Abstract
A well known result of R. R. Phelps (1960) asserts that in order that every linear continuous functional, defined on a subspace \(Y\) of a real normed space \(X\), have a unique norm preserving extension it is necessary and sufficient that its annihilator \(Y^{\perp}\) be a Chebyshevian subspace of \(X\ast\). The aim of this note is to show that this result holds also in the case of spaces with asymmetric norm.
Authors
Costica Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academ, Romania
Keywords
Asymmetric normed spaces; extensions preserving asymmetric norm; best approximation
Paper coordinates
C. Mustăţa, On the uniqueness of extension and unique best approximation in the dual of an asymmetric normed linear space, Rev. Anal. Numer. Theor. Approx. 32 (2003) no. 2, 187-192.
About this paper
Journal
Revue d’Analyse Numerique et de Theorie de l’Approximation
Publisher Name
Publising Romanian Academy
DOI
Print ISSN
2457-6794
Online ISSN
2501-059X
google scholar link
[1] Alegre, C., Ferrer, J.andGregori, V.,On the Hahn–Banach theorem in certainlinear quasi-uniform structures,Acta Math. Hungar,82, pp. 315–320, 1999.
[2] Borodin, P. A.,The Banach–Mazur Theorem for spaces with asymmetric norm andits applications in convex analysis, Mathematical Notes,69, no. 3, pp. 298–305, 2001.
[3] Cobzas S.,Phelps type duality results in best approximation,Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.
[4] Dolzhenko, E. P.andSevast’yanov, E. A.,Approximation with sign-sensitiveweights, Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 62, no. 6,pp. 59–102, 1998 and63, no. 3, pp. 77–118, 1999.
[5] Ferrer, J., Gregori, V.andAlegre, C.,Quasi-uniform structures in linear lattices,Rocky Mountain J. Math.,23, pp. 877–884, 1993.
[6] Garcia-Raffi, L. M., Romaguera, S.and Sanchez Perez, E. A.,Extension ofasymmetric norms to linear spaces, Rend. Istit. Mat. Trieste,XXXIII, pp. 113–125,2001.
[7] Garcia-Raffi, L. M., Romaguera S.and Sanchez-Perez, E. A.,The dual space ofan asymmetric normed linear space,Quaestiones Mathematicae,26, pp. 83–96, 2003.
[8] Krein, M. G.and Nudel’man, A. A.,The Markov Moment Problem and ExtremumProblems, Nauka, Moscow, 1973 (in Russian).
[9] Kopperman, R. D.,All topologies come from generalized metrics, Amer. Math.Monthly,95, pp. 89–97, 1988.
[10] McShane, E. J.,Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 847–842, 1934.
[11] Mustata, C.,Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[12] Mustata, C.,Extensions of convex semi-Lipschitz functions on quasi-metric linearspaces, Seminaire de la Theorie de la Meileure Approximation Convexite et Optimiza-tion, Cluj-Napoca, le 29 novembre, pp. 85–92, 2001.
[13] Mustata, C.,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint.Univ. Baia-Mare, Ser. B,XVIII, no. 2, pp. 275–280, 2002.
[14] Phelps, R. R.,Uniqueness of Hahn–Banach extension and unique best approximation,Trans. Amer. Math. Soc.,95, pp. 238–255, 1960.
[15] Romaguera, S.and Sanchis, M.,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[16] Singer, I.,Best approximation in normed linear spaces by elements of linear subspaces,Ed. Acad. RSR, Bucharest, 1967 (in Romanian); English Translation, Springer, Berlin,1970
[2] Borodin, P. A.,The Banach–Mazur Theorem for spaces with asymmetric norm andits applications in convex analysis, Mathematical Notes,69, no. 3, pp. 298–305, 2001.
[3] Cobzas S.,Phelps type duality results in best approximation,Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.
[4] Dolzhenko, E. P.andSevast’yanov, E. A.,Approximation with sign-sensitiveweights, Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 62, no. 6,pp. 59–102, 1998 and63, no. 3, pp. 77–118, 1999.
