Some remarks concerning norm preserving extension and best approximation

Costica Mustata
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Abstract

Let \(X,Y\) be two normed spaces, \(X_{1}\) a subspace of \(X\) and \(A:X \rightarrow Y\) a continuous linear operator. Let us denote \(Z_{1}=Ker\left( \left. A\right \vert _{X_{1}}\right) ,Z=KerA\) and for \(x\in X,E\left( x\right)=\{y\in X:Ax=Ay\) and \(\left \Vert y\right \Vert =\left \Vert Ax\right \Vert/\left \Vert A\right \Vert \}\) and \(E_{1}\left( x\right) =\{y_{1}\in X_{1}:Ax=Ay_{1}\) and \(\left \Vert y_{1}\right \Vert =\left \Vert Ax\right \Vert/\left \Vert A\right \Vert \}\). One gives the relations between the sets \(E\left( x\right)\), \(E_{1}\left(x\right)\) and \(P_{Z}\left( x\right)\), \(P_{Z_{1}}\left( x\right)\) where \(P_{C}\left( x\right) :\{y\in C:\left \Vert x-y\right \Vert =d\left(x,C\right) \}\). An application is considered.

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Costica Mustata

Costică Mustăţa


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

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C. Mustăţa, Some remarks concerning norm preserving extension and best approximation, Rev. Anal. Numer. Theor. Approx., 29 (2000) No. 2, pp. 173-180, https://ictp.acad.ro/jnaat/journal/article/view/2000-vol29-no2-art7

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Revue d’Analyse Numer.Theor. Approx.

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Publishing Romanian Academy

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https://doi.org/10.33993/jnaat292-667

 

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[1] Cobzas S., Some Applications of a Distance Formula to the Kernel of a Linear Operator (submitted)
[2] Cobzas, S., Mustata, C., Norm Preserving Extension of Convex Lipschitz Functions, J. Approx. Theory, 24, pÞ.236-244, 1978.
[3] Cobzas, S., Mustata, C., Extension of Bilinear Functionals and Best Approximation in 2-normed, Spaces, Studia Univ. “Babes-Bolyai”, Series Math., XLIII, no. 2, pp. 1-13, 1998.
[4] Czipser, J., Geher, L., Extension of Functions Satisfying a Lipschitz Condition, Acta Math. Acad. Sci. Hungar, 6, pp. 213,220, 1955.
[5] Deutsch, F., Wu Li  and Sizwe Mabizela, Helly Extensions and Best Approximation, Parametric Optimization and Related Topics III (J. Guddat, H. Th. Jongen, B. Kummer and F. No5iaeka Eds., Appromation and Optimization, vol. 3, Verlag Peter Lang, pp. 107-120, Flankfurt 1993.
[6] Engelking, R., General Topology, PWN Warszawa, 1985.
[7] McShane, E. J., Extension of Range of Functions, Bull. Amer. Math., 40, pp. 834-842, L934.
[S] Mustata., C., Best Approximation and Unique Extension of Lipschitz Functions, J. Approx. Theory, 19, pp. 222-230, 1977.
[9] Mustata, C., Norm Preserving Extension of Starshaped, Lipschitz Functions, Mathematica, 19, pp. 183-787, L977.

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2000-Mustata-Some remarks concerning norm preserving extension-Jnaat

SOME REMARKS CONCERNING NORM PRESERVING. EXTENSIONS AND BEST APPROXIMATION*

COSTICĂ MUSTĂţA

Abstract

Let X , Y X , Y X,Y\mathrm{X}, \mathrm{Y}X,Y be two normed spaces, X 1 X 1 X_(1)X_{1}X1 a subspace of X X XXX and A : X → Y A : X → Y A:X rarr YA: X \rightarrow YA:X→Y a continuous linear operator. Let us denote Z 1 = Ker ( A | X 1 ) , Z = Ker A Z 1 = Ker A X 1 , Z = Ker A Z_(1)=Ker(A|_(X_(1))),Z=Ker AZ_{1}=\operatorname{Ker}\left(\left.A\right|_{X_{1}}\right), Z=\operatorname{Ker} AZ1=Ker(A|X1),Z=KerA and for x ∈ X , E ( x ) = { y ∈ X : A x = A y x ∈ X , E ( x ) = { y ∈ X : A x = A y x in X,E(x)={y in X:Ax=Ayx \in X, E(x)= \{y \in X: A x=A yx∈X,E(x)={y∈X:Ax=Ay and ‖ y ‖ = ‖ A x ‖ / ‖ A ‖ } ‖ y ‖ = ‖ A x ‖ / ‖ A ‖ } ||y||=||Ax||//||A||}\|y\|=\|A x\| /\|A\|\}‖y‖=‖Ax‖/‖A‖} and E 1 ( x ) = { y 1 ∈ X 1 : A x = A y 1 E 1 ( x ) = y 1 ∈ X 1 : A x = A y 1 E_(1)(x)={y_(1)inX_(1):Ax=Ay_(1):}E_{1}(x)=\left\{y_{1} \in X_{1}: A x=A y_{1}\right.E1(x)={y1∈X1:Ax=Ay1 and ‖ y 1 ‖ = ‖ A x ‖ / ‖ A ‖ } y 1 = ‖ A x ‖ / ‖ A ‖ } ||y_(1)||=||Ax||//||A||}\left\|y_{1}\right\|= \|A x\| /\|A\|\}‖y1‖=‖Ax‖/‖A‖}.

One gives the relations between the sets E ( x ) , E 1 ( x ) E ( x ) , E 1 ( x ) E(x),E_(1)(x)E(x), E_{1}(x)E(x),E1(x) and P Z ( x ) , P Z 1 ( x ) P Z ( x ) , P Z 1 ( x ) P_(Z)(x),P_(Z_(1))(x)P_{Z}(x), P_{Z_{1}}(x)PZ(x),PZ1(x) where P C ( x ) := { y ∈ C : ‖ x − y ‖ = d ( x , C ) } P C ( x ) := { y ∈ C : ‖ x − y ‖ = d ( x , C ) } P_(C)(x):={y in C:||x-y||=d(x,C)}P_{C}(x):=\{y \in C:\|x-y\|=d(x, C)\}PC(x):={y∈C:‖x−y‖=d(x,C)}. An application is considered.

