\lambda F_{1}\right|_{Y}+\left.(1-\lambda) F_{2}\right|_{Y}f=λF1|Y+(1−λ)F2|Y and
‖ f | = ‖ λ F 1 | Y + ( 1 − λ ) F 2 | ≤ ‖ λ F 1 + ( 1 − λ ) F 2 | ≤ λ ‖ F 1 | + ( 1 − λ ) ‖ F 2 | = = λ ‖ f | + ( 1 − λ ) ‖ f | = ‖ f | so that λ F 1 + ( 1 − λ ) F 2 ∈ E ( f ) . f = λ F 1 Y + ( 1 − λ ) F 2 ≤ λ F 1 + ( 1 − λ ) F 2 ≤ λ F 1 + ( 1 − λ ) ‖ F 2 = = λ f + ( 1 − λ ) f | = ‖ f |  so that  λ F 1 + ( 1 − λ ) F 2 ∈ E ( f ) . {:[||f|=|| lambdaF_(1)|_(Y)+(1-lambda)F_(2)| <= ||lambdaF_(1)+(1-lambda)F_(2)| <= lambda||F_(1)|+(1-lambda)||F_(2)|=:}],[quad=lambda||f|+(1-lambda)||f|=||f|" so that "lambdaF_(1)+(1-lambda)F_(2)inE(f).:}]:}\begin{aligned} & \left\|f \left|=\left\|\left.\lambda F_{1}\right|_{Y}+(1-\lambda) F_{2}\left|\leq\left\|\lambda F_{1}+(1-\lambda) F_{2}\left|\leq \lambda\left\|F_{1}\left|+(1-\lambda) \| F_{2}\right|=\right.\right.\right.\right.\right.\right.\right. \\ & \quad=\lambda\left\|f \left|+(1-\lambda)\left\|f|=\| f| \text { so that } \lambda F_{1}+(1-\lambda) F_{2} \in \mathcal{E}(f) .\right.\right.\right. \end{aligned}‖f|=‖λF1|Y+(1−λ)F2|≤‖λF1+(1−λ)F2|≤λ‖F1|+(1−λ)‖F2|==λ‖f|+(1−λ)‖f|=‖f| so that λF1+(1−λ)F2∈E(f).

3. Extension and approximation

In [10] R.R. Phelps made a connection between the set of the extensions of a linear and continuous functional f ∈ Y ∗ ( Y ∗ f ∈ Y ∗ Y ∗ f inY^(**)(Y^(**):}f \in Y^{*}\left(Y^{*}\right.f∈Y∗(Y∗ is the algebraic - topological dual of the subspace Y Y YYY of a normed space ( X , ‖ ⋅ ‖ ) ) ( X , ‖ ⋅ ‖ ) ) (X,||*||))(X,\|\cdot\|))(X,‖⋅‖)) and the set of elements of best approximation of a functional F ∈ X ∗ F ∈ X ∗ F inX^(**)F \in X^{*}F∈X∗ by the elements of the annihilator Y ⊥ = { G ∈ X ∗ : G | Y = 0 } Y ⊥ = G ∈ X ∗ : G Y = 0 Y^(_|_)={G inX^(**):G|_(Y)=0}Y^{\perp}=\left\{G \in X^{*}:\left.G\right|_{Y}=0\right\}Y⊥={G∈X∗:G|Y=0}.