[5] Ferrer, J., Gregori, V.andAlegre, C.,Quasi-uniform structures in linear lattices,Rocky Mountain J. Math.,23, pp. 877–884, 1993.
[6] Garcia-Raffi, L. M., Romaguera, S.and Sanchez Perez, E. A.,Extension ofasymmetric norms to linear spaces, Rend. Istit. Mat. Trieste,XXXIII, pp. 113–125,2001.
[7] Garcia-Raffi, L. M., Romaguera S.and Sanchez-Perez, E. A.,The dual space ofan asymmetric normed linear space,Quaestiones Mathematicae,26, pp. 83–96, 2003.
[8] Krein, M. G.and Nudel’man, A. A.,The Markov Moment Problem and ExtremumProblems, Nauka, Moscow, 1973 (in Russian).
[9] Kopperman, R. D.,All topologies come from generalized metrics, Amer. Math.Monthly,95, pp. 89–97, 1988.
[10] McShane, E. J.,Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 847–842, 1934.
[11] Mustata, C.,Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[12] Mustata, C.,Extensions of convex semi-Lipschitz functions on quasi-metric linearspaces, Seminaire de la Theorie de la Meileure Approximation Convexite et Optimiza-tion, Cluj-Napoca, le 29 novembre, pp. 85–92, 2001.
[13] Mustata, C.,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint.Univ. Baia-Mare, Ser. B,XVIII, no. 2, pp. 275–280, 2002.
[14] Phelps, R. R.,Uniqueness of Hahn–Banach extension and unique best approximation,Trans. Amer. Math. Soc.,95, pp. 238–255, 1960.
[15] Romaguera, S.and Sanchis, M.,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[16] Singer, I.,Best approximation in normed linear spaces by elements of linear subspaces,Ed. Acad. RSR, Bucharest, 1967 (in Romanian); English Translation, Springer, Berlin,1970
Paper (preprint) in HTML form
ON THE UNIQUENESS OF EXTENSION AND UNIQUE BEST APPROXIMATION IN THE DUAL OF AN ASYMMETRIC NORMED LINEAR SPACE
Abstract
A well known result of R. R. Phelps (1960) asserts that in order that every linear continuous functional, defined on a subspace YY of a real normed space XX, have a unique norm preserving extension it is necessary and sufficient that its annihilator Y^(_|_)Y^{\perp} be a Chebyshevian subspace of X^(**)X^{*}. The aim of this note is to show that this result holds also in the case of spaces with asymmetric norm.
MSC 2000. 41A65, 46A22, 46B20.
Keywords. Asymmetric normed spaces, extensions preserving asymmetric norm, best approximation.
Keywords. Asymmetric normed spaces, extensions preserving asymmetric norm, best approximation.
1. INTRODUCTION
Let XX be a real linear space. A function ||*∣:X rarr[0,oo)\| \cdot \mid: X \rightarrow[0, \infty) is called an asymmetric norm if it satisfies all the usual axioms of a norm, excepting the absolute homogeneity, which is replaced by positive homogeneity, i.e.,
||lambda x|=lambda||x|,quad AA x in X,AA lambda >= 0.\|\lambda x|=\lambda \| x|, \quad \forall x \in X, \forall \lambda \geq 0 .
The asymmetric norm is called with extended values if there exists x in Xx \in X such that ||x∣=+oo\| x \mid=+\infty. The pair ( X,||*||X,\|\cdot\| ) is called a space with asymmetric norm (see [8], [2]).
In a space with asymmetric norm it is possible that ||x||!=||-x∣\|x\| \neq \|-x \mid for some x in Xx \in X. The asymmetric norm generates a topology having as a neighborhood base the balls B(x,r)={y in X:||y-x∣<r},x in X,r >= 0B(x, r)=\{y \in X: \| y-x \mid<r\}, x \in X, r \geq 0, but the topological space (X,tau_(||*∣))\left(X, \tau_{\| \cdot \mid}\right) is not a linear topological space, because the multiplication by scalars is not a continuous operation (see [2, p. 199]).