Let X X XXX be a real normed space and M M MMM a nonvoid closed subset of X X XXX. For x ∈ X x ∈ X x in Xx \in Xx∈X let
d ( x , M ) = inf { ‖ x − y ‖ : y ∈ M } d ( x , M ) = inf { ‖ x − y ‖ : y ∈ M } d(x,M)=i n f{||x-y||:y in M}d(x, M)=\inf \{\|x-y\|: y \in M\}d(x,M)=inf{‖x−y‖:y∈M}
be the distance from x x xxx to M M MMM and let
P M ( x ) := { y ∈ M : ‖ x − y ‖ = d ( x , M ) } P M ( x ) := { y ∈ M : ‖ x − y ‖ = d ( x , M ) } P_(M)(x):={y in M:||x-y||=d(x,M)}P_{M}(x):=\{y \in M:\|x-y\|=d(x, M)\}PM(x):={y∈M:‖x−y‖=d(x,M)}
be the set of nearest points from x x xxx in the set M M MMM.
If P M ( x ) ≠ ∅ P M ( x ) ≠ ∅ P_(M)(x)!=O/P_{M}(x) \neq \emptysetPM(x)≠∅ for every x ∈ X x ∈ X x in Xx \in Xx∈X then the set M M MMM is called proximinal, if P M ( x ) P M ( x ) P_(M)(x)P_{M}(x)PM(x) is a singleton for every x ∈ X x ∈ X x in Xx \in Xx∈X then M M MMM is called chebyshevian and if P M ( x ) = ∅ P M ( x ) = ∅ P_(M)(x)=O/P_{M}(x)=\emptysetPM(x)=∅ for every x ∈ X ∖ M x ∈ X ∖ M x in X\\Mx \in X \backslash Mx∈X∖M then the set M M MMM is called antiproximinal.
For a subspace Y Y YYY of X X XXX let
Y ⊥ = { x ∗ ∈ X ∗ : x ∗ | Y = 0 } Y ⊥ = x ∗ ∈ X ∗ : x ∗ Y = 0 Y^(_|_)={x^(**)inX^(**):x^(**)|_(Y)=0}Y^{\perp}=\left\{x^{*} \in X^{*}:\left.x^{*}\right|_{Y}=0\right\}Y⊥={x∗∈X∗:x∗|Y=0}
be the annihilator of the subspace Y Y YYY in the conjugate space X ∗ X ∗ X^(**)X^{*}X∗ of X X XXX.
R.R. Phelps [13] studied the relation between the norm-preserving extension properties of the space Y ∗ Y ∗ Y^(**)Y^{*}Y∗ with respect to X ∗ X ∗ X^(**)X^{*}X∗ and the best approximation properties of Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥. Namely, he proved that every y ∗ ∈ Y ∗ y ∗ ∈ Y ∗ y^(**)inY^(**)y^{*} \in Y^{*}y∗∈Y∗ has a unique normpreserving extension x ∗ ∈ X ∗ x ∗ ∈ X ∗ x^(**)inX^(**)x^{*} \in X^{*}x∗∈X∗ if and only if Y ⊥ Y ⊥ Y^(_|_)Y^{\perp}Y⊥ is a chebyshevian subspace of X ∗ X ∗ X^(**)X^{*}X∗. By the Hahn-Banach extension theorem, every y ∗ ∈ Y ∗ y ∗ ∈ Y ∗ y^(**)inY^(**)y^{*} \in Y^{*}y∗∈Y∗ has at least
one norm-preserving extension x ∗ ∈ X ∗ x ∗ ∈ X ∗ x^(**)inX^(**)x^{*} \in X^{*}x∗∈X∗. Since then, there have been proved a lot of theorems emphasizing the relations between the extension and best approximation properties for special classes of functions. These results correspond to various extension theorems, such as Tietze extension theorem for continuous function [6], Mc Shane's extension theorem for Lipschitz functions [7], extension theorems for bilinear functionals on 2 -normed spaces [3].
S. Cobzas [1] proved that all the above mentioned results can be derived from a formula for the distance to the kernel of a continuous linear operator.
For normed spaces X , Y X , Y X,YX, YX,Y and A : X → Y A : X → Y A:X rarr YA: X \rightarrow YA:X→Y a continuous linear operator, let
(1) Z = Ker A = { x ∈ X : A x = 0 } (1) Z = Ker A = { x ∈ X : A x = 0 } {:(1)Z=Ker A={x in X:Ax=0}:}\begin{equation*} Z=\operatorname{Ker} A=\{x \in X: A x=0\} \tag{1} \end{equation*}(1)Z=KerA={x∈X:Ax=0}
be the kernel of the operator A A AAA. Obviously that Z Z ZZZ is a closed subspace of X X XXX.
For x ∈ X x ∈ X x in Xx \in Xx∈X put
(2) E ( x ) = { y ∈ X : A y = A x and ‖ y ‖ = ‖ A x ‖ ‖ A ‖ } (2) E ( x ) = y ∈ X : A y = A x  and  ‖ y ‖ = ‖ A x ‖ ‖ A ‖ {:(2)E(x)={y in X:Ay=Ax" and "||y||=(||Ax||)/(||A||)}:}\begin{equation*} E(x)=\left\{y \in X: A y=A x \text { and }\|y\|=\frac{\|A x\|}{\|A\|}\right\} \tag{2} \end{equation*}(2)E(x)={y∈X:Ay=Ax and ‖y‖=‖Ax‖‖A‖}
