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.
Authors
Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
C. Mustăţa, Some remarks concerning norm preserving extension and best approximation, Rev. Anal. Numer. Theor. Approx., 29 (2000) No. 2, pp. 173-180.
[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.
SOME REMARKS CONCERNING NORM PRESERVING. EXTENSIONS AND BEST APPROXIMATION*
COSTICĂ MUSTĂţA
Abstract
Let X,Y\mathrm{X}, \mathrm{Y} be two normed spaces, X_(1)X_{1} a subspace of XX and A:X rarr YA: X \rightarrow Y a continuous linear operator. Let us denote Z_(1)=Ker(A|_(X_(1))),Z=Ker AZ_{1}=\operatorname{Ker}\left(\left.A\right|_{X_{1}}\right), Z=\operatorname{Ker} A and for x in X,E(x)={y in X:Ax=Ayx \in X, E(x)= \{y \in X: A x=A y and ||y||=||Ax||//||A||}\|y\|=\|A x\| /\|A\|\} and 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. and ||y_(1)||=||Ax||//||A||}\left\|y_{1}\right\|= \|A x\| /\|A\|\}.
One gives the relations between the sets E(x),E_(1)(x)E(x), E_{1}(x) and P_(Z)(x),P_(Z_(1))(x)P_{Z}(x), P_{Z_{1}}(x) where P_(C)(x):={y in C:||x-y||=d(x,C)}P_{C}(x):=\{y \in C:\|x-y\|=d(x, C)\}. An application is considered.
Let XX be a real normed space and MM a nonvoid closed subset of XX. For x in Xx \in X let
d(x,M)=i n f{||x-y||:y in M}d(x, M)=\inf \{\|x-y\|: y \in M\}
be the distance from xx to MM and let
P_(M)(x):={y in M:||x-y||=d(x,M)}P_{M}(x):=\{y \in M:\|x-y\|=d(x, M)\}
be the set of nearest points from xx in the set MM.
If P_(M)(x)!=O/P_{M}(x) \neq \emptyset for every x in Xx \in X then the set MM is called proximinal, if P_(M)(x)P_{M}(x) is a singleton for every x in Xx \in X then MM is called chebyshevian and if P_(M)(x)=O/P_{M}(x)=\emptyset for every x in X\\Mx \in X \backslash M then the set MM is called antiproximinal.
be the annihilator of the subspace YY in the conjugate space X^(**)X^{*} of XX.
R.R. Phelps [13] studied the relation between the norm-preserving extension properties of the space Y^(**)Y^{*} with respect to X^(**)X^{*} and the best approximation properties of Y^(_|_)Y^{\perp}. Namely, he proved that every y^(**)inY^(**)y^{*} \in Y^{*} has a unique normpreserving extension x^(**)inX^(**)x^{*} \in X^{*} if and only if Y^(_|_)Y^{\perp} is a chebyshevian subspace of X^(**)X^{*}. By the Hahn-Banach extension theorem, every y^(**)inY^(**)y^{*} \in Y^{*} has at least
one norm-preserving extension x^(**)inX^(**)x^{*} \in 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,YX, Y and A:X rarr YA: X \rightarrow Y a continuous linear operator, let
{:(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*}
be the kernel of the operator AA. Obviously that ZZ is a closed subspace of XX.
For x in Xx \in X put
{:(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*}
Theorem 1. S. Cobzas, [1]. The following assertions hold: 1^(@)1^{\circ}
then z_(0)inP_(Z)(x)z_{0} \in P_{Z}(x) and (4) and (6) hold.
By specializing the spaces X,YX, Y and the operator AA, 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] ) 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,YX, Y be normed spaces over the same field K(R\mathbb{K}(\mathbb{R} or C)\mathbb{C}), and let A:X rarr YA: X \rightarrow Y be a continuous linear operator. For a subspace X_(1)X_{1} of XX let
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)
For x in Xx \in X let
{:(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*}
and
{:(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*}
Obviously, Z_(1)Z_{1} is a subspace of ZZ and E_(1)(x)sube E(x)E_{1}(x) \subseteq E(x), for x inX_(1)x \in X_{1}.