Let X^(#)X^{\#} be the algebraic dual of XX, i.e. the space of all linear functional on XX. We say that f inX^(#)f \in X^{\#} is bounded on XX if
{:(1)s u p{f(x):x in X","quad||x∣≤1} < oo:}\begin{equation*}
\sup \{f(x): x \in X, \quad \| x \mid \leq 1\}<\infty \tag{1}
\end{equation*}
It is immediate that if f,g inX^(#)f, g \in X^{\#} are bounded then their sum f+gf+g and the product lambda f\lambda f for lambda >= 0\lambda \geq 0 are bounded too. This shows that the set of all bounded linear functionals on XX is a cone (see [2]), or an ac-space, according to [7].
In general, it is possible that for a bounded linear functional ff on ( X,||*||X,\|\cdot\| ) the linear functional -f-f be not bounded. Such an example is given by the functional f(x)=x(1)f(x)=x(1) defined on the space
X={x:[0,1]rarrR∣x" continuous and "int_(0)^(1)x(t)dt=0}X=\left\{x:[0,1] \rightarrow \mathbb{R} \mid x \text { continuous and } \int_{0}^{1} x(t) \mathrm{d} t=0\right\}
equipped with the asymmetric norm
||x∣=max{x(t):t in[0,1]},x in X.\| x \mid=\max \{x(t): t \in[0,1]\}, x \in X .
Taking x_(n)(t)=1-nt^(n-1)x_{n}(t)=1-n t^{n-1} it follows ||x_(n)||=1\left\|x_{n}\right\|=1 and (-f)(x_(n))=f(-x_(n))=(-x_(n))(1)=n-1(-f)\left(x_{n}\right)=f\left(-x_{n}\right)= \left(-x_{n}\right)(1)=n-1, implying
s u p{(-f)(x):||x∣≤1} >= s u p{(-f)(x_(n)):n inN}=+oo.\sup \{(-f)(x): \| x \mid \leq 1\} \geq \sup \left\{(-f)\left(x_{n}\right): n \in \mathbb{N}\right\}=+\infty .
For a bounded linear functional ff put
{:(2)||f∣=s u p{f(x):x in X","||x∣≤1}:}\begin{equation*}
\| f \mid=\sup \{f(x): x \in X, \| x \mid \leq 1\} \tag{2}
\end{equation*}
It follows that the function ||*∣\| \cdot \mid defined by (2) satisfies the axioms of an asymmetric norm on the cone of all bounded linear functionals on XX (see [2]).
Observe that
f(x) <= ||f|*||x|,quad x in Xf(x) \leq\|f|\cdot \| x|, \quad x \in X
and, since f(-x) <= ||f|*||-x|f(-x) \leq\|f|\cdot \|-x|, one obtains the inequalities
-||f|*||-x| <= f(x) <= ||f|*||x|,quad x in X.-\|f|\cdot\|-x|\leq f(x) \leq\|f|\cdot \| x|, \quad x \in X .
The inequality |f(x)| <= ||f|*||x||f(x)| \leq\|f|\cdot \| x| is not true in general, but if we consider the symmetric norm ||x||=max{||x|,||-x|}\|x\|=\max \{\|x|, \|-x|\} on XX, then |f(x)| <= ||f∣||x||,x in X|f(x)| \leq\|f \mid\| x \|, x \in X. This shows that a bounded linear functional is always continuous with respect to the topology generated by the symmetric norm ||x||=max{||x|,||-x|}\|x\|=\max \{\|x|, \|-x|\} associated to an asymmetric norm ||*∣\| \cdot \mid.
If both ff and -f-f are bounded then the linear functional ff is continuous with respect to the topology generated by the asymmetric norm.
