Common selections for the metric projections

Costica Mustata
(original), Approximation Theory, paper

Abstract

Authors

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

Keywords

Paper coordinates

C. Mustăţa, Common selections for the metric projections, Mathematica 35 (58) (1993) 2, 193-200.

PDF

https://ictp.acad.ro/wp-content/uploads/2025/07/1993-Mustata-Mathematica-Common-selections-for-the-metric-projections.pdf

About this paper

Journal

Mathematica

Publisher Name

Romanian Academy

DOI
Print ISSN

1222-9016

 

Online ISSN

 2601-744X

MR # 96b: 46018

google scholar link

[1] Czipser, J. and Geher, L., Extension of Funcitons Satisfying a Lipschitz Condition, Acta Math. Acad. Sci. Hungar, 6 (1955), 213-220.
[2] Deutsch, F., Linear Selections for Metric Projeciton, J. Funct. Analysis 49, 3 (1982), 269-292.
[3] Deutsch, F., Wu Li, Sung-Ho Park, Tietze Extensions and Continuous Selections for Metric Projections, J. Approx. Theory 64, 1 (1991), 55-68.
[4] Mc Shane, E.J., Extension of Range of Funcitons, bull. Amer. Math. Soc. 40 (1934), 837-842.
[5] Mustata, C., Asupra unor subspații cebîșeniene din spațiul normal al funcțiilor lipschitziene, Rev. Anal. Numer. Teoria Aproximației 2 (1973), 81-87.
[6] Mustata, C., Best Approximation and Unique Extension of Lipschitz Functions, J. Approx. Theory 19, 3, (1977), 222-230.
[7] Mustata, C., M-ideal in Metric Spaces, “Babes-Bolyai” Univ. Research Seminars, Seminar on Mathematical Analysis, Preprint nr.7, (1988), 65-74.
[8] Mustata, C., Extension of Holder Functions and Some Related Problems of Best Approximation, “Babes-Bolyai” Univ. Research Seminars, Seminar on Mathematical analysis, Preprint, nr.7 (1991), 71-86.
[9] Mustata C., Selections Associated to Mc Shane’s Extension Theorem for Lipschitz Functions, Revue d’Analyse Numer. et de la Theorie de l’Approximation 21, 2 (1992), 135-145.