Theorem 2. 1^(@)1^{\circ}. For every x inX_(1)x \in X_{1} we have
for every x inX_(1)x \in X_{1}. 2^(@)2^{\circ} If d(x,Z_(1))=(||Ax||)/(||A||)d\left(x, Z_{1}\right)=\frac{\|A x\|}{\|A\|} then, by the definition of d(x,Z_(1))d\left(x, Z_{1}\right), there exists a sequence (z_(n))\left(z_{n}\right) in Z_(1)Z_{1} such that
Conversely, if (z_(n))\left(z_{n}\right) is a sequence in Z_(1)Z_{1} such that ||x-z_(n)||rarr(||Ax||)/(||A||)\left\|x-z_{n}\right\| \rightarrow \frac{\|A x\|}{\|A\|}, then, since ||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}, we get
(a) If x inX_(1)x \in X_{1} is such that d(x,Z_(1))=(||Ax||)/(||A||)d\left(x, Z_{1}\right)=\frac{\|A x\|}{\|A\|} then the following equivalences hold
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\|}
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) .
Taking into account (11) one obtains the equivalences
{:[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}
It follows that (12) holds for z_(n)=z_(0),n=1,2,dotsz_{n}=z_{0}, n=1,2, \ldots, so that by the point 2^(@)2^{\circ} of the theorem, (11), (13) and (14) hold.
Application
Let XX be a real normed space and YY a nonvoid convex subset of XX containing 0 .
Consider the space
(16) quadLip_(0)Y={f:Y rarrR:f\quad \operatorname{Lip}_{0} Y=\{f: Y \rightarrow \mathbb{R}: f is a Lipschitz on YY and f(0)=0}f(0)=0\}
equipped with the Lipschitz norm
(17)
||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\} .
The space Lip_(0)XL i p_{0} X and the Lipschitz norm ||*||_(X)\|\cdot\|_{X} are defined similarly.
By the theorem of McShane [7], [4], the space Lip_(0)Y\operatorname{Lip}_{0} Y has the extension property with respect to Lip_(0)X\operatorname{Lip}_{0} X, i.e., for every f inLip_(0)Yf \in \operatorname{Lip}_{0} Y there exists f inf \in Lip _(0)X_{0} X such that
F|_(Y)=f" and "||F||_(X)=||f||_(Y).\left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y} .
Definition 1. A function f inf \in Lip _(0)Y_{0} Y is called convex if
for every y in Yy \in Y and every alpha in[0,1]\alpha \in[0,1].
Obviously that every convex function f inf \in Lip _(0)Y{ }_{0} Y is starshaped.
Definition 2. A subset CC of a vector space XX is called a convex cone if
(a) x+y in Cx+y \in C for every x,y in Cx, y \in C, and
(b) lambda x in C\lambda x \in C for every x in Cx \in C and lambda >= 0\lambda \geq 0.
Denoting by K_(Y)K_{Y} (respectively by S_(Y)S_{Y} ) the sets of all convex (respectively starshaped) functions in Lip_(0)Y\operatorname{Lip}_{0} Y, it follows that K_(Y)K_{Y} and S_(Y)S_{Y} are convex cones in Lip_(0)YL i p_{0} Y.
The sets of convex (starshaped) Lipschitz functions in Lip_(0)X\operatorname{Lip}_{0} X are denoted by K_(X)K_{X} (respectively by S_(X)S_{X} ). Again they are convex cones in Lip_(0)XL i p_{0} X.