Theorem 1. S. Cobzas, [1]. The following assertions hold: 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘
(3) d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ (3) d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ {:(3)d(x","Z) >= (||Ax||)/(||A||):}\begin{equation*} d(x, Z) \geqslant \frac{\|A x\|}{\|A\|} \tag{3} \end{equation*}(3)d(x,Z)⩾‖Ax‖‖A‖
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘
(4) d ( x , Z ) = ‖ A x ‖ ‖ A ‖ (4) d ( x , Z ) = ‖ A x ‖ ‖ A ‖ {:(4)d(x","Z)=(||Ax||)/(||A||):}\begin{equation*} d(x, Z)=\frac{\|A x\|}{\|A\|} \tag{4} \end{equation*}(4)d(x,Z)=‖Ax‖‖A‖
if and only if there exists a sequence ( z n z n z_(n)z_{n}zn ) in Z Z ZZZ such that
(5) ‖ x − z n ‖ → ‖ A x ‖ ‖ A ‖ (5) x − z n → ‖ A x ‖ ‖ A ‖ {:(5)||x-z_(n)||rarr(||Ax||)/(||A||):}\begin{equation*} \left\|x-z_{n}\right\| \rightarrow \frac{\|A x\|}{\|A\|} \tag{5} \end{equation*}(5)‖x−zn‖→‖Ax‖‖A‖
(a) If (4) holds then
(6)
P Z ( x ) = x − E ( x ) P Z ( x ) = x − E ( x ) P_(Z)(x)=x-E(x)P_{Z}(x)=x-E(x)PZ(x)=x−E(x)
(b) If there exists z 0 ∈ Z z 0 ∈ Z z_(0)in Zz_{0} \in Zz0∈Z such that
(7) ‖ x − z 0 ‖ = ‖ A x ‖ ‖ A ‖ (7) x − z 0 = ‖ A x ‖ ‖ A ‖ {:(7)||x-z_(0)||=(||Ax||)/(||A||):}\begin{equation*} \left\|x-z_{0}\right\|=\frac{\|A x\|}{\|A\|} \tag{7} \end{equation*}(7)‖x−z0‖=‖Ax‖‖A‖
then z 0 ∈ P Z ( x ) z 0 ∈ P Z ( x ) z_(0)inP_(Z)(x)z_{0} \in P_{Z}(x)z0∈PZ(x) and (4) and (6) hold.
By specializing the spaces X , Y X , Y X,YX, YX,Y and the operator A A AAA, S. Cobzas obtained in the above quoted paper a lot of duality results of Phelps type as well as other results on best approximation.
There are also some duality results as, e.g., those concerning normpreserving extensions of convex or star-shaped Lipschitz functions (see [2], [ 9 ] [ 9 ] [9][9][9] ) which cannot be derived from the theorem mentioned above. The aim of this paper is to prove a slight extension of Theorem 1 such as to cover these extension results, too.
Let X , Y X , Y X,YX, YX,Y be normed spaces over the same field K ( R K ( R K(R\mathbb{K}(\mathbb{R}K(R or C ) C ) C)\mathbb{C})C), and let A : X → Y A : X → Y A:X rarr YA: X \rightarrow YA:X→Y be a continuous linear operator. For a subspace X 1 X 1 X_(1)X_{1}X1 of X X XXX let
Z = Ker A and Z 1 = Ker ( A ∣ x 1 ) Z = Ker A  and  Z 1 = Ker A ∣ x 1 Z=Ker A" and "Z_(1)=Ker(A∣x_(1))Z=\operatorname{Ker} A \text { and } Z_{1}=\operatorname{Ker}\left(A \mid \mathrm{x}_{1}\right)Z=KerA and Z1=Ker(A∣x1)
For x ∈ X x ∈ X x in Xx \in Xx∈X let
(8) E ( x ) = { y ∈ X : A x = A y and ‖ y ‖ = ‖ A x ‖ ‖ A ‖ } (8) E ( x ) = y ∈ X : A x = A y  and  ‖ y ‖ = ‖ A x ‖ ‖ A ‖ {:(8)E(x)={y in X:Ax=Ay" and "||y||=(||Ax||)/(||A||)}:}\begin{equation*} E(x)=\left\{y \in X: A x=A y \text { and }\|y\|=\frac{\|A x\|}{\|A\|}\right\} \tag{8} \end{equation*}(8)E(x)={y∈X:Ax=Ay and ‖y‖=‖Ax‖‖A‖}
and
(9) E 1 ( x ) = { y ∈ X 1 : A x = A y and ‖ y ‖ = ‖ A x ‖ ‖ A ‖ } = X 1 ∩ E ( x ) (9) E 1 ( x ) = y ∈ X 1 : A x = A y  and  ‖ y ‖ = ‖ A x ‖ ‖ A ‖ = X 1 ∩ E ( x ) {:(9)E_(1)(x)={y inX_(1):Ax=Ay" and "||y||=(||Ax||)/(||A||)}=X_(1)nn E(x):}\begin{equation*} E_{1}(x)=\left\{y \in X_{1}: A x=A y \text { and }\|y\|=\frac{\|A x\|}{\|A\|}\right\}=X_{1} \cap E(x) \tag{9} \end{equation*}(9)E1(x)={y∈X1:Ax=Ay and ‖y‖=‖Ax‖‖A‖}=X1∩E(x)
Obviously, Z 1 Z 1 Z_(1)Z_{1}Z1 is a subspace of Z Z ZZZ and E 1 ( x ) ⊆ E ( x ) E 1 ( x ) ⊆ E ( x ) E_(1)(x)sube E(x)E_{1}(x) \subseteq E(x)E1(x)⊆E(x), for x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1.