Paper (preprint) in HTML form

1993-Mustata-Mathematica-Common-selections-for-the-metric-projections

COMMON SELECTIONS FOR THE METRIC PROJECTIONS

COSTICĂ MUSTĂTA

  1. Let X X XXX be a normed space and M M MMM a closed subspace of X X XXX. The subspace M M MMM is called proximinal (Chebyshev) if for every x ∈ X x ∈ X x in Xx \in Xx∈X, the set of the elements of best approximation for x x xxx in M M MMM, given by
(1) P M ( x ) = { y ∈ M : ‖ x − y ‖ = d ( x , M ) } (1) P M ( x ) = { y ∈ M : ‖ x − y ‖ = d ( x , M ) } {:(1)P_(M)(x)={y in M:||x-y||=d(x","M)}:}\begin{equation*} P_{M}(x)=\{y \in M:\|x-y\|=d(x, M)\} \tag{1} \end{equation*}(1)PM(x)={y∈M:‖x−y‖=d(x,M)}
where
(2) d ( x , M ) = inf { ‖ x − y ‖ : y ∈ M } (2) d ( x , M ) = inf { ‖ x − y ‖ : y ∈ M } {:(2)d(x","M)=i n f{||x-y||:y in M}:}\begin{equation*} d(x, M)=\inf \{\|x-y\|: y \in M\} \tag{2} \end{equation*}(2)d(x,M)=inf{‖x−y‖:y∈M}
is nonvoid (respectively a one-point set).
The quantity d ( x , M ) d ( x , M ) d(x,M)d(x, M)d(x,M) is called the distance from x x xxx to M M MMM.
If M M MMM is a proximinal subspace of X X XXX, then the operator P M : X → 2 M P M : X → 2 M P_(M):X rarr2^(M)P_{M}: X \rightarrow 2^{M}PM:X→2M is called the metric projection on M M MMM, and the set
(3) ker P M = { x ∈ X : 0 ∈ P M ( x ) } = { x ∈ X : ‖ x ‖ = d ( x , M ) } (3)  ker  P M = x ∈ X : 0 ∈ P M ( x ) = { x ∈ X : ‖ x ‖ = d ( x , M ) } {:(3)" ker "P_(M)={x in X:0inP_(M)(x)}={x in X:||x||=d(x","M)}:}\begin{equation*} \text { ker } P_{M}=\left\{x \in X: 0 \in P_{M}(x)\right\}=\{x \in X:\|x\|=d(x, M)\} \tag{3} \end{equation*}(3) ker PM={x∈X:0∈PM(x)}={x∈X:‖x‖=d(x,M)}
is called the kernel of the metric projection P M P M P_(M)P_{M}PM.
Definition 1. A function p : X → M p : X → M p:X rarr Mp: X \rightarrow Mp:X→M is called a selection for the metric projection P M P M P_(M)P_{M}PM, if p ( x ) ∈ P M ( x ) p ( x ) ∈ P M ( x ) p(x)inP_(M)(x)p(x) \in P_{M}(x)p(x)∈PM(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X.
The existence of continuous (and eventualy linear) selections and characterizations of continuous or linear selections for P M P M P_(M)P_{M}PM have been studied in [2], for arbitrary normed spaces X X XXX.
The finding of continuous or linear metric selections in concrete normed spaces is a problem specific to each considered case. Two such concrete cases were considered in [3] and [91,
This paper will be concerned with the following natural problem: if ‖ ‖ 1 ‖ 1 ||||_(1):}\left\|\|_{1}\right.‖‖1 and ‖ ‖ 2 ‖ ‖ 2 ||||_(2)\| \|_{2}‖‖2 are two norms on a linear space X X XXX and M M MMM is a subspace of X X XXX which is proximinal with respect to each of these norms, find a common selection for the metric projections P M 1 P M 1 P_(M)^(1)P_{M}^{1}PM1 and P M 2 P M 2 P_(M)^(2)P_{M}^{2}PM2; i.e. an application p : X → M p : X → M p:X rarr Mp: X \rightarrow Mp:X→M such that p ( x ) ∈ P M 1 ( x ) ∩ P M 2 ( x ) p ( x ) ∈ P M 1 ( x ) ∩ P M 2 ( x ) p(x)inP_(M)^(1)(x)nnP_(M)^(2)(x)p(x) \in P_{M}^{1}(x) \cap P_{M}^{2}(x)p(x)∈PM1(x)∩PM2(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X.
The following characterization result of common linear selections for two metric projections is an immediate consequence of Theorem 2.2 in [2].
Theorem A. Let M M MMM be a subspace of the linear space X X XXX which is proximinal with respect to each of the norms ‖ ‖ 1 , ‖ ‖ 2 1 , 2 ||||_(1),||||_(2)\left\|\left\|_{1},\right\|\right\|_{2}‖‖1,‖‖2 on X X XXX. Then the following assertions are equivalent:
1 ∘ P M 1 1 ∘ P M 1 1^(@)P_(M)^(1)1^{\circ} P_{M}^{1}1∘PM1 and P M 2 P M 2 P_(M)^(2)P_{M}^{2}PM2 admit a common linear selection;
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The set ker P M 1 ∩ P M 1 ∩ P_(M)^(1)nnP_{M}^{1} \capPM1∩ ker P M 2 P M 2 P_(M)^(2)P_{M}^{2}PM2 contains a closed subspace N N NNN such that X = M ⊕ N X = M ⊕ N X=M o+NX=M \oplus NX=M⊕N (algebraic direct sum);
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘ The set ker P M 1 ∩ ker P M 2 P M 1 ∩ ker P M 2 P_(M)^(1)nn kerP_(M)^(2)P_{M}^{1} \cap \operatorname{ker} P_{M}^{2}PM1∩kerPM2 contains a closed subspace N N NNN such that X = M + N X = M + N X=M+NX=M+NX=M+N (algebraic and topological direct sum).
In the following we shall iluustrate Theorem A in a concrete setting.
First, we shall define a linear and continuous selection for the extension operator which preserves both the Lipschitz and uniform norms and then, using a Phelps' type result ([8], Theorem 3) we shall define a common selection for the operators of metric projection in the Lipschitz and in the uniform norms.
2. Let a , b , c , d ∈ R a , b , c , d ∈ R a,b,c,d in Ra, b, c, d \in Ra,b,c,d∈R be such that c < a < b < d c < a < b < d c < a < b < dc<a<b<dc<a<b<d and let X = [ c , d ] Y = [ a , b ] , x 0 ∈ [ a , b ] X = [ c , d ] Y = [ a , b ] , x 0 ∈ [ a , b ] X=[c,d]Y=[a,b],x_(0)in[a,b]X=[c, d] Y=[a, b], x_{0} \in[a, b]X=[c,d]Y=[a,b],x0∈[a,b] fixed and d ( x , y ) = | x − y | d ( x , y ) = | x − y | d(x,y)=|x-y|d(x, y)=|x-y|d(x,y)=|x−y|.
A function f : Y → R f : Y → R f:Y rarr Rf: Y \rightarrow Rf:Y→R is called Lipschitz (on Y Y YYY ) if there exists K ⩾ 0 K ⩾ 0 K >= 0K \geqslant 0K⩾0 such that
(4) | f ( x ) − f ( y ) | ⩽ K ⋅ d ( x , y ) (4) | f ( x ) − f ( y ) | ⩽ K ⋅ d ( x , y ) {:(4)|f(x)-f(y)| <= K*d(x","y):}\begin{equation*} |f(x)-f(y)| \leqslant K \cdot d(x, y) \tag{4} \end{equation*}(4)|f(x)−f(y)|⩽K⋅d(x,y)
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y. The smallest constant K K KKK for which (4) holds is
(5) ‖ f ‖ L = sup { | f ( x ) − f ( y ) | / d ( x , y ) : x , y ∈ Y , x ≠ y } (5) ‖ f ‖ L = sup { | f ( x ) − f ( y ) | / d ( x , y ) : x , y ∈ Y , x ≠ y } {:(5)||f||_(L)=s u p{|f(x)-f(y)|//d(x","y):x","y inY","x!=y}:}\begin{equation*} \|f\|_{L}=\sup \{|f(x)-f(y)| / d(x, y): x, y \in \mathbf{Y}, x \neq y\} \tag{5} \end{equation*}(5)‖f‖L=sup{|f(x)−f(y)|/d(x,y):x,y∈Y,x≠y}
and is called the Lipschitz norm of the function f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y, where
(6) Lip 0 Y = { f ; f : [ a , b ] → R , f is Lipschitz on Y , f ( x 0 ) = 0 } (6) Lip 0 Y = f ; f : [ a , b ] → R , f  is Lipschitz on  Y , f x 0 = 0 {:(6)Lip_(0)Y={f;f:[a,b]rarr R,f" is Lipschitz on "Y,f(x_(0))=0}:}\begin{equation*} \operatorname{Lip}_{0} Y=\left\{f ; f:[a, b] \rightarrow R, f \text { is Lipschitz on } Y, f\left(x_{0}\right)=0\right\} \tag{6} \end{equation*}(6)Lip0Y={f;f:[a,b]→R,f is Lipschitz on Y,f(x0)=0}
is the Banach space of all real valued Lipschitz functions defined on Y Y YYY and vanishing at the fixed point x 0 ∈ Y x 0 ∈ Y x_(0)in Yx_{0} \in Yx0∈Y.
The Banach space Lip 0 X 0 X _(0)X{ }_{0} X0X is defined in a similar way (the fixed point x 0 x 0 x_(0)x_{0}x0 is the same as for Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y )
Since Y Y YYY is a compact subset of R R RRR one can also define the uniform norm on Lip 0 I Lip 0 I Lip_(0)I\operatorname{Lip}_{0} \boldsymbol{I}Lip0I by
(7) ‖ f ‖ u = max { | f ( y ) | : y ∈ I } (7) ‖ f ‖ u = max { | f ( y ) | : y ∈ I } {:(7)||f||_(u)=max quad{|f(y)|:y in I}:}\begin{equation*} \|f\|_{u}=\max \quad\{|f(y)|: y \in I\} \tag{7} \end{equation*}(7)‖f‖u=max{|f(y)|:y∈I}
for f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y. The uniform norm on Lip Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X is defined similarly.

It holds:

Theorem 1. (a) For every f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y there exists F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X such that
(8) F | X ′ = f and ‖ F ‖ L = ‖ f ‖ L ; (8) F X ′ = f  and  ‖ F ‖ L = ‖ f ‖ L ; {:(8)F|_(X^('))=f" and "||F||_(L)=||f||_(L);:}\begin{equation*} \left.F\right|_{X^{\prime}}=f \text { and }\|F\|_{L}=\|f\|_{L} ; \tag{8} \end{equation*}(8)F|X′=f and ‖F‖L=‖f‖L;
(b) For every f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y there exists F ¯ ∈ Lip 0 X F ¯ ∈ Lip 0 X bar(F)inLip_(0)X\bar{F} \in \operatorname{Lip}_{0} XF¯∈Lip0X such that
(9) F ¯ | Y = f and ‖ F ¯ ‖ u = ‖ f ‖ u . (9) F ¯ Y = f  and  ‖ F ¯ ‖ u = ‖ f ‖ u . {:(9)( bar(F))|_(Y)=f" and "|| bar(F)||_(u)=||f||_(u).:}\begin{equation*} \left.\bar{F}\right|_{Y}=f \text { and }\|\bar{F}\|_{u}=\|f\|_{u} . \tag{9} \end{equation*}(9)F¯|Y=f and ‖F¯‖u=‖f‖u.
Proof. The assertion (a) is a particular case of a theorem of Mc Shane [4] (see also [1], [6]).
Denoting by
(10) E L ( f ) = { F ∈ Lip 0 X : F | Y = f and ‖ F ‖ L = ‖ f ‖ L { , (10) E L ( f ) = F ∈ Lip 0 X : F Y = f  and  ‖ F ‖ L = ‖ f ‖ L { , {:(10)E_(L)(f)={F inLip_(0)X:F|_(Y)=f" and "||F||_(L)=||f||_(L){,:}:}\begin{equation*} E_{L}(f)=\left\{F \in \operatorname{Lip}_{0} X:\left.F\right|_{Y}=f \text { and }\|F\|_{L}=\|f\|_{L}\{,\right. \tag{10} \end{equation*}(10)EL(f)={F∈Lip0X:F|Y=f and ‖F‖L=‖f‖L{,
the non-void set of all extensions of the function f f fff, preserving the Lipschitz norm, let
T : E L ( f ) → E L ( f ) , T : E L ( f ) → E L ( f ) , T:E_(L)(f)rarrE_(L)(f),T: E_{L}(f) \rightarrow E_{L}(f),T:EL(f)→EL(f),
be the truncation operator defined, for F ∈ E L ( f ) F ∈ E L ( f ) F inE_(L)(f)F \in E_{L}(f)F∈EL(f), by T ( F ) = F ¯ T ( F ) = F ¯ T(F)= bar(F)T(F)=\bar{F}T(F)=F¯, where
(11) F ¯ ( x ) = ‖ f ‖ u if F ( x ) > ‖ f ‖ u , = F ( x ) if − ‖ f ‖ u ⩽ F ( x ) ⩽ ‖ f ‖ w , = − ‖ f ‖ u if F ′ ( x ) < − ‖ f ‖ u . (11) F ¯ ( x ) = ‖ f ‖ u  if  F ( x ) > ‖ f ‖ u , = F ( x )  if  − ‖ f ‖ u ⩽ F ( x ) ⩽ ‖ f ‖ w , = − ‖ f ‖ u  if  F ′ ( x ) < − ‖ f ‖ u . {:[(11) bar(F)(x)=||f||_(u)quad" if "quad F(x) > ||f||_(u)","],[=F(x)quad" if "-||f||_(u) <= F(x) <= ||f||_(w)","],[=-||f||_(u)" if "quadF^(')(x) < -||f||_(u).]:}\begin{align*} \bar{F}(x) & =\|f\|_{u} \quad \text { if } \quad F(x)>\|f\|_{u}, \tag{11}\\ & =F(x) \quad \text { if }-\|f\|_{u} \leqslant F(x) \leqslant\|f\|_{w}, \\ & =-\|f\|_{u} \text { if } \quad F^{\prime}(x)<-\|f\|_{u} . \end{align*}(11)F¯(x)=‖f‖u if F(x)>‖f‖u,=F(x) if −‖f‖u⩽F(x)⩽‖f‖w,=−‖f‖u if F′(x)<−‖f‖u.
Obviously, T ( F ′ ) = F ¯ ∈ E L ( f ) T F ′ = F ¯ ∈ E L ( f ) T(F^('))= bar(F)inE_(L)(f)T\left(F^{\prime}\right)=\bar{F} \in E_{L}(f)T(F′)=F¯∈EL(f) and ‖ F ¯ ‖ u = ‖ f ‖ u ‖ F ¯ ‖ u = ‖ f ‖ u || bar(F)||_(u)=||f||_(u)\|\bar{F}\|_{u}=\|f\|_{u}‖F¯‖u=‖f‖u, proving the assertion (b) of the Theorem.
Denote by
(12) E u ( f ) = { F ¯ ∈ Lip 0 X : F ¯ | Y = f and ‖ F ¯ ‖ u = ‖ f ‖ u } , (12) E u ( f ) = F ¯ ∈ Lip 0 X : F ¯ Y = f  and  ‖ F ¯ ‖ u = ‖ f ‖ u , {:(12)E_(u)(f)={( bar(F))inLip_(0)X:( bar(F))|_(Y)=f" and "||( bar(F))||_(u)=||f||_(u)}",":}\begin{equation*} E_{u}(f)=\left\{\bar{F} \in \operatorname{Lip}_{0} X:\left.\bar{F}\right|_{Y}=f \text { and }\|\bar{F}\|_{u}=\|f\|_{u}\right\}, \tag{12} \end{equation*}(12)Eu(f)={F¯∈Lip0X:F¯|Y=f and ‖F¯‖u=‖f‖u},
the set of all extensions of the function f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y which preserve the uniform norm of f f fff (this set is non-void by the assertion (b) of the above theorem).
Since the truncation F ¯ F ¯ bar(F)\bar{F}F¯ of the extension F F FFF of a function f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y. is ⋅ ⋅ *\cdot⋅ in E L ( f ) ∩ E u ( f ) E L ( f ) ∩ E u ( f ) E_(L)(f)nnE_(u)(f)E_{L}(f) \cap E_{u}(f)EL(f)∩Eu(f), it follows that
(13) E L ( f ) ∩ E u ( f ) ≠ ∅ , (13) E L ( f ) ∩ E u ( f ) ≠ ∅ , {:(13)E_(L)(f)nnE_(u)(f)!=O/",":}\begin{equation*} E_{L}(f) \cap E_{u}(f) \neq \emptyset, \tag{13} \end{equation*}(13)EL(f)∩Eu(f)≠∅,
for every f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)quad Yf \in \operatorname{Lip}_{0} \quad Yf∈Lip0Y and it holds the inclusion
(14) T ( E L ( f ) ) ⊂ E u ( f ) , (14) T E L ( f ) ⊂ E u ( f ) , {:(14)T(E_(L)(f))subE_(u)(f)",":}\begin{equation*} T\left(E_{L}(f)\right) \subset E_{u}(f), \tag{14} \end{equation*}(14)T(EL(f))⊂Eu(f),
(the example given at the end of this paper shows that the inclusion can be strict).
A function e L : Lip 0 Y → Lip 0 X e L : Lip 0 Y → Lip 0 X e_(L):Lip_(0)Y rarrLip_(0)Xe_{L}: \operatorname{Lip}_{0} Y \rightarrow \operatorname{Lip}_{0} XeL:Lip0Y→Lip0X is called a selection associated to the extension operator
E L : Lip 0 Y → 2 Lip 0 X E L : Lip 0 Y → 2 Lip 0 X E_(L):Lip_(0)Y rarr2^(Lip_(0)X)E_{L}: \operatorname{Lip}_{0} Y \rightarrow 2^{\operatorname{Lip}_{0} X}EL:Lip0Y→2Lip0X
if e L ( f ) ∈ E L ( f ) e L ( f ) ∈ E L ( f ) e_(L)(f)inE_(L)(f)e_{L}(f) \in E_{L}(f)eL(f)∈EL(f), for every f ∈ Lip p 0 Y f ∈ Lip p 0 Y f in Lipp_(0)Yf \in \operatorname{Lip} p_{0} Yf∈Lipp0Y. A selection e u e u e_(u)e_{u}eu associated to the extension operator E u E u E_(u)E_{u}Eu is defined in a similar way.