By McShane's theorem ([7], [4]), for every f in Lip_(0)Yf \in L i p_{0} Y the function is a norm preserving extensions of ff, i.e.,
(21)
F|_(Y)=f quad" and "quad||F||_(X)=||f||_(Y)\left.F\right|_{Y}=f \quad \text { and } \quad\|F\|_{X}=\|f\|_{Y}
a) if f inK_(Y)f \in K_{Y} then the function FF given by (20) belongs to K_(X)K_{X}, i.e., K_(Y)K_{Y} has the extension property with respect to K_(X)K_{X};
b) if f inS_(Y)f \in S_{Y} then F inS_(X)F \in S_{X}, so that S_(Y)S_{Y} has the extension property with respect to S_(X)S_{X}, too.
generated by the cones K_(X)K_{X} and S_(X)\mathrm{S}_{X}, respectively. We have X_(c)subX_(s)sub Lipp_(0)XX_{c} \subset X_{s} \subset \operatorname{Lip} p_{0} X.
Take in Theorem 2, AA to be the restriction operator r:Lip_(0)X rarrLip_(0)Yr: \operatorname{Lip}_{0} X \rightarrow \operatorname{Lip}_{0} Y defined by
{:(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*}
it follows ||r|| <= 1\|r\| \leq 1.
For f inLip_(0)Yf \in \operatorname{Lip}_{0} Y put
{:(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*}
By McShane's theorem [7] the set E(f)E(f) is nonempty for every f inLip_(0)Yf \in \operatorname{Lip}_{0} Y.
If f inK_(Y)f \in K_{Y} then
{:(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*}
{:(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*}
{:[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}
We have the following result
Proposition 1. (a) If F inS_(X)F \in S_{X} then 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 \varnothing and ||r|_(X_(s))||=1\left\|\left.r\right|_{X_{s}}\right\|=1. Furthermore
(28)
(b) If F inK_(X)F \in K_{X} then 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 \varnothing and ||r|_(X_(c))||=1\left\|\left.r\right|_{X_{c}}\right\|=1. Furthermore
implying ||r|_(X_(s))||=1\left\|\left.r\right|_{X_{s}}\right\|=1. By Theorem 2.(b), G inP_(Y_(s)^(_|_))(F)G \in P_{Y_{s}^{\perp}}(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)
[2] COBZAŞ, S., MUSTĂTA, C., Norm Preserving Extension of Convex Lipschitz Functions, J. Approx. Theory, 24, pp. 236-244, 1978.
[3] COBZAŞ, S., MUSTÃTA, C., Extension of Bilinear Functionals and Best Approximation in 2-normed Spaces, Studia Univ. "Babeş-Bolyai", Series Math., XLIII, no. 2, pp. 1-13, 1998.
[4] CZIPSER, J., GÉHÉR, 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. Nošiaeka Eds., Appromation and Optimization, vol. 3, Verlag Peter Lang, pp. 107-120, Frankfurt 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, 1934.
[8] MUSTATA, C., Best Approximation and Unique Extension of Lipschitz Functions, J. Approx. Theory, 19, pp. 222-230, 1977.
[9] MUSTĂTA, C., Norm Preserving Extension of Starshaped Lipschitz Functions, Mathematica, 19, pp. 183-187, 1977.
[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.
[11] MUSTĂTA, C., Extension of Holder Functions and Some Related Problems of Best Approximation, Seminar on Mathematic:al Analysis "Babeş-Bolyai" Univ. Preprint, no. 7, pp. 71-86, 1991.
[12] SUNG-HO, PARK, Quotient Mappings, Helly Extensions, Hahn-Banach Extension, Tietze Extensions, Lipschitz Extension and Best Approximation, J. Koreean Math. Soc., 29, no. 2, pp. 239-250, 1992.
[13] PHELPS, R. R., Uniqueness of Hahn-Banach Extensions and Best Approximations, Trans. Amer. Math. Soc., 95, pp. 238-255, 1960.
Received September 21, 1999
"T. Popoviciu" Institute
of Numerical Analysis
С. P. 68,
str. Gh. Bilaşcu 37
3400 Cluj-Napoca, Romania
2000 AMS Subject Classification: 41A65.
*Supported by the Ministry of Research and Technology (GR 4122/1999).