Consider on R\mathbb{R} the asymmetric norm u(x)=x vv0=max{x,0}u(x)=x \vee 0=\max \{x, 0\} (see [7]). A functional f:(X,||*||)rarr(R,u)f:(X,\|\cdot\|) \rightarrow(\mathbb{R}, u) is continuous in x_(0)in Xx_{0} \in X if for every epsi > 0\varepsilon>0 there exists delta > 0\delta>0 such that for every x in Xx \in X with ||x-x_(0)∣<delta\| x-x_{0} \mid<\delta we have (f(x)-f(x_(0)))vv0 < epsi\left(f(x)-f\left(x_{0}\right)\right) \vee 0<\varepsilon.
It is clear that a linear functional f:(X,||*||)rarr(R,u)f:(X,\|\cdot\|) \rightarrow(\mathbb{R}, u) is continuous if and only if there exists M > 0M>0 such that f(x)vv0 < M||x∣f(x) \vee 0<M \| x \mid.
According to [7], the set
{:(3)X^(**)={f:(X","||*||)rarr(R","u)","f" linear continuous "}:}\begin{equation*}
X^{*}=\{f:(X,\|\cdot\|) \rightarrow(\mathbb{R}, u), f \text { linear continuous }\} \tag{3}
\end{equation*}
is called the asymmetric dual of the space with asymmetric norm ( X,||*||X,\|\cdot\| ).
If ||x||=max{||x|,||-x|}\|x\|=\max \{\|x|, \|-x|\} is the norm generated by ||*∣\| \cdot \mid on XX and R\mathbb{R} is equipped with the usual absolute-value norm |*||\cdot|, then the set
If ||x||=max{||x|,||-x|}\|x\|=\max \{\|x|, \|-x|\} is the norm generated by ||*∣\| \cdot \mid on XX and R\mathbb{R} is equipped with the usual absolute-value norm |*||\cdot|, then the set
{:(4)X^(**s)={f:(X","||*||)rarr(R","|*|)","f" linear continuous "}:}\begin{equation*}
X^{* s}=\{f:(X,\|\cdot\|) \rightarrow(\mathbb{R},|\cdot|), f \text { linear continuous }\} \tag{4}
\end{equation*}
is the (symmetric) dual of ( X,||*||X,\|\cdot\| ). In other words, X^(**s)X^{* s} is the usual topological algebraic dual of the normed space (X,||*||)(X,\|\cdot\|), where ||x||=max{||x|,||-x|}\|x\|=\max \{\|x|, \|-x|\}.
Observe that X^(**)X^{*} is a cone in the linear space X^(**s)X^{* s}.
Equip the linear space X^(**s)X^{* s} with the extended asymmetric norm
Equip the linear space X^(**s)X^{* s} with the extended asymmetric norm
||f|^(**s)=s u p{f(x):||x∣≤1}\|\left. f\right|^{* s}=\sup \{f(x): \| x \mid \leq 1\}
whose restriction to the asymmetric dual X^(**)X^{*} is
||f|^(**)=s u p{f(x)vv0:||x∣≤1}.\|\left. f\right|^{*}=\sup \{f(x) \vee 0: \| x \mid \leq 1\} .
It is important to remark the fact that a linear functional ff belongs to X^(**)X^{*} if and only if it is an upper semicontinuous linear functional on ( X,||*||X,\|\cdot\| ), and that X^(**)={f inX^(**s):||f|^(**s) < +oo}X^{*}=\left\{f \in X^{* s}: \|\left. f\right|^{* s}<+\infty\right\} (see [7]).
Let (Y,||*||)(Y,\|\cdot\|) be a subspace of the space with asymmetric norm ( X,||*||X,\|\cdot\| ), and let Y^(**)Y^{*} and Y^(**s)Y^{* s} be the dual cone and the (symmetric) dual of YY, respectively.