If F ∈ X ∗ F ∈ X ∗ F inX^(**)F \in X^{*}F∈X∗ then the set of elements of best approximation of F F FFF in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is P Y ⊥ ( F ) = F − E ( F | Y ) P Y ⊥ ( F ) = F − E F Y P_(Y^(_|_))(F)=F-E(F|_(Y))P_{Y^{\perp}}(F)= F-\mathcal{E}\left(\left.F\right|_{Y}\right)PY⊥(F)=F−E(F|Y) where E ( F | Y ) = { H ∈ X ∗ : H | Y = F | Y E F Y = H ∈ X ∗ : H Y = F Y E(F|_(Y))={H inX^(**):H|_(Y)=F|_(Y):}\mathcal{E}\left(\left.F\right|_{Y}\right)=\left\{H \in X^{*}:\left.H\right|_{Y}=\left.F\right|_{Y}\right.E(F|Y)={H∈X∗:H|Y=F|Y and ‖ H ‖ = ‖ F | Y ‖ } ‖ H ‖ = F Y {:||H||=||F|_(Y)||}\left.\|H\|=\left\|\left.F\right|_{Y}\right\|\right\}‖H‖=‖F|Y‖}. The extension of a functional f ∈ Y ∗ f ∈ Y ∗ f inY^(**)f \in Y^{*}f∈Y∗ is unique if and only if Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is a Chebyshevian subspace of X ∗ X ∗ X^(**)X^{*}X∗.
In the proof of R.R. Phelps'result one uses an essential fact: together with F ∈ X ∗ F ∈ X ∗ F inX^(**)F \in X^{*}F∈X∗ the functional F − G F − G F-GF-GF−G belongs to X ∗ X ∗ X^(**)X^{*}X∗, for every G ∈ E ( F | Y ) G ∈ E F Y G inE(F|_(Y))G \in \mathcal{E}\left(\left.F\right|_{Y}\right)G∈E(F|Y), i.e. the fact that X ∗ X ∗ X^(**)X^{*}X∗ has a structure of linear space.
Because X + ∗ X + ∗ X_(+)^(**)X_{+}^{*}X+∗ has only a structure of a convex cone, it could exist a linear and bounded functional F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗, such that for certain extensions G G GGG from E ( F | Y ) E F Y E(F|_(Y))\mathcal{E}\left(\left.F\right|_{Y}\right)E(F|Y), or for all of them, we could have F − G F − G F-GF-GF−G unbounded, i.e. F − G ∉ X + ∗ F − G ∉ X + ∗ F-G!inX_(+)^(**)F-G \notin X_{+}^{*}F−G∉X+∗. Some additional definitions are necessary. For a cone K K K\mathcal{K}K in a linear space V V V\mathcal{V}V and x , y ∈ V x , y ∈ V x,y inVx, y \in \mathcal{V}x,y∈V, we will write x ≤ y x ≤ y x <= yx \leq yx≤y if and only if y − x ∈ K y − x ∈ K y-x inKy-x \in \mathcal{K}y−x∈K.
Let M M M\mathcal{M}M be a non-empty subset of the cone X + ∗ X + ∗ X_(+)^(**)X_{+}^{*}X+∗ and F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗. We say that F F FFF admits minorants in M M M\mathcal{M}M if there exists G ∈ M G ∈ M G inMG \in \mathcal{M}G∈M such that F ≥ G F ≥ G F >= GF \geq GF≥G (i.e. F − G ∈ X + ∗ F − G ∈ X + ∗ F-G inX_(+)^(**)F-G \in X_{+}^{*}F−G∈X+∗ ) and we say that F F FFF majorizes the set M M M\mathcal{M}M if F ≥ G F ≥ G F >= GF \geq GF≥G for every G ∈ M G ∈ M G inMG \in \mathcal{M}G∈M. (i.e. F − M ⊂ X + ∗ F − M ⊂ X + ∗ F-MsubX_(+)^(**)F-\mathcal{M} \subset X_{+}^{*}F−M⊂X+∗ ). Obviously, if F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗ and majorizes M M M\mathcal{M}M, then F F FFF admits minorants in M M M\mathcal{M}M.