Theorem 2. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. For every x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1 we have
(10) d ( x , Z 1 ) ⩾ d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ (10) d x , Z 1 ⩾ d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ {:(10)d(x,Z_(1)) >= d(x","Z) >= (||Ax||)/(||A||):}\begin{equation*} d\left(x, Z_{1}\right) \geqslant d(x, Z) \geqslant \frac{\|A x\|}{\|A\|} \tag{10} \end{equation*}(10)d(x,Z1)⩾d(x,Z)⩾‖Ax‖‖A‖
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘. For x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1 we have
(11) d ( x , Z 1 ) = d ( x , Z ) = ‖ A x ‖ ‖ A ‖ (11) d x , Z 1 = d ( x , Z ) = ‖ A x ‖ ‖ A ‖ {:(11)d(x,Z_(1))=d(x","Z)=(||Ax||)/(||A||):}\begin{equation*} d\left(x, Z_{1}\right)=d(x, Z)=\frac{\|A x\|}{\|A\|} \tag{11} \end{equation*}(11)d(x,Z1)=d(x,Z)=‖Ax‖‖A‖
if and only if there exists a sequence ( z n z n z_(n)z_{n}zn ) in Z 1 Z 1 Z_(1)Z_{1}Z1 such that
(12) ‖ x − z n ‖ → ‖ A x ‖ ‖ A ‖ . (12) x − z n → ‖ A x ‖ ‖ A ‖ . {:(12)||x-z_(n)||rarr(||Ax||)/(||A||).:}\begin{equation*} \left\|x-z_{n}\right\| \rightarrow \frac{\|A x\|}{\|A\|} . \tag{12} \end{equation*}(12)‖x−zn‖→‖Ax‖‖A‖.
3 ∘ 3 ∘ 3^(@)3^{\circ}3∘. (a) If the equalities (11) hold then
(13) P Z 1 ( x ) = x − E 1 ( x ) (13) P Z 1 ( x ) = x − E 1 ( x ) {:(13)P_(Z_(1))(x)=x-E_(1)(x):}\begin{equation*} P_{Z_{1}}(x)=x-E_{1}(x) \tag{13} \end{equation*}(13)PZ1(x)=x−E1(x)
and
(14) P Z ( x ) = x − E ( x ) (14) P Z ( x ) = x − E ( x ) {:(14)P_(Z)(x)=x-E(x):}\begin{equation*} P_{Z}(x)=x-E(x) \tag{14} \end{equation*}(14)PZ(x)=x−E(x)
(b) If there exists z 0 ∈ Z 1 z 0 ∈ Z 1 z_(0)inZ_(1)z_{0} \in Z_{1}z0∈Z1 such that
(15) ‖ x − z 0 ‖ = ‖ A x ‖ ‖ A ‖ (15) x − z 0 = ‖ A x ‖ ‖ A ‖ {:(15)||x-z_(0)||=(||Ax||)/(||A||):}\begin{equation*} \left\|x-z_{0}\right\|=\frac{\|A x\|}{\|A\|} \tag{15} \end{equation*}(15)‖x−z0‖=‖Ax‖‖A‖
then z 0 ∈ P Z ( x ) z 0 ∈ P Z ( x ) z_(0)inP_(Z)(x)z_{0} \in P_{Z}(x)z0∈PZ(x) and the equalities (11), (13) and (14) hold.
Proof. 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘. Let x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1. For every z ∈ Z 1 z ∈ Z 1 z inZ_(1)z \in Z_{1}z∈Z1 we have
‖ A x ‖ = ‖ A x − A z ‖ = ‖ A ( x − z ) ‖ ⩽ ‖ A ‖ ‖ x − z ‖ ‖ A x ‖ = ‖ A x − A z ‖ = ‖ A ( x − z ) ‖ ⩽ ‖ A ‖ ‖ x − z ‖ ||Ax||=||Ax-Az||=||A(x-z)|| <= ||A||||x-z||\|A x\|=\|A x-A z\|=\|A(x-z)\| \leqslant\|A\|\|x-z\|‖Ax‖=‖Ax−Az‖=‖A(x−z)‖⩽‖A‖‖x−z‖
implying
‖ x − z ‖ ⩾ ‖ A x ‖ ‖ A ‖ , for all z ∈ Z 1 ‖ x − z ‖ ⩾ ‖ A x ‖ ‖ A ‖ ,  for all  z ∈ Z 1 ||x-z|| >= (||Ax||)/(||A||)," for all "z inZ_(1)\|x-z\| \geqslant \frac{\|A x\|}{\|A\|}, \text { for all } z \in Z_{1}‖x−z‖⩾‖Ax‖‖A‖, for all z∈Z1
so that
d ( x , Z 1 ) ⩾ ‖ A x ‖ ‖ A ‖ d x , Z 1 ⩾ ‖ A x ‖ ‖ A ‖ d(x,Z_(1)) >= (||Ax||)/(||A||)d\left(x, Z_{1}\right) \geqslant \frac{\|A x\|}{\|A\|}d(x,Z1)⩾‖Ax‖‖A‖
Similarly
d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ d(x,Z) >= (||Ax||)/(||A||)d(x, Z) \geqslant \frac{\|A x\|}{\|A\|}d(x,Z)⩾‖Ax‖‖A‖
Since Z 1 Z 1 Z_(1)Z_{1}Z1 is a subspace of Z Z ZZZ it follows that
d ( x , Z 1 ) ⩾ d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ d x , Z 1 ⩾ d ( x , Z ) ⩾ ‖ A x ‖ ‖ A ‖ d(x,Z_(1)) >= d(x,Z) >= (||Ax||)/(||A||)d\left(x, Z_{1}\right) \geqslant d(x, Z) \geqslant \frac{\|A x\|}{\|A\|}d(x,Z1)⩾d(x,Z)⩾‖Ax‖‖A‖
for every x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1.