The following Hahn-Banach type theorem holds:
Theorem 1. Let ( X,||*||X,\|\cdot\| ) be a real space with asymmetric norm and ( Y,||*||Y,\|\cdot\| ) a subspace of it. Then for every f inY^(**)f \in Y^{*} there exists F inX^(**)F \in X^{*} such that
Theorem 1. Let ( X,||*||X,\|\cdot\| ) be a real space with asymmetric norm and ( Y,||*||Y,\|\cdot\| ) a subspace of it. Then for every f inY^(**)f \in Y^{*} there exists F inX^(**)F \in X^{*} such that
F|_(Y)=f quad" and " quad||F|^(**)=||f|^(**).\left.F\right|_{Y}=f \quad \text { and }\left.\quad\left\|\left.F\right|^{*}=\right\| f\right|^{*} .
Proof. For f inY^(**)f \in Y^{*} let p:X rarrRp: X \rightarrow \mathbb{R} be defined by p(x)=||f|^(**)*||x∣,x in Xp(x)=\left\|\left.f\right|^{*} \cdot\right\| x \mid, x \in X.
The functional pp is convex, positively homogeneous and f(y) <= ||f|^(**)*||y∣f(y) \leq\left\|\left.f\right|^{*} \cdot\right\| y \mid, for every y in Yy \in Y. By the Hahn-Banach extension theorem ([8, p. 484]) there exists a linear functional F:X rarrRF: X \rightarrow \mathbb{R} such that F|_(Y)=f\left.F\right|_{Y}=f and F(x) <= ||f|^(**)||x∣F(x) \leq\left\|\left.f\right|^{*}\right\| x \mid, for every x in Xx \in X. It follows
The functional pp is convex, positively homogeneous and f(y) <= ||f|^(**)*||y∣f(y) \leq\left\|\left.f\right|^{*} \cdot\right\| y \mid, for every y in Yy \in Y. By the Hahn-Banach extension theorem ([8, p. 484]) there exists a linear functional F:X rarrRF: X \rightarrow \mathbb{R} such that F|_(Y)=f\left.F\right|_{Y}=f and F(x) <= ||f|^(**)||x∣F(x) \leq\left\|\left.f\right|^{*}\right\| x \mid, for every x in Xx \in X. It follows
||F|^(**)=s u p{F(x)vv0:||x∣≤1} <= ||f|^(**).\left.\left\|\left.F\right|^{*}=\sup \{F(x) \vee 0: \| x \mid \leq 1\} \leq\right\| f\right|^{*} .
On the other hand
{:[||F|^(**)=s u p{F(x)vv0:||x∣≤1","x in X}],[ >= s u p{F(y)vv0:||y∣≤1","y in Y}],[=s u p{f(y)vv0:||y∣≤1","y in Y}],[=||f|^(**)","]:}\begin{aligned}
\|\left. F\right|^{*} & =\sup \{F(x) \vee 0: \| x \mid \leq 1, x \in X\} \\
& \geq \sup \{F(y) \vee 0: \| y \mid \leq 1, y \in Y\} \\
& =\sup \{f(y) \vee 0: \| y \mid \leq 1, y \in Y\} \\
& =\|\left. f\right|^{*},
\end{aligned}
showing that ||F||^(**)=||f|^(**)\|F\|^{*}=\|\left. f\right|^{*}.
For f inY^(**)f \in Y^{*} denote by
E(f)={F inX^(**):F|_(Y)=f" and "||F|^(**)=||f|^(**)}\mathcal{E}(f)=\left\{F \in X^{*}:\left.F\right|_{Y}=f \text { and }\left.\left\|\left.F\right|^{*}=\right\| f\right|^{*}\right\}
the set of all extensions that preserve the asymmetric norm.
By Theorem 1, the set E(f)\mathcal{E}(f) is always nonempty.
The problem of finding necessary or/and sufficient conditions in order that every f inY^(**)f \in Y^{*} have a unique norm preserving extension is closely related to a best approximation problem in the space X^(**s)X^{* s} equipped with the asymmetric norm ||F|^(**s)=s u p{F(x):||x∣≤1}\|\left. F\right|^{* s}=\sup \{F(x): \| x \mid \leq 1\}.