For a subspace Y Y YYY of the space X X XXX with asymmetric norm, we denote by Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ the annihilator of Y Y YYY in X + ∗ X + ∗ X_(+)^(**)X_{+}^{*}X+∗ i.e., the set
(7) Y + ⊥ = { G ∈ X + ∗ : G | Y = 0 } . (7) Y + ⊥ = G ∈ X + ∗ : G Y = 0 . {:(7)Y_(+)^(_|_)={G inX_(+)^(**):G|_(Y)=0}.:}\begin{equation*} Y_{+}^{\perp}=\left\{G \in X_{+}^{*}:\left.G\right|_{Y}=0\right\} . \tag{7} \end{equation*}(7)Y+⊥={G∈X+∗:G|Y=0}.
We state the following problem of best approximation:
For F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗ find G 0 ∈ Y + ⊥ G 0 ∈ Y + ⊥ G_(0)inY_(+)^(_|_)G_{0} \in Y_{+}^{\perp}G0∈Y+⊥ such that ‖ F − G 0 ∣= d + ( F , Y + ⊥ ) ‖ F − G 0 ∣= d + F , Y + ⊥ ||F-G_(0)∣=d_(+)(F,Y_(+)^(_|_))\| F-G_{0} \mid=d_{+}\left(F, Y_{+}^{\perp}\right)‖F−G0∣=d+(F,Y+⊥) where
(8) d + ( F , Y + ∗ ) = inf { ‖ F − G ∣: G ∈ Y + ⊥ , F ≥ G } . (8) d + F , Y + ∗ = inf ‖ F − G ∣: G ∈ Y + ⊥ , F ≥ G . {:(8)d_(+)(F,Y_(+)^(**))=i n f{||F-G∣:G inY_(+)^(_|_),F >= G}.:}\begin{equation*} d_{+}\left(F, Y_{+}^{*}\right)=\inf \left\{\| F-G \mid: G \in Y_{+}^{\perp}, F \geq G\right\} . \tag{8} \end{equation*}(8)d+(F,Y+∗)=inf{‖F−G∣:G∈Y+⊥,F≥G}.
Let
(9) P Y + ⊥ ( F ) := { G ∈ Y + ∗ : F ≥ G , ‖ F − G ∣= d + ( F , Y + ⊥ ) } . (9) P Y + ⊥ ( F ) := G ∈ Y + ∗ : F ≥ G , ‖ F − G ∣= d + F , Y + ⊥ . {:(9)P_(Y_(+)^(_|_))(F):={G inY_(+)^(**):F >= G,||F-G∣=d_(+)(F,Y_(+)^(_|_))}.:}\begin{equation*} P_{Y_{+}^{\perp}}(F):=\left\{G \in Y_{+}^{*}: F \geq G, \| F-G \mid=d_{+}\left(F, Y_{+}^{\perp}\right)\right\} . \tag{9} \end{equation*}(9)PY+⊥(F):={G∈Y+∗:F≥G,‖F−G∣=d+(F,Y+⊥)}.
We say that Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ is F F FFF - proximinal if P Y + ⊥ ( F ) ≠ ∅ P Y + ⊥ ( F ) ≠ ∅ P_(Y_(+)^(_|_))(F)!=O/P_{Y_{+}^{\perp}}(F) \neq \emptysetPY+⊥(F)≠∅. If, in addition, card P Y + ⊥ ( F ) = 1 P Y + ⊥ ( F ) = 1 P_(Y_(+)^(_|_))(F)=1P_{Y_{+}^{\perp}}(F)=1PY+⊥(F)=1 then Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ is called F F FFF - Chebyshevian.
The following result is similar to Phelps'result([10]).