Now we shall consider the following problem : there exists a common linear and continuous selection associated to the operators E L E L E_(L)E_{L}EL and E u E u E_(u)E_{u}Eu ?
The answer is given by the following theorem:
Theorem 2. The function e : Lip 0 Y → Lip 9 X e : Lip 0 Y → Lip 9 X e:Lip_(0)Y rarrLip_(9)Xe: \operatorname{Lip}_{0} Y \rightarrow \operatorname{Lip}_{9} Xe:Lip0Y→Lip9X, given by the equality
(15) e ( f ) = 1 2 ( F 1 + F 2 ) , j ∈ Lip 0 Y , (15) e ( f ) = 1 2 F 1 + F 2 , j ∈ Lip 0 Y , {:(15)e(f)=(1)/(2)(F_(1)+F_(2))","quad j inLip_(0)Y",":}\begin{equation*} e(f)=\frac{1}{2}\left(F_{1}+F_{2}\right), \quad j \in \operatorname{Lip}_{0} Y, \tag{15} \end{equation*}(15)e(f)=12(F1+F2),j∈Lip0Y,
where
F 1 ( x ) = max { f ( y ) − ‖ f ‖ L ⋅ | x − y | : y ∈ Y } , F 1 ( x ) = max f ( y ) − ‖ f ‖ L ⋅ | x − y | : y ∈ Y , F_(1)(x)=max{f(y)-||f||_(L)*|x-y|:y inY},F_{1}(x)=\max \left\{f(y)-\|f\|_{L} \cdot|x-y|: y \in \mathbf{Y}\right\},F1(x)=max{f(y)−‖f‖L⋅|x−y|:y∈Y},
(16) F 2 ( x ) = min { f ( y ) + ‖ f ‖ L ⋅ | x − y | : y ∈ Y } , (16) F 2 ( x ) = min f ( y ) + ‖ f ‖ L ⋅ | x − y | : y ∈ Y , {:(16)F_(2)(x)=min{f(y)+||f||_(L)*|x-y|:y in Y}",":}\begin{equation*} F_{2}(x)=\min \left\{f(y)+\|f\|_{L} \cdot|x-y|: y \in Y\right\}, \tag{16} \end{equation*}(16)F2(x)=min{f(y)+‖f‖L⋅|x−y|:y∈Y},
is a common linear and continuous selection for the operators E L E L E_(L)E_{L}EL and E w E w E_(w)E_{w}Ew.
Proof. For f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y, the functions given by (16) are extensions of the function f f fff, preserving the Lipschitz norm (see [1], [4], [8] for the properties of the functions F 1 , F 2 F 1 , F 2 F_(1),F_(2)F_{1}, F_{2}F1,F2 and of the set E L ( f ) E L ( f ) E_(L)(f)E_{L}(f)EL(f) ).
For x ∈ X x ∈ X x in Xx \in Xx∈X we find
(17) c ( f ) ( x ) = f ( a ) for x ∈ [ c , a ) = f ( x ) for x ∈ [ a , b ] = Y = f ( b ) for x ∈ ( b , d ] (17) c ( f ) ( x ) = f ( a )  for  x ∈ [ c , a ) = f ( x )  for  x ∈ [ a , b ] = Y = f ( b )  for  x ∈ ( b , d ] {:[(17)c(f)(x)=f(a)" for "x in[c","a)],[=f(x)" for "x in[a","b]=Y],[=f(b)" for "x in(b","d]]:}\begin{align*} c(f)(x) & =f(a) \text { for } x \in[c, a) \tag{17}\\ & =f(x) \text { for } x \in[a, b]=\mathbf{Y} \\ & =f(b) \text { for } x \in(b, d] \end{align*}(17)c(f)(x)=f(a) for x∈[c,a)=f(x) for x∈[a,b]=Y=f(b) for x∈(b,d]
Obviously that ‖ e ( f ) ‖ L = ‖ f ‖ L ‖ e ( f ) ‖ L = ‖ f ‖ L ||e(f)||_(L)=||f||_(L)\|e(f)\|_{L}=\|f\|_{L}‖e(f)‖L=‖f‖L so that e ( f ) ∈ E L ( f ) e ( f ) ∈ E L ( f ) e(f)inE_(L)(f)e(f) \in E_{L}(f)e(f)∈EL(f). Furthermore, ‖ e ( f ) ‖ u = ‖ f ‖ u ‖ e ( f ) ‖ u = ‖ f ‖ u ||e(f)||_(u)=||f||_(u)\|e(f)\|_{u}=\|f\|_{u}‖e(f)‖u=‖f‖u so that e ( f ) e ( f ) e(f)e(f)e(f) belongs to the set E u ( f ) E u ( f ) E_(u)(f)E_{u}(f)Eu(f), too.
In [9] Theorem 4 and Corollary 5 , it was proved that e e eee is a linear selection, continuous with respect to the topology generated by the Lipschitz norm.
We shall show that e e eee is continuous with respect to the topology generated by the uniform norm, too. To this end, let ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and let 0 < δ < ε 0 < δ < ε 0 < delta < epsi0<\delta<\varepsilon0<δ<ε. If f , g ∈ Lip 0 Y f , g ∈ Lip 0 Y f,g inLip_(0)Yf, g \in \operatorname{Lip}_{0} Yf,g∈Lip0Y are such that ‖ f − g ‖ u < δ ‖ f − g ‖ u < δ ||f-g||_(u) < delta\|f-g\|_{u}<\delta‖f−g‖u<δ, then ‖ e 1 f ) − e ( g ) ‖ u == ‖ f − g ‖ u < δ < ε ‖ e 1 f − e ( g ) ‖ u == ‖ f − g ‖ u < δ < ε {:||e_(1)f)-e(g)||_(u)==||f-g||_(u) < delta < epsi\left.\| e_{1} f\right)-e(g) \|_{u}= =\|f-g\|_{u}<\delta<\varepsilon‖e1f)−e(g)‖u==‖f−g‖u<δ<ε, proving the continuity of e e eee with respect to the uniform norm. Theorem is proved.
There is a close relation between the selections associated to the extensions operators E L E L E_(L)E_{L}EL and E u E u E_(u)E_{u}Eu and those associated to the operators of metric projection (in the Lipschitz norm and respectively in the uniform norm) on the annihilator of the set Y Y YYY in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X, i.e. on the subspace.
(18) Y ⊥ = { G ∈ Lip 0 X : G | Y = 0 } . (18) Y ⊥ = G ∈ Lip 0 X : G Y = 0 . {:(18)Y^(_|_)={G inLip_(0)X:G|_(Y)=0}.:}\begin{equation*} Y^{\perp}=\left\{G \in \operatorname{Lip}_{0} X:\left.G\right|_{Y}=0\right\} . \tag{18} \end{equation*}(18)Y⊥={G∈Lip0X:G|Y=0}.
Let P Y L ⊥ , P Y n ⊥ P Y L ⊥ , P Y n ⊥ P_(Y)^(L)_|_,P_(Y)^(n)_|_P_{Y}^{L} \perp, P_{Y}^{n} \perpPYL⊥,PYn⊥ Lip 0 X → 2 Y ⊥ 0 X → 2 Y ⊥ 0X rarr2^(Y _|_)0 X \rightarrow 2^{Y \perp}0X→2Y⊥ denote the operators of metric projection on Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ (in the Lipschitz respectively in the uniform norm) and let
(19) d L ( F , Y ⊥ ) = inf { ‖ F − G ‖ L : G ∈ Y ⊥ } , d u ( F ′ , Y ⊥ ) = inf { ‖ F − G ‖ A : G ∈ Y ⊥ } , (19) d L F , Y ⊥ = inf ‖ F − G ‖ L : G ∈ Y ⊥ , d u F ′ , Y ⊥ = inf ‖ F − G ‖ A : G ∈ Y ⊥ , {:[(19)d_(L)(F,Y^(_|_))=i n f{||F-G||_(L):G inY^(_|_)}","],[d_(u)(F^('),Y^(_|_))=i n f{||F-G||_(A):G inY^(_|_)}","]:}\begin{gather*} d_{L}\left(F, Y^{\perp}\right)=\inf \left\{\|F-G\|_{L}: G \in Y^{\perp}\right\}, \tag{19}\\ d_{u}\left(F^{\prime}, Y^{\perp}\right)=\inf \left\{\|F-G\|_{A}: G \in Y^{\perp}\right\}, \end{gather*}(19)dL(F,Y⊥)=inf{‖F−G‖L:G∈Y⊥},du(F′,Y⊥)=inf{‖F−G‖A:G∈Y⊥},