2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ If d ( x , Z 1 ) = ‖ A x ‖ ‖ A ‖ d x , Z 1 = ‖ A x ‖ ‖ A ‖ d(x,Z_(1))=(||Ax||)/(||A||)d\left(x, Z_{1}\right)=\frac{\|A x\|}{\|A\|}d(x,Z1)=‖Ax‖‖A‖ then, by the definition of d ( x , Z 1 ) d x , Z 1 d(x,Z_(1))d\left(x, Z_{1}\right)d(x,Z1), there exists a sequence ( z n ) z n (z_(n))\left(z_{n}\right)(zn) in Z 1 Z 1 Z_(1)Z_{1}Z1 such that
‖ x − z π ‖ → ‖ A x ‖ ‖ A ‖ x − z π → ‖ A x ‖ ‖ A ‖ ||x-z_(pi)||rarr(||Ax||)/(||A||)\left\|x-z_{\pi}\right\| \rightarrow \frac{\|A x\|}{\|A\|}‖x−zπ‖→‖Ax‖‖A‖
Conversely, if ( z n ) z n (z_(n))\left(z_{n}\right)(zn) is a sequence in Z 1 Z 1 Z_(1)Z_{1}Z1 such that ‖ x − z n ‖ → ‖ A x ‖ ‖ A ‖ x − z n → ‖ A x ‖ ‖ A ‖ ||x-z_(n)||rarr(||Ax||)/(||A||)\left\|x-z_{n}\right\| \rightarrow \frac{\|A x\|}{\|A\|}‖x−zn‖→‖Ax‖‖A‖, then, since ‖ x − z n ‖ ⩾ d ( x , Z 1 ) , n ∈ N x − z n ⩾ d x , Z 1 , n ∈ N ||x-z_(n)|| >= d(x,Z_(1)),n inN\left\|x-z_{n}\right\| \geqslant d\left(x, Z_{1}\right), n \in \mathbb{N}‖x−zn‖⩾d(x,Z1),n∈N, we get
d ( x , Z 1 ) ⩽ lim n → ∞ ‖ x − z n ‖ = ‖ A x ‖ ‖ A ‖ d x , Z 1 ⩽ lim n → ∞   x − z n = ‖ A x ‖ ‖ A ‖ d(x,Z_(1)) <= lim_(n rarr oo)||x-z_(n)||=(||Ax||)/(||A||)d\left(x, Z_{1}\right) \leqslant \lim _{n \rightarrow \infty}\left\|x-z_{n}\right\|=\frac{\|A x\|}{\|A\|}d(x,Z1)⩽limn→∞‖x−zn‖=‖Ax‖‖A‖
which, combined with (10), gives
d ( x , Z 1 ) = ‖ A x ‖ ‖ A ‖ d x , Z 1 = ‖ A x ‖ ‖ A ‖ d(x,Z_(1))=(||Ax||)/(||A||)d\left(x, Z_{1}\right)=\frac{\|A x\|}{\|A\|}d(x,Z1)=‖Ax‖‖A‖
(a) If x ∈ X 1 x ∈ X 1 x inX_(1)x \in X_{1}x∈X1 is such that d ( x , Z 1 ) = ‖ A x ‖ ‖ A ‖ d x , Z 1 = ‖ A x ‖ ‖ A ‖ d(x,Z_(1))=(||Ax||)/(||A||)d\left(x, Z_{1}\right)=\frac{\|A x\|}{\|A\|}d(x,Z1)=‖Ax‖‖A‖ then the following equivalences hold
z ∈ P Z 1 ( x ) ⟺ z ∈ Z 1 and ‖ x − z ‖ = d ( x , Z 1 ) = ‖ A x ‖ ‖ A ‖ z ∈ P Z 1 ( x ) ⟺ z ∈ Z 1  and  ‖ x − z ‖ = d x , Z 1 = ‖ A x ‖ ‖ A ‖ z inP_(Z_(1))(x)Longleftrightarrow z inZ_(1)" and "||x-z||=d(x,Z_(1))=(||Ax||)/(||A||)z \in P_{Z_{1}}(x) \Longleftrightarrow z \in Z_{1} \text { and }\|x-z\|=d\left(x, Z_{1}\right)=\frac{\|A x\|}{\|A\|}z∈PZ1(x)⟺z∈Z1 and ‖x−z‖=d(x,Z1)=‖Ax‖‖A‖
⟺ x − z ∈ E 1 ( x ) ⟺ z ∈ x − E 1 ( x ) . ⟺ x − z ∈ E 1 ( x ) ⟺ z ∈ x − E 1 ( x ) . Longleftrightarrow x-z inE_(1)(x)Longleftrightarrow z in x-E_(1)(x).\Longleftrightarrow x-z \in E_{1}(x) \Longleftrightarrow z \in x-E_{1}(x) .⟺x−z∈E1(x)⟺z∈x−E1(x).
Taking into account (11) one obtains the equivalences
z ∈ P Z ( x ) ⟺ z ∈ Z and ‖ x − z ‖ = d ( x , Z ) = ‖ A x ‖ ‖ A ‖ ⟺ x − z ∈ E ( x ) ⟺ z ∈ x − E ( x ) . z ∈ P Z ( x ) ⟺ z ∈ Z  and  ‖ x − z ‖ = d ( x , Z ) = ‖ A x ‖ ‖ A ‖ ⟺ x − z ∈ E ( x ) ⟺ z ∈ x − E ( x ) . {:[z inP_(Z)(x) Longleftrightarrow z in Z" and "||x-z||=d(x","Z)=(||Ax||)/(||A||)],[ Longleftrightarrow x-z in E(x)Longleftrightarrow z in x-E(x).]:}\begin{aligned} z \in P_{Z}(x) & \Longleftrightarrow z \in Z \text { and }\|x-z\|=d(x, Z)=\frac{\|A x\|}{\|A\|} \\ & \Longleftrightarrow x-z \in E(x) \Longleftrightarrow z \in x-E(x) . \end{aligned}z∈PZ(x)⟺z∈Z and ‖x−z‖=d(x,Z)=‖Ax‖‖A‖⟺x−z∈E(x)⟺z∈x−E(x).
(b) Let z 0 ∈ Z 1 z 0 ∈ Z 1 z_(0)inZ_(1)z_{0} \in Z_{1}z0∈Z1 be such that
‖ x − z 0 ‖ = ‖ A x ‖ ‖ A ‖ = d ( x , Z 1 ) = d ( x , Z ) x − z 0 = ‖ A x ‖ ‖ A ‖ = d x , Z 1 = d ( x , Z ) ||x-z_(0)||=(||Ax||)/(||A||)=d(x,Z_(1))=d(x,Z)\left\|x-z_{0}\right\|=\frac{\|A x\|}{\|A\|}=d\left(x, Z_{1}\right)=d(x, Z)‖x−z0‖=‖Ax‖‖A‖=d(x,Z1)=d(x,Z)
It follows that (12) holds for z n = z 0 , n = 1 , 2 , … z n = z 0 , n = 1 , 2 , … z_(n)=z_(0),n=1,2,dotsz_{n}=z_{0}, n=1,2, \ldotszn=z0,n=1,2,…, so that by the point 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘ of the theorem, (11), (13) and (14) hold.