Theorem 2. Let X X XXX be a space with asymmetric norm, Y Y YYY a subspace of X X XXX, and F ∈ X + ∗ F ∈ X + ∗ F inX_(+)^(**)F \in X_{+}^{*}F∈X+∗. Let
(10) E ( F | Y ) = { H ∈ X + ∗ : H | Y = F | Y and ‖ H | = ‖ F | } (10) E F Y = H ∈ X + ∗ : H Y = F Y  and  ‖ H | = ‖ F | } {:(10)E(F|_(Y))={H inX_(+)^(**):H|_(Y)=F|_(Y)" and "||H|=||F|}:}:}\begin{equation*} \mathcal{E}\left(\left.F\right|_{Y}\right)=\left\{H \in X_{+}^{*}:\left.H\right|_{Y}=\left.F\right|_{Y} \text { and }\|H|=\| F|\}\right. \tag{10} \end{equation*}(10)E(F|Y)={H∈X+∗:H|Y=F|Y and ‖H|=‖F|}
and
(11) E + ( F | Y ) = { H ∈ E ( F | Y ) : H ≤ F } (11) E + F Y = H ∈ E F Y : H ≤ F {:(11)E_(+)(F|_(Y))={H inE(F|_(Y)):H <= F}:}\begin{equation*} \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)=\left\{H \in \mathcal{E}\left(\left.F\right|_{Y}\right): H \leq F\right\} \tag{11} \end{equation*}(11)E+(F|Y)={H∈E(F|Y):H≤F}
a) If E + ( F | Y ) ≠ ∅ E + F Y ≠ ∅ E_(+)(F|_(Y))!=O/\mathcal{E}_{+}\left(\left.F\right|_{Y}\right) \neq \emptysetE+(F|Y)≠∅ then Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ is F F FFF - proximinal and the following equality holds:
(12) d + ( F , Y + ⊥ ) = ‖ F | Y ∣ (12) d + F , Y + ⊥ = ‖ F Y ∣ {:(12)d_(+)(F,Y_(+)^(_|_))=||F|_(Y)∣:}\begin{equation*} d_{+}\left(F, Y_{+}^{\perp}\right)=\|\left. F\right|_{Y} \mid \tag{12} \end{equation*}(12)d+(F,Y+⊥)=‖F|Y∣
b) If G 0 ∈ P Y + ⊥ ( F ) G 0 ∈ P Y + ⊥ ( F ) G_(0)inP_(Y_(+)^(_|_))(F)G_{0} \in P_{Y_{+}^{\perp}}(F)G0∈PY+⊥(F) then F − G 0 ∈ E + ( F | Y ) F − G 0 ∈ E + F Y F-G_(0)inE_(+)(F|_(Y))F-G_{0} \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)F−G0∈E+(F|Y).
c) We have E + ( F | Y ) ≠ ∅ E + F Y ≠ ∅ E_(+)(F|_(Y))!=O/\mathcal{E}_{+}\left(\left.F\right|_{Y}\right) \neq \emptysetE+(F|Y)≠∅ if and only if P Y + ⊥ ( F ) ≠ ∅ P Y + ⊥ ( F ) ≠ ∅ P_(Y_(+)^(_|_))(F)!=O/P_{Y_{+}^{\perp}}(F) \neq \emptysetPY+⊥(F)≠∅ and the following equality holds:
(13) F − E + ( F | Y ) = P Y + ⊥ ( F ) (13) F − E + F Y = P Y + ⊥ ( F ) {:(13)F-E_(+)(F|_(Y))=P_(Y_(+)^(_|_))(F):}\begin{equation*} F-\mathcal{E}_{+}\left(\left.F\right|_{Y}\right)=P_{Y_{+}^{\perp}}(F) \tag{13} \end{equation*}(13)F−E+(F|Y)=PY+⊥(F)
d) Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ is F F FFF - Chebyshevian if and only quad\quad card E + ( F | Y ) = 1 E + F Y = 1 E_(+)(F|_(Y))=1\mathcal{E}_{+}\left(\left.F\right|_{Y}\right)=1E+(F|Y)=1.
e) F ∈ E + ( F | Y ) F ∈ E + F Y F inE_(+)(F|_(Y))F \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)F∈E+(F|Y) if and only if 0 ∈ P Y + ⊥ ( F ) 0 ∈ P Y + ⊥ ( F ) 0inP_(Y_(+)^(_|_))(F)0 \in P_{Y_{+}^{\perp}}(F)0∈PY+⊥(F).