By Theorem 1, the set E(f)\mathcal{E}(f) is always nonempty.
The problem of finding necessary or/and sufficient conditions in order that every f inY^(**)f \in Y^{*} have a unique norm preserving extension is closely related to a best approximation problem in the space X^(**s)X^{* s} equipped with the asymmetric norm ||F|^(**s)=s u p{F(x):||x∣≤1}\|\left. F\right|^{* s}=\sup \{F(x): \| x \mid \leq 1\}.
Concerning the following notions, in the case of usual spaces, one can consult Singer's book [16].
Let (Y,||*||)(Y,\|\cdot\|) be subspace of (X,||*||)(X,\|\cdot\|) and let
Y^(_|_)={G inX^(**s):G|_(Y)=0}Y^{\perp}=\left\{G \in X^{* s}:\left.G\right|_{Y}=0\right\}
the annihilator of YY in the space ( X^(**s),||*|^(**s)X^{* s}, \|\left.\cdot\right|^{* s} ).
For F inX^(**s)F \in X^{* s}, an element G_(0)inY^(_|_)G_{0} \in Y^{\perp} is called a best approximation element for FF in Y^(_|_)Y^{\perp} if
For F inX^(**s)F \in X^{* s}, an element G_(0)inY^(_|_)G_{0} \in Y^{\perp} is called a best approximation element for FF in Y^(_|_)Y^{\perp} if
||F-G_(0)||^(**s)=i n f{||F-g|^(**s):G inY^(_|_)}=d(F,Y^(_|_))\left\|F-G_{0}\right\|^{* s}=\inf \left\{\| F-\left.g\right|^{* s}: G \in Y^{\perp}\right\}=d\left(F, Y^{\perp}\right)
The set of all best approximation elements for FF in Y^(_|_)Y^{\perp} is denoted by P_(Y^(_|_))(F)P_{Y^{\perp}}(F). If P_(Y^(_|_))(F)!=O/P_{Y^{\perp}}(F) \neq \emptyset for every F inX^(**s)F \in X^{* s} one says that Y^(_|_)Y^{\perp} is proximinal, and if cardP_(Y^(_|_))(F)=1\operatorname{card} P_{Y^{\perp}}(F)=1 for every F inX^(**s)F \in X^{* s}, then one says that Y^(_|_)Y^{\perp} is Chebyshevian.
Theorem 2. Let ( X,||*||X,\|\cdot\| ) be a space with asymmetric norm and ( Y,||*||Y,\|\cdot\| ) a subspace of it. Then
a) The annihilator Y^(_|_)Y^{\perp} of YY is a proximinal subspace of X^(**s)X^{* s} and, for every F inX^(**)F \in X^{*} we have
a) The annihilator Y^(_|_)Y^{\perp} of YY is a proximinal subspace of X^(**s)X^{* s} and, for every F inX^(**)F \in X^{*} we have
d(F,Y^(_|_))=||F|_(Y)|^(**)d\left(F, Y^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|^{*}
b) An element G inY^(_|_)G \in Y^{\perp} is in P_(Y^(_|_))(F)P_{Y^{\perp}}(F) if and only if G=F-HG=F-H for some H inE(F|_(Y))H \in \mathcal{E}\left(\left.F\right|_{Y}\right), i.e.,
P_(Y^(_|_))(F)=F-E(F|_(Y)).P_{Y^{\perp}}(F)=F-\mathcal{E}\left(\left.F\right|_{Y}\right) .
c) The subspace Y^(_|_)Y^{\perp} is Chebyshevian in X^(**s)X^{* s} if and only if every functional f inY^(**)f \in Y^{*} has a unique norm preserving extension in X^(**s)X^{* s}.