Application

Let X X XXX be a real normed space and Y Y YYY a nonvoid convex subset of X X XXX containing 0 .
Consider the space
(16) Lip 0 Y = { f : Y → R : f Lip 0 Y = { f : Y → R : f quadLip_(0)Y={f:Y rarrR:f\quad \operatorname{Lip}_{0} Y=\{f: Y \rightarrow \mathbb{R}: fLip0Y={f:Y→R:f is a Lipschitz on Y Y YYY and f ( 0 ) = 0 } f ( 0 ) = 0 } f(0)=0}f(0)=0\}f(0)=0}
equipped with the Lipschitz norm
(17)
‖ f ‖ Y = sup { | f ( y 1 ) − f ( y 2 ) | ‖ y 1 − y 2 ‖ y 1 , y 2 ∈ Y , y 1 ≠ y 2 } . ‖ f ‖ Y = sup f y 1 − f y 2 y 1 − y 2 y 1 , y 2 ∈ Y , y 1 ≠ y 2 . ||f||_(Y)=s u p{(|f(y_(1))-f(y_(2))|)/(||y_(1)-y_(2)||)y_(1),y_(2)in Y,y_(1)!=y_(2)}.\|f\|_{Y}=\sup \left\{\frac{\left|f\left(y_{1}\right)-f\left(y_{2}\right)\right|}{\left\|y_{1}-y_{2}\right\|} y_{1}, y_{2} \in Y, y_{1} \neq y_{2}\right\} .‖f‖Y=sup{|f(y1)−f(y2)|‖y1−y2‖y1,y2∈Y,y1≠y2}.
The space L i p 0 X L i p 0 X Lip_(0)XL i p_{0} XLip0X and the Lipschitz norm ‖ ⋅ ‖ X ‖ ⋅ ‖ X ||*||_(X)\|\cdot\|_{X}‖⋅‖X are defined similarly.
By the theorem of McShane [7], [4], the space Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y has the extension property with respect to Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X, i.e., for every f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y there exists f ∈ f ∈ f inf \inf∈ Lip 0 X 0 X _(0)X_{0} X0X such that
F | Y = f and ‖ F ‖ X = ‖ f ‖ Y . F Y = f  and  ‖ F ‖ X = ‖ f ‖ Y . F|_(Y)=f" and "||F||_(X)=||f||_(Y).\left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y} .F|Y=f and ‖F‖X=‖f‖Y.
Definition 1. A function f ∈ f ∈ f inf \inf∈ Lip 0 Y 0 Y _(0)Y_{0} Y0Y is called convex if
(18) f ( α y 1 + ( 1 − α ) y 2 ) ≤ α f ( y 1 ) + ( 1 − α ) f ( y 2 ) (18) f α y 1 + ( 1 − α ) y 2 ≤ α f y 1 + ( 1 − α ) f y 2 {:(18)f(alphay_(1)+(1-alpha)y_(2)) <= alpha f(y_(1))+(1-alpha)f(y_(2)):}\begin{equation*} f\left(\alpha y_{1}+(1-\alpha) y_{2}\right) \leq \alpha f\left(y_{1}\right)+(1-\alpha) f\left(y_{2}\right) \tag{18} \end{equation*}(18)f(αy1+(1−α)y2)≤αf(y1)+(1−α)f(y2)
for all y 1 , y 2 ∈ Y y 1 , y 2 ∈ Y y_(1),y_(2)in Yy_{1}, y_{2} \in Yy1,y2∈Y and all α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1], and starshaped if
(19) f ( α y ) ≤ α f ( y ) (19) f ( α y ) ≤ α f ( y ) {:(19)f(alpha y) <= alpha f(y):}\begin{equation*} f(\alpha y) \leq \alpha f(y) \tag{19} \end{equation*}(19)f(αy)≤αf(y)
for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y and every α ∈ [ 0 , 1 ] α ∈ [ 0 , 1 ] alpha in[0,1]\alpha \in[0,1]α∈[0,1].
Obviously that every convex function f ∈ f ∈ f inf \inf∈ Lip 0 Y 0 Y _(0)Y{ }_{0} Y0Y is starshaped.
Definition 2. A subset C C CCC of a vector space X X XXX is called a convex cone if
(a) x + y ∈ C x + y ∈ C x+y in Cx+y \in Cx+y∈C for every x , y ∈ C x , y ∈ C x,y in Cx, y \in Cx,y∈C, and
(b) λ x ∈ C λ x ∈ C lambda x in C\lambda x \in Cλx∈C for every x ∈ C x ∈ C x in Cx \in Cx∈C and λ ≥ 0 λ ≥ 0 lambda >= 0\lambda \geq 0λ≥0.
Denoting by K Y K Y K_(Y)K_{Y}KY (respectively by S Y S Y S_(Y)S_{Y}SY ) the sets of all convex (respectively starshaped) functions in Lip 0 Y Lip 0 Y Lip_(0)Y\operatorname{Lip}_{0} YLip0Y, it follows that K Y K Y K_(Y)K_{Y}KY and S Y S Y S_(Y)S_{Y}SY are convex cones in L i p 0 Y L i p 0 Y Lip_(0)YL i p_{0} YLip0Y.
The sets of convex (starshaped) Lipschitz functions in Lip 0 X Lip 0 X Lip_(0)X\operatorname{Lip}_{0} XLip0X are denoted by K X K X K_(X)K_{X}KX (respectively by S X S X S_(X)S_{X}SX ). Again they are convex cones in L i p 0 X L i p 0 X Lip_(0)XL i p_{0} XLip0X.
By McShane's theorem ([7], [4]), for every f ∈ L i p 0 Y f ∈ L i p 0 Y f in Lip_(0)Yf \in L i p_{0} Yf∈Lip0Y the function is a norm preserving extensions of f f fff, i.e.,
(21)