Proof. Let G 0 G 0 G_(0)G_{0}G0 be a minorant of F F FFF in E ( F | Y ) ( G 0 E F Y G 0 E(F|_(Y))(G_(0):}\mathcal{E}\left(\left.F\right|_{Y}\right)\left(G_{0}\right.E(F|Y)(G0 exists, because E + ( F | Y ≠ ∅ ) E + F Y ≠ ∅ E_(+)(F|_(Y)!=O/)\mathcal{E}_{+}\left(\left.F\right|_{Y} \neq \emptyset\right)E+(F|Y≠∅). Then, F − G 0 ∈ X + ∗ F − G 0 ∈ X + ∗ F-G_(0)inX_(+)^(**)F-G_{0} \in X_{+}^{*}F−G0∈X+∗ and
‖ F | Y | = ‖ G 0 | = ‖ F − ( F − G 0 ) | ≥ d + ( F , Y + ⊥ ) . F | Y = G 0 = ‖ F − F − G 0 ≥ d + F , Y + ⊥ . ||F|_(Y)|=||G_(0)|=||F-(F-G_(0))| >= d_(+)(F,Y_(+)^(_|_)).:}\left\|F | _ { Y } \left|=\left\|G_{0}\left|=\| F-\left(F-G_{0}\right)\right| \geq d_{+}\left(F, Y_{+}^{\perp}\right) .\right.\right.\right.‖F|Y|=‖G0|=‖F−(F−G0)|≥d+(F,Y+⊥).
On the other hand, for every G ∈ Y + ⊥ ( F ≥ G ) G ∈ Y + ⊥ ( F ≥ G ) G inY_(+)^(_|_)(F >= G)G \in Y_{+}^{\perp}(F \geq G)G∈Y+⊥(F≥G) we have
‖ F | Y | = ‖ F | Y − G | Y | ≤ ‖ F − G | F | Y = F Y − G Y | ≤ ‖ F − G | ||F|_(Y)|=||F|_(Y)-G|_(Y)| <= ||F-G|:}\left\|F | _ { Y } \left|=\left\|\left.F\right|_{Y}-\left.G\right|_{Y}|\leq \| F-G|\right.\right.\right.‖F|Y|=‖F|Y−G|Y|≤‖F−G|
Taking the infimum with respect to G ∈ Y + ⊥ ( F ≥ G ) G ∈ Y + ⊥ ( F ≥ G ) G inY_(+)^(_|_)(F >= G)G \in Y_{+}^{\perp}(F \geq G)G∈Y+⊥(F≥G) we find
‖ F | Y ∣≤ d + ( F , Y + ⊥ ) ‖ F Y ∣≤ d + F , Y + ⊥ ||F|_(Y)∣≤d_(+)(F,Y_(+)^(_|_))\|\left. F\right|_{Y} \mid \leq d_{+}\left(F, Y_{+}^{\perp}\right)‖F|Y∣≤d+(F,Y+⊥)
Therefore, the formula (12) holds, and Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ is F F FFF - proximinal.