be the distances from an element F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X to the subspace Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ with respect to the Lipschitz and the uniform norm, respectively.
An element G 1 ∈ Y ⊥ G 1 ∈ Y ⊥ G_(1)inY^(_|_)G_{1} \in Y^{\perp}G1∈Y⊥ such that ‖ F − G 1 ‖ L = d L ( F , Y ⊥ ) F − G 1 L = d L F , Y ⊥ ||F-G_(1)||_(L)=d_(L)(F,Y^(_|_))\left\|F-G_{1}\right\|_{L}=d_{L}\left(F, Y^{\perp}\right)‖F−G1‖L=dL(F,Y⊥) is called an L L LLL - nearest point to F F FFF in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ and en element G 2 ∈ Y ⊥ G 2 ∈ Y ⊥ G_(2)inY^(_|_)G_{2} \in Y^{\perp}G2∈Y⊥ for which ‖ F − G 2 ‖ ‖ == d u ( F , Y ⊥ ) F − G 2 ‖ == d u F , Y ⊥ ||F-G_(2)||_(||)==d_(u)(F,Y^(_|_))\left\|F-G_{2}\right\|_{\|}= =d_{u}\left(F, Y^{\perp}\right)‖F−G2‖‖==du(F,Y⊥) is called a u u uuu - nearest point to F F FFF in Y ⊥ Y ⊥ Y^(_|_)\boldsymbol{Y}^{\perp}Y⊥.
Theorem 3. (a) The equalities
(20) d L ( F ′ , Y ⊥ ) = ‖ F ′ | Y ‖ L ; d n ( F ′ , Y ⊥ ) = ‖ F ′ | Y ‖ N (20) d L F ′ , Y ⊥ = F ′ Y L ; d n F ′ , Y ⊥ = F ′ Y N {:(20)d_(L)(F^('),Y^(_|_))=||F^(')|_(Y)||_(L);quadd_(n)(F^('),Y^(_|_))=||F^(')|_(Y)||_(N):}\begin{equation*} \boldsymbol{d}_{L}\left(\boldsymbol{F}^{\prime}, \boldsymbol{Y}^{\perp}\right)=\left\|\left.\boldsymbol{F}^{\prime}\right|_{Y}\right\|_{L} ; \quad \boldsymbol{d}_{n}\left(\boldsymbol{F}^{\prime}, \boldsymbol{Y}^{\perp}\right)=\left\|\left.\boldsymbol{F}^{\prime}\right|_{Y}\right\|_{N} \tag{20} \end{equation*}(20)dL(F′,Y⊥)=‖F′|Y‖L;dn(F′,Y⊥)=‖F′|Y‖N
hold, for every F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X;
(b) A function G 1 ∈ Y ⊥ G 1 ∈ Y ⊥ G_(1)inY^(_|_)G_{1} \in Y^{\perp}G1∈Y⊥ is an L L LLL-nearest point to F ⊤ F ⊤ F^(TT)F^{\top}F⊤ in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ if and only if G 1 = F − H 1 G 1 = F − H 1 G_(1)=F-H_(1)G_{1}=F-H_{1}G1=F−H1, for a function H 1 ∈ E L ( F | X ) H 1 ∈ E L F X H_(1)inE_(L)(F|_(X))H_{1} \in E_{L}\left(\left.F\right|_{X}\right)H1∈EL(F|X);
(c) A function G 2 ∈ Y ⊥ G 2 ∈ Y ⊥ G_(2)inY^(_|_)G_{2} \in Y^{\perp}G2∈Y⊥ is a u-nearest point to F F FFF in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ if and only iff G 2 = F − H 2 G 2 = F − H 2 G_(2)=F-H_(2)G_{2}=F-H_{2}G2=F−H2, for a function H 2 ∈ E n ( F | Y ) H 2 ∈ E n F Y H_(2)inE_(n)(F|_(Y))H_{2} \in E_{n}\left(\left.F\right|_{Y}\right)H2∈En(F|Y).
Proof. The first equality in (20) was proved in [5], Theorem 2 and Lemma 1. To prove the second one, observe that
‖ F | Y ‖ H = ‖ F | Y − G | Y ‖ n ⩽ ‖ F − G ‖ u , F Y H = F Y − G Y n ⩽ ‖ F − G ‖ u , ||F|_(Y)||_(H)=||F|_(Y)-G|_(Y)||_(n) <= ||F-G||_(u),\left\|\left.F\right|_{Y}\right\|_{H}=\left\|\left.F\right|_{Y}-\left.G\right|_{Y}\right\|_{n} \leqslant\|F-G\|_{u},‖F|Y‖H=‖F|Y−G|Y‖n⩽‖F−G‖u,
for every G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ and, taking the infimum with respect to G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥, one obtains the inequality
‖ F ′ | Y ‖ N ⩽ d n ( F ′ , Y ⊥ ) . F ′ Y N ⩽ d n F ′ , Y ⊥ . ||F^(')|_(Y)||_(N) <= d_(n)(F^('),Y^(_|_)).\left\|\left.F^{\prime}\right|_{Y}\right\|_{N} \leqslant d_{n}\left(F^{\prime}, \boldsymbol{Y}^{\perp}\right) .‖F′|Y‖N⩽dn(F′,Y⊥).
On the other hand
d n ( F , Y ⊥ ) ⩽ ‖ F − ( F − H ) ‖ n = ‖ H ‖ N = ‖ F | Y ‖ N , d n F , Y ⊥ ⩽ ‖ F − ( F − H ) ‖ n = ‖ H ‖ N = F Y N , d_(n)(F,Y^(_|_)) <= ||F-(F-H)||_(n)=||H||_(N)=||F|_(Y)||_(N),d_{n}\left(F, Y^{\perp}\right) \leqslant\|F-(F-H)\|_{n}=\|H\|_{N}=\left\|\left.F\right|_{Y}\right\|_{N},dn(F,Y⊥)⩽‖F−(F−H)‖n=‖H‖N=‖F|Y‖N,
for every H ∈ E n ( F | Y r ) H ∈ E n F Y r H inE_(n)(F|_(Y^(r)))H \in E_{n}\left(\left.F\right|_{Y^{r}}\right)H∈En(F|Yr), implying ‖ F | Y ∗ ‖ n ⩾ d n ( F ′ , Y ⊥ ) F Y ∗ n ⩾ d n F ′ , Y ⊥ ||F|_(Y^(**))||_(n) >= d_(n)(F^('),Y^(_|_))\left\|\left.F\right|_{Y^{*}}\right\|_{n} \geqslant d_{n}\left(F^{\prime}, \boldsymbol{Y}^{\perp}\right)‖F|Y∗‖n⩾dn(F′,Y⊥).
The fact that every L L LLL-nearest point G 1 G 1 G_(1)G_{1}G1 to F ⊥ F ⊥ F^(_|_)F^{\perp}F⊥ in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ has the form G 1 == F ′ − H 1 G 1 == F ′ − H 1 G_(1)==F^(')-H_(1)G_{1}= =F^{\prime}-H_{1}G1==F′−H1, for an H 1 ∈ E L ( F ′ | Y ) H 1 ∈ E L F ′ Y H_(1)inE_(L)(F^(')|_(Y))H_{1} \in E_{L}\left(\left.F^{\prime}\right|_{Y}\right)H1∈EL(F′|Y) was proved in [6], Lemma 1.
The assertion (c) can be proved in a similar way.
From Theorem 3 it follows that
(21) P Y L ⊥ ( F ) = F − E L ( F ′ | Y ) ; P Y N ⊥ F ′ ) = F − E n ( F | Y ) , (21) P Y L ⊥ ( F ) = F − E L F ′ Y ; P Y N ⊥ F ′ = F − E n F Y , {:(21){:P_(Y)^(L)_|_(F)=F-E_(L)(F^(')|_(Y));quadP_(Y)^(N)_|_F^('))=F-E_(n)(F|_(Y))",":}\begin{equation*} \left.P_{Y}^{L} \perp(F)=F-E_{L}\left(\left.F^{\prime}\right|_{Y}\right) ; \quad P_{Y}^{N} \perp F^{\prime}\right)=F-E_{n}\left(\left.F\right|_{Y}\right), \tag{21} \end{equation*}(21)PYL⊥(F)=F−EL(F′|Y);PYN⊥F′)=F−En(F|Y),
and, taking into account relations (13) and (14), we find