F | Y = f and ‖ F ‖ X = ‖ f ‖ Y F Y = f  and  ‖ F ‖ X = ‖ f ‖ Y F|_(Y)=f quad" and "quad||F||_(X)=||f||_(Y)\left.F\right|_{Y}=f \quad \text { and } \quad\|F\|_{X}=\|f\|_{Y}F|Y=f and ‖F‖X=‖f‖Y
a) if f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY then the function F F FFF given by (20) belongs to K X K X K_(X)K_{X}KX, i.e., K Y K Y K_(Y)K_{Y}KY has the extension property with respect to K X K X K_(X)K_{X}KX;
b) if f ∈ S Y f ∈ S Y f inS_(Y)f \in S_{Y}f∈SY then F ∈ S X F ∈ S X F inS_(X)F \in S_{X}F∈SX, so that S Y S Y S_(Y)S_{Y}SY has the extension property with respect to S X S X S_(X)S_{X}SX, too.
Consider the subspaces
(22) X c = K X − K X (22) X c = K X − K X {:(22)X_(c)=K_(X)-K_(X):}\begin{equation*} X_{c}=K_{X}-K_{X} \tag{22} \end{equation*}(22)Xc=KX−KX
and
(23) X s = S X − S X (23) X s = S X − S X {:(23)X_(s)=S_(X)-S_(X):}\begin{equation*} X_{s}=S_{X}-S_{X} \tag{23} \end{equation*}(23)Xs=SX−SX
generated by the cones K X K X K_(X)K_{X}KX and S X S X S_(X)\mathrm{S}_{X}SX, respectively. We have X c ⊂ X s ⊂ Lip p 0 X X c ⊂ X s ⊂ Lip p 0 X X_(c)subX_(s)sub Lipp_(0)XX_{c} \subset X_{s} \subset \operatorname{Lip} p_{0} XXc⊂Xs⊂Lipp0X.
Take in Theorem 2, A A AAA to be the restriction operator r : Lip 0 X → Lip 0 Y r : Lip 0 X → Lip 0 Y r:Lip_(0)X rarrLip_(0)Yr: \operatorname{Lip}_{0} X \rightarrow \operatorname{Lip}_{0} Yr:Lip0X→Lip0Y defined by
(24) r ( F ) = F | Y , F ∈ L i p 0 X (24) r ( F ) = F Y , F ∈ L i p 0 X {:(24)r(F)=F|_(Y)","F in Lip_(0)X:}\begin{equation*} r(F)=\left.F\right|_{Y}, F \in L i p_{0} X \tag{24} \end{equation*}(24)r(F)=F|Y,F∈Lip0X
The operator r r rrr is linear, continuous, and, since
‖ r ( F ) ‖ Y = ‖ F | Y ‖ Y ≤ ‖ F ‖ X , ‖ r ( F ) ‖ Y = F Y Y ≤ ‖ F ‖ X , ||r(F)||_(Y)=||F|_(Y)||_(Y) <= ||F||_(X),\|r(F)\|_{Y}=\left\|\left.F\right|_{Y}\right\|_{Y} \leq\|F\|_{X},‖r(F)‖Y=‖F|Y‖Y≤‖F‖X,
it follows ‖ r ‖ ≤ 1 ‖ r ‖ ≤ 1 ||r|| <= 1\|r\| \leq 1‖r‖≤1.
For f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y put
(25) E ( f ) = { F ∈ L i p 0 X : F | Y = f and ‖ F ‖ X = ‖ f ‖ Y } . (25) E ( f ) = F ∈ L i p 0 X : F Y = f  and  ‖ F ‖ X = ‖ f ‖ Y . {:(25)E(f)={F in Lip_(0)X:F|_(Y)=f" and "||F||_(X)=||f||_(Y)}.:}\begin{equation*} E(f)=\left\{F \in L i p_{0} X:\left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y}\right\} . \tag{25} \end{equation*}(25)E(f)={F∈Lip0X:F|Y=f and ‖F‖X=‖f‖Y}.
By McShane's theorem [7] the set E ( f ) E ( f ) E(f)E(f)E(f) is nonempty for every f ∈ Lip 0 Y f ∈ Lip 0 Y f inLip_(0)Yf \in \operatorname{Lip}_{0} Yf∈Lip0Y.
If f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY then
(26) E c ( f ) = { F ∈ E ( f ) : F ∈ K X } ≠ ∅ (26) E c ( f ) = F ∈ E ( f ) : F ∈ K X ≠ ∅ {:(26)E_(c)(f)={F in E(f):F inK_(X)}!=O/:}\begin{equation*} E_{c}(f)=\left\{F \in E(f): F \in K_{X}\right\} \neq \varnothing \tag{26} \end{equation*}(26)Ec(f)={F∈E(f):F∈KX}≠∅
(27) E s ( f ) = { F ∈ E ( f ) : F ∈ S X } ≠ ∅ (27) E s ( f ) = F ∈ E ( f ) : F ∈ S X ≠ ∅ {:(27)E_(s)(f)={F in E(f):F inS_(X)}!=O/:}\begin{equation*} E_{s}(f)=\left\{F \in E(f): F \in S_{X}\right\} \neq \varnothing \tag{27} \end{equation*}(27)Es(f)={F∈E(f):F∈SX}≠∅
It follows that for f ∈ S Y f ∈ S Y f inS_(Y)f \in S_{Y}f∈SY
E s ( f ) ⊆ E ( f ) E s ( f ) ⊆ E ( f ) E_(s)(f)sube E(f)E_{s}(f) \subseteq E(f)Es(f)⊆E(f)
and for f ∈ K Y f ∈ K Y f inK_(Y)f \in K_{Y}f∈KY
E c ( f ) ⊆ E s ( f ) ⊆ E ( f ) E c ( f ) ⊆ E s ( f ) ⊆ E ( f ) E_(c)(f)subeE_(s)(f)sube E(f)E_{c}(f) \subseteq E_{s}(f) \subseteq E(f)Ec(f)⊆Es(f)⊆E(f)
Let
Y ⊥ = { F ∈ L i p 0 X : F | Y = 0 } Y c ⊥ = Y ⊥ ∩ X c Y ⊥ = F ∈ L i p 0 X : F Y = 0 Y c ⊥ = Y ⊥ ∩ X c {:[Y^(_|_)={F in Lip_(0)X:F|_(Y)=0}],[Y_(c)^(_|_)=Y^(_|_)nnX_(c)]:}\begin{gathered} Y^{\perp}=\left\{F \in L i p_{0} X:\left.F\right|_{Y}=0\right\} \\ Y_{c}^{\perp}=Y^{\perp} \cap X_{c} \end{gathered}Y⊥={F∈Lip0X:F|Y=0}Yc⊥=Y⊥∩Xc