Proof. a) Let F inX^(**)F \in X^{*}. Then F|_(Y)inY^(**)\left.F\right|_{Y} \in Y^{*} and, by Theorem 1, E(F|_(Y))!=O/\mathcal{E}\left(\left.F\right|_{Y}\right) \neq \emptyset. If H inE(F|_(Y))H \in \mathcal{E}\left(\left.F\right|_{Y}\right) then
F|_(Y)=H|_(Y)quad" and " quad||F|_(Y)|^(**)=||H|^(**),\left.F\right|_{Y}=\left.H\right|_{Y} \quad \text { and }\left.\quad\left\|\left.\left.F\right|_{Y}\right|^{*}=\right\| H\right|^{*},
so that F-H inY^(_|_)F-H \in Y^{\perp}. We have
||F|_(Y)|^(**)=||H|^(**)=||H|^(**s)=||F-(F-H)|^(**s) >= d(F,Y^(_|_))\left.\left\|\left.\left.F\right|_{Y}\right|^{*}=\right\| H\right|^{*}=\left\|\left.H\right|^{* s}=\right\| F-\left.(F-H)\right|^{* s} \geq d\left(F, Y^{\perp}\right)
and, for any G inY^(_|_)G \in Y^{\perp},
||F|_(Y)|^(**)=||F|_(Y)-G|_(Y)|^(**) <= ||F-G|^(**s).\left.\left\|\left.\left.F\right|_{Y}\right|^{*}=\right\| F\right|_{Y}-\left.\left.G\right|_{Y}\right|^{*} \leq \| F-\left.G\right|^{* s} .
Taking the infimum with respect to G inY^(_|_)G \in Y^{\perp} we get
||F|_(Y)|^(**) <= d(F,Y^(_|_))\|\left.\left. F\right|_{Y}\right|^{*} \leq d\left(F, Y^{\perp}\right)
It follows that d(F,Y^(_|_))=||F|_(Y)|^(**)d\left(F, Y^{\perp}\right)=\|\left.\left. F\right|_{Y}\right|^{*}.
b) If G inP_(Y^(_|_))(F)G \in P_{Y^{\perp}}(F) then
b) If G inP_(Y^(_|_))(F)G \in P_{Y^{\perp}}(F) then
||F-G|^(**s)=||F|_(Y)|^(**)\left.\left.\left\|F-\left.G\right|^{* s}=\right\| F\right|_{Y}\right|^{*}
and
(F-G)|_(Y)=F|_(Y)\left.(F-G)\right|_{Y}=\left.F\right|_{Y}
which shows that F-G inE(F|_(Y))F-G \in \mathcal{E}\left(\left.F\right|_{Y}\right). The conclusion holds with H=F-GH=F-G.
c) Follows from b).
c) Follows from b).
Remark. 1^(@)1^{\circ} Let X^(**)X^{*} be the usual topological algebraic dual of the normed space ( X,||*||X,\|\cdot\| ), and Y^(**)Y^{*} the topological algebraic dual of ( Y,||*||Y,\|\cdot\| ), where YY is a subspace of XX. R. R. Phelps [14] showed that Y^(_|_)Y^{\perp} (the annihilator of YY in {:X^(**))\left.X^{*}\right) is Chebyshevian if and only if every f inY^(**)f \in Y^{*} has a unique norm-preserving extension F inX^(**)F \in X^{*}.
2^(@)2^{\circ} Some Phelps type duality results and applications can be found in [3], and in the bibliography quoted there.
2^(@)2^{\circ} Some Phelps type duality results and applications can be found in [3], and in the bibliography quoted there.
REFERENCES
[1] Alegre, C., Ferrer, J. and Gregori, V., On the Hahn-Banach theorem in certain linear quasi-uniform structures, Acta Math. Hungar, 82, pp. 315-320, 1999.
[2] Borodin, P. A., The Banach-Mazur Theorem for spaces with asymmetric norm and its applications in convex analysis, Mathematical Notes, 69, no. 3, pp. 298-305, 2001.