(22) P Y u ⊥ ( F ′ ) ∩ P Y L ⊥ ( F ) ⊃ F − T ( E L ( F ′ | Y ) ) , (22) P Y u ⊥ F ′ ∩ P Y L ⊥ ( F ) ⊃ F − T E L F ′ Y , {:(22)P_(Y)^(u)_|_(F^('))nnP_(Y)^(L)_|_(F)sup F-T(E_(L)(F^(')|_(Y)))",":}\begin{equation*} P_{Y}^{u} \perp\left(F^{\prime}\right) \cap P_{Y}^{L} \perp(F) \supset F-T\left(E_{L}\left(\left.F^{\prime}\right|_{Y}\right)\right), \tag{22} \end{equation*}(22)PYu⊥(F′)∩PYL⊥(F)⊃F−T(EL(F′|Y)),
for every F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X.
The following Corollary holds:
Corollary 1. (a) For every F ∈ F ∈ F inF \inF∈ Lip 0 X 0 X _(0)X_{0} X0X there exists a function G 0 ∈ Y ⊥ G 0 ∈ Y ⊥ G_(0)inY^(_|_)G_{0} \in Y^{\perp}G0∈Y⊥ which is simultaneously an L L LLL-nearest point and a u u uuu-nearest point for F F FFF in Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥;
(b) If F ′ ∈ Lip 0 X F ′ ∈ Lip 0 X F^(')inLip_(0)XF^{\prime} \in \operatorname{Lip}_{0} XF′∈Lip0X is such that ‖ F | Y ‖ L = ‖ F | Y ‖ u F Y L = F Y u ||F|_(Y)||_(L)=||F|_(Y)||_(u)\left\|\left.F\right|_{Y}\right\|_{L}=\left\|\left.F\right|_{Y}\right\|_{u}‖F|Y‖L=‖F|Y‖u, then
(23) d L ( F ′ , Y ⊥ ) = d u ( F ′ , Y ⊥ ) ; (23) d L F ′ , Y ⊥ = d u F ′ , Y ⊥ ; {:(23)d_(L)(F^('),Y^(_|_))=d_(u)(F^('),Y^(_|_));:}\begin{equation*} d_{L}\left(F^{\prime}, Y^{\perp}\right)=d_{u}\left(F^{\prime}, Y^{\perp}\right) ; \tag{23} \end{equation*}(23)dL(F′,Y⊥)=du(F′,Y⊥);
(c) The function p : Lip 0 X → Y ⊥ p : Lip 0 X → Y ⊥ p:Lip_(0)X rarrY^(_|_)p: \operatorname{Lip}_{0} X \rightarrow Y^{\perp}p:Lip0X→Y⊥ given by
(24) p ( F ) = F − e ( F | Y ) (24) p ( F ) = F − e F Y {:(24)p(F)=F-e(F|_(Y)):}\begin{equation*} p(F)=F-e\left(\left.F\right|_{Y}\right) \tag{24} \end{equation*}(24)p(F)=F−e(F|Y)
is a common linear and continuous setection of the metric projection operators P Y L ⊥ P Y L ⊥ P_(Y)^(L)_|_P_{Y}^{L} \perpPYL⊥ and P Y u ⊥ P Y u ⊥ P_(Y)^(u)_|_P_{Y}^{u} \perpPYu⊥.
(d) The subspace W = { e ( F | Y ) : F ∈ Lip 0 X } W = e F Y : F ∈ Lip 0 X W={e(F|_(Y)):F inLip_(0)X}W=\left\{e\left(\left.F\right|_{Y}\right): F \in \operatorname{Lip}_{0} X\right\}W={e(F|Y):F∈Lip0X} is the algebraic and topological complement of Y ⊥ Y ⊥ Y _|_Y \perpY⊥ in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X.
Furthermore, the assertions (c) and (d) are equivalent.
Proof. The assertions (a) and (b) are immediate consequences of Theorem 3. It is obvious that the selection p p ppp, given by (24) is linear and continuous (both in the Lipschitz and in the uniform norms). The fact that W W WWW is the complement of Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ (with respect to the Lipschitz norm) was proved in [9], Corollary 7, but it follows also from Theorem 2.2 in [2]. The same theorem implies the equivalence of the assertions (c) and (d). Every function F ∈ Lip 0 X F ∈ Lip 0 X F inLip_(0)XF \in \operatorname{Lip}_{0} XF∈Lip0X can be uniquely written in the form F == G + H F == G + H F==G+HF= =G+HF==G+H, with G ∈ Y ⊥ G ∈ Y ⊥ G inY^(_|_)G \in Y^{\perp}G∈Y⊥ and H ∈ W H ∈ W H in WH \in WH∈W, where the functions G G GGG and H H HHH can be explicitly given by
G ( x ) = 0 for x ∈ [ a , b ] ≐ F ( x ) − F ( a ) for c ⩽ x < a = F ( x ) − F ( b ) for b < x ⩽ d G ( x ) = 0  for  x ∈ [ a , b ] ≐ F ( x ) − F ( a )  for  c ⩽ x < a = F ( x ) − F ( b )  for  b < x ⩽ d {:[G(x)=0quad" for "x in[a","b]],[≐F(x)-F(a)" for "c <= x < a],[=F(x)-F(b)" for "b < x <= d]:}\begin{aligned} G(x) & =0 \quad \text { for } x \in[a, b] \\ & \doteq F(x)-F(a) \text { for } c \leqslant x<a \\ & =F(x)-F(b) \text { for } b<x \leqslant d \end{aligned}G(x)=0 for x∈[a,b]≐F(x)−F(a) for c⩽x<a=F(x)−F(b) for b<x⩽d
and
H 1 x ) = F ( x ) for x ∈ [ a , b ] = F ′ ( a ) for c ⩽ x < a = F ′ ( b ) for b < x ⩽ d H 1 x = F ( x )  for  x ∈ [ a , b ] = F ′ ( a )  for  c ⩽ x < a = F ′ ( b )  for  b < x ⩽ d {:[{:H_(1)x)=F(x)" for "x in[a","b]],[=F^(')(a)" for "c <= x < a],[=F^(')(b)" for "b < x <= d]:}\begin{aligned} \left.H_{1} x\right) & =F(x) \text { for } x \in[a, b] \\ & =F^{\prime}(a) \text { for } c \leqslant x<a \\ & =F^{\prime}(b) \text { for } b<x \leqslant d \end{aligned}H1x)=F(x) for x∈[a,b]=F′(a) for c⩽x<a=F′(b) for b<x⩽d
Kemarks. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘ The assertions (c) and (d) from Corollary 1 are illustrations to the Theorem 2.2 in [2] (assertions (1), (2) and (4)).
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ The kernels of the applications P Y ′ L P Y ′ L P_(Y^('))^(L)P_{Y^{\prime}}^{L}PY′L, and P Y ′ u P Y ′ u P_(Y^('))^(u)P_{Y^{\prime}}^{u}PY′u are given by ker P Y L = { F ∈ Lip 0 X : ‖ F ‖ L = ‖ F ′ | Y ‖ L } , P Y L = F ∈ Lip 0 X : ‖ F ‖ L = F ′ Y L , quadP_(Y)^(L)={F inLip_(0)X:||F||_(L)=||F^(')|_(Y)||_(L)},quad\quad P_{Y}^{L}=\left\{F \in \operatorname{Lip}_{0} X:\|F\|_{L}=\left\|\left.F^{\prime}\right|_{Y}\right\|_{L}\right\}, \quadPYL={F∈Lip0X:‖F‖L=‖F′|Y‖L}, respectively ker P L ′ ′ ⊥== { F ∈ Lip 0 X : ‖ F ‖ u = ‖ F | Y ‖ u } P L ′ ′ ⊥== F ∈ Lip 0 X : ‖ F ‖ u = F Y u quadP_(L)^('')⊥=={F inLip_(0)X:||F||_(u)=||F|_(Y)||_(u)}\quad P_{L}^{\prime \prime} \perp= =\left\{F \in \operatorname{Lip}_{0} X:\|F\|_{u}=\left\|\left.F\right|_{Y}\right\|_{u}\right\}PL′′⊥=={F∈Lip0X:‖F‖u=‖F|Y‖u}, and W W WWW is a closed subspace of Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X contained in ker P Y L ⊥ ∩ ker P Y u ⊥ P Y L ⊥ ∩ ker P Y u ⊥ P_(Y)^(L)^(_|_)nn kerP_(Y)^(u)_|_P_{Y}^{L}{ }^{\perp} \cap \operatorname{ker} P_{Y}^{u} \perpPYL⊥∩kerPYu⊥.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘ As Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ and W W WWW are closed subspaces of Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X, which is a Banach space with respect to both of Lipschitz and uniform norms, it follows that their algebraic sum is also topological (a consequence of the open mapping theorem)