and
Y s ⊥ = Y ⊥ ∩ X s Y s ⊥ = Y ⊥ ∩ X s Y_(s)^(_|_)=Y^(_|_)nnX_(s)Y_{s}^{\perp}=Y^{\perp} \cap X_{s}Ys⊥=Y⊥∩Xs
We have the following result
Proposition 1. (a) If F ∈ S X F ∈ S X F inS_(X)F \in S_{X}F∈SX then F | Y ∈ S Y , E s ( F | Y ) ≠ ∅ F Y ∈ S Y , E s F Y ≠ ∅ F|_(Y)inS_(Y),E_(s)(F|_(Y))!=O/\left.F\right|_{Y} \in S_{Y}, E_{s}\left(\left.F\right|_{Y}\right) \neq \varnothingF|Y∈SY,Es(F|Y)≠∅ and ‖ r | X s ‖ = 1 r X s = 1 ||r|_(X_(s))||=1\left\|\left.r\right|_{X_{s}}\right\|=1‖r|Xs‖=1. Furthermore
(28)
d ( F , Y s ⊥ ) = d ( F , Y ⊥ ) = ‖ F | Y ‖ d F , Y s ⊥ = d F , Y ⊥ = F Y d(F,Y_(s)^(_|_))=d(F,Y^(_|_))=||F|_(Y)||d\left(F, Y_{s}^{\perp}\right)=d\left(F, Y^{\perp}\right)=\left\|\left.F\right|_{Y}\right\|d(F,Ys⊥)=d(F,Y⊥)=‖F|Y‖
and
(29) F − E s ( F | Y ) = P Y s ⊥ ( F ) ⊆ P Y ⊥ ( F ) = F − E ( F | Y ) (29) F − E s F Y = P Y s ⊥ ( F ) ⊆ P Y ⊥ ( F ) = F − E F Y {:(29)F-E_(s)(F|_(Y))=P_(Y_(s)^(_|_))(F)subeP_(Y^(_|_))(F)=F-E(F|_(Y)):}\begin{equation*} F-E_{s}\left(\left.F\right|_{Y}\right)=P_{Y_{s}^{\perp}}(F) \subseteq P_{Y^{\perp}}(F)=F-E\left(\left.F\right|_{Y}\right) \tag{29} \end{equation*}(29)F−Es(F|Y)=PYs⊥(F)⊆PY⊥(F)=F−E(F|Y)
(b) If F ∈ K X F ∈ K X F inK_(X)F \in K_{X}F∈KX then F | Y ∈ K Y , E c ( F | Y ) ≠ ∅ F Y ∈ K Y , E c F Y ≠ ∅ F|_(Y)inK_(Y),E_(c)(F|_(Y))!=O/\left.F\right|_{Y} \in K_{Y}, E_{c}\left(\left.F\right|_{Y}\right) \neq \varnothingF|Y∈KY,Ec(F|Y)≠∅ and ‖ r | X c ‖ = 1 r X c = 1 ||r|_(X_(c))||=1\left\|\left.r\right|_{X_{c}}\right\|=1‖r|Xc‖=1. Furthermore
(30) F − E c ( F | Y ) = P Y c ⊥ ( F ) ⊆ P Y s ⊥ ( F ) = F − E s ( F | Y ) ⊆ P Y ⊥ ( F ) = F − − E ( f ) (30) F − E c F Y = P Y c ⊥ ( F ) ⊆ P Y s ⊥ ( F ) = F − E s F Y ⊆ P Y ⊥ ( F ) = F − − E ( f ) {:[(30)F-E_(c)(F|_(Y))=P_(Y_(c)^(_|_))(F)subeP_(Y_(s)^(_|_))(F)],[=F-E_(s)(F|_(Y))subeP_(Y^(_|_))(F)=F^(-)-E(f)]:}\begin{gather*} F-E_{c}\left(\left.F\right|_{Y}\right)=P_{Y_{c}^{\perp}}(F) \subseteq P_{Y_{s}^{\perp}}(F) \tag{30}\\ =F-E_{s}\left(\left.F\right|_{Y}\right) \subseteq P_{Y^{\perp}}(F)=F^{-}-E(f) \end{gather*}(30)F−Ec(F|Y)=PYc⊥(F)⊆PYs⊥(F)=F−Es(F|Y)⊆PY⊥(F)=F−−E(f)
Proof. (a) If F ∈ E s ( F | Y ) F ∈ E s F Y F inE_(s)(F|_(Y))F \in E_{s}\left(\left.F\right|_{Y}\right)F∈Es(F|Y) then G = F − F ¯ ∈ Y s ⊥ G = F − F ¯ ∈ Y s ⊥ G=F- bar(F)inY_(s)^(_|_)G=F-\bar{F} \in Y_{s}^{\perp}G=F−F¯∈Ys⊥ and
‖ F − G ‖ X = ‖ F ¯ ‖ X = ‖ F | Y ‖ Y = ‖ r ( F − G ) ‖ Y ≤ ‖ r | X s ‖ ‖ F − G ‖ Y ‖ F − G ‖ X = ‖ F ¯ ‖ X = F Y Y = ‖ r ( F − G ) ‖ Y ≤ r X s ‖ F − G ‖ Y ||F-G||_(X)=|| bar(F)||_(X)=||F|_(Y)||_(Y)=||r(F-G)||_(Y) <= ||r|_(X_(s))||||F-G||_(Y)\|F-G\|_{X}=\|\bar{F}\|_{X}=\left\|\left.F\right|_{Y}\right\|_{Y}=\|r(F-G)\|_{Y} \leq\left\|\left.r\right|_{X_{s}}\right\|\|F-G\|_{Y}‖F−G‖X=‖F¯‖X=‖F|Y‖Y=‖r(F−G)‖Y≤‖r|Xs‖‖F−G‖Y
implying ‖ r | X s ‖ = 1 r X s = 1 ||r|_(X_(s))||=1\left\|\left.r\right|_{X_{s}}\right\|=1‖r|Xs‖=1. By Theorem 2.(b), G ∈ P Y s ⊥ ( F ) G ∈ P Y s ⊥ ( F ) G inP_(Y_(s)^(_|_))(F)G \in P_{Y_{s}^{\perp}}(F)G∈PYs⊥(F) so that (28) and (29) hold. The proof of (b) is similar.

REFERENCES

[1] COBZAŞ, S., Some Applications of a Distance Formula to the Kernel of a Linear Operator (submitted)
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[10] MUSTĂTA, C., Extension of Bounded Lipschitz Functions and Some Related Problems of Best Approximation, "Babeş-Bolyai" Univ., Seminar on Func. Anal. and Numerical Methods Preprint, no. 4, pp. 93-99, 1981.
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Received September 21, 1999
"T. Popoviciu" Institute
of Numerical Analysis
С. P. 68,
str. Gh. Bilaşcu 37
3400 Cluj-Napoca, Romania
  1. 2000 AMS Subject Classification: 41A65.
    *Supported by the Ministry of Research and Technology (GR 4122/1999).
2000

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