b) Let G 0 ∈ P Y + ⊥ ( F ) G 0 ∈ P Y + ⊥ ( F ) G_(0)inP_(Y_(+)^(_|_))(F)G_{0} \in P_{Y_{+}^{\perp}}(F)G0∈PY+⊥(F). Then F ≥ G 0 F ≥ G 0 F >= G_(0)F \geq G_{0}F≥G0 (according to the definition of P Y + ⊥ ( F ) P Y + ⊥ ( F ) P_(Y_(+)^(_|_))(F)P_{Y_{+}^{\perp}}(F)PY+⊥(F) ), ( F − G 0 ) | Y = F | Y F − G 0 Y = F Y (F-G_(0))|_(Y)=F|_(Y)\left.\left(F-G_{0}\right)\right|_{Y}=\left.F\right|_{Y}(F−G0)|Y=F|Y and
‖ F − G 0 | = inf { ‖ F − G ∣: G ∈ Y + ⊥ , F ≥ G } = d + ( F , Y + ⊥ ) = ‖ F | Y ∣ F − G 0 = inf ‖ F − G ∣: G ∈ Y + ⊥ , F ≥ G = d + F , Y + ⊥ = ‖ F Y ∣ ||F-G_(0)|=i n f{||F-G∣:G inY_(+)^(_|_),F >= G}=d_(+)(F,Y_(+)^(_|_))=||F|_(Y)∣:}\left\|F-G_{0}\left|=\inf \left\{\| F-G \mid: G \in Y_{+}^{\perp}, F \geq G\right\}=d_{+}\left(F, Y_{+}^{\perp}\right)=\| F\right|_{Y} \mid\right.‖F−G0|=inf{‖F−G∣:G∈Y+⊥,F≥G}=d+(F,Y+⊥)=‖F|Y∣
(according to a)). Thus F − G 0 ∈ E + ( F | Y ) F − G 0 ∈ E + F Y F-G_(0)inE_(+)(F|_(Y))F-G_{0} \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)F−G0∈E+(F|Y).
c) Follows from a ) and b ).
If H ∈ E + ( F | Y ) H ∈ E + F Y H inE_(+)(F|_(Y))H \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)H∈E+(F|Y) then F ≥ H , ( F − H ) | Y = 0 F ≥ H , ( F − H ) Y = 0 F >= H,(F-H)|_(Y)=0F \geq H,\left.(F-H)\right|_{Y}=0F≥H,(F−H)|Y=0 and
‖ F − ( F − H ) | = ‖ H | = ‖ F | Y ∣= d + ( F , Y + ⊥ ) F − ( F − H ) = H | = ‖ F | Y ∣= d + F , Y + ⊥ ||F-(F-H)|=||H|=||F|_(Y)∣=d_(+)(F,Y_(+)^(_|_)):}\left\|F-(F-H)\left|=\left\|H|=\| F|_{Y} \mid=d_{+}\left(F, Y_{+}^{\perp}\right)\right.\right.\right.‖F−(F−H)|=‖H|=‖F|Y∣=d+(F,Y+⊥)
and then F − H ∈ P Y + ⊥ ( F ) F − H ∈ P Y + ⊥ ( F ) F-H inP_(Y_(+)^(_|_))(F)F-H \in P_{Y_{+}^{\perp}}(F)F−H∈PY+⊥(F).
Conversely, G ∈ P Y + ⊥ ( F ) G ∈ P Y + ⊥ ( F ) G inP_(Y_(+)^(_|_))(F)G \in P_{Y_{+}^{\perp}}(F)G∈PY+⊥(F) implies F ≥ G F ≥ G F >= GF \geq GF≥G, so that F − G ∈ X + ∗ , ( F − G ) | Y = F | Y F − G ∈ X + ∗ , ( F − G ) Y = F Y F-G inX_(+)^(**),(F-G)|_(Y)=F|_(Y)F-G \in X_{+}^{*},\left.(F-G)\right|_{Y}=\left.F\right|_{Y}F−G∈X+∗,(F−G)|Y=F|Y, and
‖ F − G | = ‖ F | Y ∣= d + ( F , P Y + ⊥ ) F − G | = ‖ F | Y ∣= d + F , P Y + ⊥ ||F-G|=||F|_(Y)∣=d_(+)(F,P_(Y_(+)^(_|_))):}\left\|F-G|=\| F|_{Y} \mid=d_{+}\left(F, P_{Y_{+}^{\perp}}\right)\right.‖F−G|=‖F|Y∣=d+(F,PY+⊥)
It follows that F − G ∈ E + ( F | Y ) F − G ∈ E + F Y F-G inE_(+)(F|_(Y))F-G \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)F−G∈E+(F|Y), i.e. G ∈ F − E + ( F | Y ) G ∈ F − E + F Y G in F-E_(+)(F|_(Y))G \in F-\mathcal{E}_{+}\left(\left.F\right|_{Y}\right)G∈F−E+(F|Y).