Example. Let Y = [ a , b ] = [ 1 , 3 ] , X = [ − 2 , 5 ~ ] , x 0 = 2  Example. Let  Y = [ a , b ] = [ 1 , 3 ] , X = [ − 2 , 5 ~ ] , x 0 = 2 " Example. Let "Y=[a,b]=[1,3],X=[-2, tilde(5)],x_(0)=2\text { Example. Let } Y=[a, b]=[1,3], X=[-2, \tilde{5}], x_{0}=2 Example. Let Y=[a,b]=[1,3],X=[−2,5~],x0=2
The function
f ( x ) = − x + 2 fer x ∈ [ 1 , 2 ] , = 2 x − 4 for x ∈ ( 2 , 3 ⌉ f ( x ) = − x + 2  fer  x ∈ [ 1 , 2 ] , = 2 x − 4  for  x ∈ ( 2 , 3 ⌉ {:[f(x)=-x+2" fer "x in[1","2]","],[=2x-4quad" for "x in(2","3~|]:}\begin{aligned} f(x) & =-x+2 \text { fer } x \in[1,2], \\ & =2 x-4 \quad \text { for } x \in(2,3\rceil \end{aligned}f(x)=−x+2 fer x∈[1,2],=2x−4 for x∈(2,3⌉
is in Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y and ‖ f ‖ L = ‖ f ‖ n = 2 ‖ f ‖ L = ‖ f ‖ n = 2 ||f||_(L)=||f||_(n)=2\|f\|_{L}=\|f\|_{n}=2‖f‖L=‖f‖n=2
The functions (16) from Theorem 2 are
F 1 ( x ) = 2 x − 1 , x ∈ [ − 2 , 1 ] and F 2 ( x ) = − 2 x + 3 , x ∈ [ − 2 , 1 ] = f ( x ) , x ∈ ( 1 , 3 ] = f ( x ) , x ∈ ( 1 , 3 ] = − 2 x + 8 , x ∈ ( 3 , 5 ] = 2 x − 4 , x ∈ ( 3 , 5 ] F 1 ( x ) = 2 x − 1 , x ∈ [ − 2 , 1 ]  and  F 2 ( x ) = − 2 x + 3 , x ∈ [ − 2 , 1 ] = f ( x ) , x ∈ ( 1 , 3 ] = f ( x ) , x ∈ ( 1 , 3 ] = − 2 x + 8 , x ∈ ( 3 , 5 ] = 2 x − 4 , x ∈ ( 3 , 5 ] {:[F_(1)(x)=2x-1","x in[-2","1]" and "F_(2)(x)=-2x+3","x in[-2","1]],[=f(x)","x in(1","3]=f(x)","x in(1","3]],[=-2x+8","x in(3","5]=2x-4","x in(3","5]]:}\begin{aligned} F_{1}(x) & =2 x-1, x \in[-2,1] \text { and } & F_{2}(x) & =-2 x+3, x \in[-2,1] \\ & =f(x), x \in(1,3] & & =f(x), x \in(1,3] \\ & =-2 x+8, x \in(3,5] & & =2 x-4, x \in(3,5] \end{aligned}F1(x)=2x−1,x∈[−2,1] and F2(x)=−2x+3,x∈[−2,1]=f(x),x∈(1,3]=f(x),x∈(1,3]=−2x+8,x∈(3,5]=2x−4,x∈(3,5]
Every function F ∈ E L ( f ) F ∈ E L ( f ) F inE_(L)(f)F \in E_{L}(f)F∈EL(f) verifies the inequalities
F 1 ( x ) ⩽ F ( x ) ⩽ F 2 ( x ) , x ∈ [ − 2 , 5 ] . F 1 ( x ) ⩽ F ( x ) ⩽ F 2 ( x ) , x ∈ [ − 2 , 5 ] . F_(1)(x) <= F(x) <= F_(2)(x),quad x in[-2,5].F_{1}(x) \leqslant F(x) \leqslant F_{2}(x), \quad x \in[-2,5] .F1(x)⩽F(x)⩽F2(x),x∈[−2,5].
implying that
T ( F 1 ) ⩽ T ( F ′ ) ⩽ T ( F 2 ) . T F 1 ⩽ T F ′ ⩽ T F 2 . T(F_(1)) <= T(F^(')) <= T(F_(2)).T\left(F_{1}\right) \leqslant T\left(F^{\prime}\right) \leqslant T\left(F_{2}\right) .T(F1)⩽T(F′)⩽T(F2).
We have T ( F 2 ) ( x ) = F ¯ 2 ( x ) = 2 , x ∈ [ − 2 , 1 / 2 ] T F 2 ( x ) = F ¯ 2 ( x ) = 2 , x ∈ [ − 2 , 1 / 2 ] T(F_(2))(x)= bar(F)_(2)(x)=2,quad x in[-2,1//2]T\left(F_{2}\right)(x)=\bar{F}_{2}(x)=2, \quad x \in[-2,1 / 2]T(F2)(x)=F¯2(x)=2,x∈[−2,1/2]
= − 2 x + 3 , x ( 1 / 2 , 1 ) = f ( x ) , x ∈ [ 1 , 3 ] = 2 , x ∈ ( 3 , 5 ] . = − 2 x + 3 ,      x ( 1 / 2 , 1 ) = f ( x ) ,      x ∈ [ 1 , 3 ] = 2 ,      x ∈ ( 3 , 5 ] . {:[=-2x+3",",x(1//2","1)],[=f(x)",",x in[1","3]],[=2",",x in(3","5].]:}\begin{array}{ll} =-2 x+3, & x(1 / 2,1) \\ =f(x), & x \in[1,3] \\ =2, & x \in(3,5] . \end{array}=−2x+3,x(1/2,1)=f(x),x∈[1,3]=2,x∈(3,5].
Let
H ( x ) = 2 , x ∈ [ − 2 , 3 / 4 ] = − 4 x + 5 , x ∈ ( 3 / 4 , 1 ] = F ¯ 2 ( x ) , x ∈ ( 1 , 5 ] . H ( x ) = 2 , x ∈ [ − 2 , 3 / 4 ] = − 4 x + 5 , x ∈ ( 3 / 4 , 1 ] = F ¯ 2 ( x ) , x ∈ ( 1 , 5 ] . {:[H(x)=2","x in[-2","3//4]],[=-4x+5","x in(3//4","1]],[= bar(F)_(2)(x)","x in(1","5].]:}\begin{aligned} H(x) & =2, & & x \in[-2,3 / 4] \\ & =-4 x+5, & & x \in(3 / 4,1] \\ & =\bar{F}_{2}(x), & & x \in(1,5] . \end{aligned}H(x)=2,x∈[−2,3/4]=−4x+5,x∈(3/4,1]=F¯2(x),x∈(1,5].
Then ‖ H ‖ L = 4 ‖ H ‖ L = 4 ||H||_(L)=4\|H\|_{L}=4‖H‖L=4 and ‖ H ‖ u = 2 ‖ H ‖ u = 2 ||H||_(u)=2\|H\|_{u}=2‖H‖u=2 so that H ∉ T ( E L ( f ) ) H ∉ T E L ( f ) H!in T(E_(L)(f))H \notin T\left(E_{L}(f)\right)H∉T(EL(f)) but H ∈ E u ( f ) H ∈ E u ( f ) H inE_(u)(f)H \in E_{u}(f)H∈Eu(f). This example shows that the inclusion (11) can be strict.
Also
Y ⊥ = { G ∈ Lip 0 [ − 2 , 5 ] : G | [ 1 , 3 ] = 0 } . Y ⊥ = G ∈ Lip 0 [ − 2 , 5 ] : G [ 1 , 3 ] = 0 . Y^(_|_)={G inLip_(0)[-2,5]:G|_([1,3])=0}.Y^{\perp}=\left\{G \in \operatorname{Lip}_{0}[-2,5]:\left.G\right|_{[1,3]}=0\right\} .Y⊥={G∈Lip0[−2,5]:G|[1,3]=0}.
If F ∈ Lip ) 0 [ − 2 , 5 ] F ∈ Lip ) 0 [ − 2 , 5 ] F in Lip)_(0)[-2,5]F \in \operatorname{Lip})_{0}[-2,5]F∈Lip)0[−2,5] is such that F | [ 1 , 3 ] = f F [ 1 , 3 ] = f F|_([1,3])=f\left.F\right|_{[1,3]}=fF|[1,3]=f then ‖ F | Y ‖ L = ‖ F | Y ‖ L F Y L = F Y L ||F|_(Y)||_(L)=||F|_(Y)||_(L)\left\|\left.F\right|_{Y}\right\|_{L}=\left\|\left.F\right|_{Y}\right\|_{L}‖F|Y‖L=‖F|Y‖L and, consequently,
d L ( F ′ , Y ⊥ ) = d n ( F ′ , Y ⊥ ) = 2 , d L F ′ , Y ⊥ = d n F ′ , Y ⊥ = 2 , d_(L)(F^('),Y^(_|_))=d_(n)(F^('),Y^(_|_))=2,d_{L}\left(F^{\prime}, \boldsymbol{Y}^{\perp}\right)=d_{n}\left(F^{\prime}, \boldsymbol{Y}^{\perp}\right)=2,dL(F′,Y⊥)=dn(F′,Y⊥)=2,
showing that condition (23) from Corollary 1 is fulfilled.

REFERENCES

1 Czipser, J. and Géher, L., Extension of Funclions Satisfying a Lipschilz Condition, Acta Math. Acad. Sci. Hungar. 6 (1955), 213-220.
2 Deutsch, F., Linear Selections for Melric Projection, J. Funct. Analysis 49, 3(1982), 269--292.
3. Deutsch, F., Wu Li, Sung-Ho Park, Tielze Extensions and Conlinuous Selections for Metric Projections, J. Approx. Theory 64, 1 (1991), 55-68.
4. Mc Shane, E. J., Extension of Range of Functions, Bull. Amer. Math. Soc. 40 (1934), 837--842.
5. Mustăta, C., Asupra unor subspalii cebişeviene din spatiul normal al functillor lipschitziene, Hev. Anal. Numer. Teoria Aproximatiei 2 (1973), 81-87.
6. Mustăta, C., Best Approximation and Unique Extension of Lipschilz Funclions, J. Approx. Theory 19, 3 (1977), 222-230.
7. Mustăta, C., M-ideal in Metric Spaces, „Babeş-Bolyai" Univ. Research Seminars, Seminar on Mathematical Analysis, Preprint Nr. 7 (1988), 65-74.
8. Mustăta, C., Extension of Hölder Functions and Some Related Problems of Best Approximation, „Babeş-Bolyai" Univ., Research Seminars, Seminar on Mathematical Analysis, Preprint Nr. 7 (1991), 71-86.
9. Mustăta, C., Selections Associated to Mc Shane's Extension Theorem for Lipschitz Functions, Revue d'Analyse Numér. et de la Théorie de l'Approximation 21, 2 (1992), 135-145.

Receined 4.V. 1993

Inslilutul de CalculOficiul Postal 13400 Ciluj-NapocaRomania

1993

Related Posts