d) If Y + ⊥ Y + ⊥ Y_(+)^(_|_)Y_{+}^{\perp}Y+⊥ is F F FFF - Chebyshevian, it results that there exists only one element G ∈ P Y + ⊥ ( F ) G ∈ P Y + ⊥ ( F ) G inP_(Y_(+)^(_|_))(F)G \in P_{Y_{+}^{\perp}}(F)G∈PY+⊥(F) such that F ≥ G F ≥ G F >= GF \geq GF≥G, so that F − G ∈ X + ∗ , ( F − G ) | Y = F | Y F − G ∈ X + ∗ , ( F − G ) Y = F Y F-G inX_(+)^(**),(F-G)|_(Y)=F|_(Y)F-G \in X_{+}^{*},\left.(F-G)\right|_{Y}=\left.F\right|_{Y}F−G∈X+∗,(F−G)|Y=F|Y and
‖ F − G | = d + ( F , Y + ⊥ ) = ‖ F | Y ∣ , F − G = d + F , Y + ⊥ = ‖ F Y ∣ , ||F-G|=d_(+)(F,Y_(+)^(_|_))=||F|_(Y)∣,:}\left\|F-G\left|=d_{+}\left(F, Y_{+}^{\perp}\right)=\| F\right|_{Y} \mid,\right.‖F−G|=d+(F,Y+⊥)=‖F|Y∣,
i.e. E + ( F | Y ) E + F Y E_(+)(F|_(Y))\mathcal{E}_{+}\left(\left.F\right|_{Y}\right)E+(F|Y) contains only one element, namely F − G F − G F-GF-GF−G.
e) If F ∈ E + ( F | Y ) F ∈ E + F Y F inE_(+)(F|_(Y))F \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)F∈E+(F|Y) then there exists H ∈ E + ( F | Y ) H ∈ E + F Y H inE_(+)(F|_(Y))H \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)H∈E+(F|Y) such that F = H F = H F=HF=HF=H. Thus, according to c) F − H = F − F = 0 ∈ P Y + ⊥ ( F ) F − H = F − F = 0 ∈ P Y + ⊥ ( F ) F-H=F-F=0inP_(Y_(+)^(_|_))(F)F-H=F-F=0 \in P_{Y_{+}^{\perp}}(F)F−H=F−F=0∈PY+⊥(F).
If 0 ∈ P Y + ⊥ ( F ) 0 ∈ P Y + ⊥ ( F ) 0inP_(Y_(+)^(_|_))(F)0 \in P_{Y_{+}^{\perp}}(F)0∈PY+⊥(F) then ‖ F | = d + ( F , Y + ⊥ ) = ‖ F | Y ∣ F = d + F , Y + ⊥ = ‖ F Y ∣ ||F|=d_(+)(F,Y_(+)^(_|_))=||F|_(Y)∣:}\left\|F\left|=d_{+}\left(F, Y_{+}^{\perp}\right)=\| F\right|_{Y} \mid\right.‖F|=d+(F,Y+⊥)=‖F|Y∣, so F ∈ E + ( F | Y ) F ∈ E + F Y F inE_(+)(F|_(Y))F \in \mathcal{E}_{+}\left(\left.F\right|_{Y}\right)F∈E+(F|Y).

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Received: 1.09.2002
Department of Mathematics and Computer Science
North University of Baia Mare, Str. Victoriei nr. 76
4800 Baia Mare ROMANIA;
Email: mmustata@ubbcluj.ro
2002

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