[3] Cobzaş, S., Phelps type duality results in best approximation, Rev. Anal. Numér. Théor. Approx., 31, no. 1, pp. 29-43, 2002. /_\\triangle
[4] Dolzhenko, E. P. and Sevast'yanov, E. A., Approximation with sign-sensitive weights, Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 62, no. 6, pp. 59-102, 1998 and 63, no. 3, pp. 77-118, 1999.
[5] Ferrer, J., Gregori, V. and Alegre, C., Quasi-uniform structures in linear lattices, Rocky Mountain J. Math., 23, pp. 877-884, 1993.
[6] Garćia-Raffi, L. M., Romaguera, S. and Sanchez Pérez, E. A., Extension of asymmetric norms to linear spaces, Rend. Istit. Mat. Trieste, XXXIII, pp. 113-125, 2001.
[7] Garćia-Raffi, L. M., Romaguera S. and Sánchez-Pérez, E. A., The dual space of an asymmetric normed linear space, Quaestiones Mathematicae, 26, pp. 83-96, 2003.
[8] Krein, M. G. and Nudel'man, A. A., The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973 (in Russian).
[9] Kopperman, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89-97, 1988.
[10] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 847842, 1934.
[11] Mustăţa, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 61-67, 2001. ©
[12] Mustăţa, C., Extensions of convex semi-Lipschitz functions on quasi-metric linear spaces, Séminaire de la Théorie de la Meileure Approximation Convexité et Optimization, Cluj-Napoca, le 29 novembre, pp. 85-92, 2001.
[13] Mustăţa, C., A Phelps type theorem for spaces with asymmetric norms, Bul. Ştiint. Univ. Baia-Mare, Ser. B, XVIII, no. 2, pp. 275-280, 2002.
[14] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238-255, 1960.
[15] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000.
[16] Singer, I., Best approximation in normed linear spaces by elements of linear subspaces, Ed. Acad. RSR, Bucharest, 1967 (in Romanian); English Translation, Springer, Berlin, 1970.
[2] Borodin, P. A., The Banach-Mazur Theorem for spaces with asymmetric norm and its applications in convex analysis, Mathematical Notes, 69, no. 3, pp. 298-305, 2001.
[3] Cobzaş, S., Phelps type duality results in best approximation, Rev. Anal. Numér. Théor. Approx., 31, no. 1, pp. 29-43, 2002. /_\\triangle
[4] Dolzhenko, E. P. and Sevast'yanov, E. A., Approximation with sign-sensitive weights, Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 62, no. 6, pp. 59-102, 1998 and 63, no. 3, pp. 77-118, 1999.
[5] Ferrer, J., Gregori, V. and Alegre, C., Quasi-uniform structures in linear lattices, Rocky Mountain J. Math., 23, pp. 877-884, 1993.
[6] Garćia-Raffi, L. M., Romaguera, S. and Sanchez Pérez, E. A., Extension of asymmetric norms to linear spaces, Rend. Istit. Mat. Trieste, XXXIII, pp. 113-125, 2001.
[7] Garćia-Raffi, L. M., Romaguera S. and Sánchez-Pérez, E. A., The dual space of an asymmetric normed linear space, Quaestiones Mathematicae, 26, pp. 83-96, 2003.
[8] Krein, M. G. and Nudel'man, A. A., The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973 (in Russian).
[9] Kopperman, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89-97, 1988.
[10] McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 847842, 1934.
[11] Mustăţa, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 61-67, 2001. ©
[12] Mustăţa, C., Extensions of convex semi-Lipschitz functions on quasi-metric linear spaces, Séminaire de la Théorie de la Meileure Approximation Convexité et Optimization, Cluj-Napoca, le 29 novembre, pp. 85-92, 2001.
[13] Mustăţa, C., A Phelps type theorem for spaces with asymmetric norms, Bul. Ştiint. Univ. Baia-Mare, Ser. B, XVIII, no. 2, pp. 275-280, 2002.
[14] Phelps, R. R., Uniqueness of Hahn-Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238-255, 1960.
[15] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000.
[16] Singer, I., Best approximation in normed linear spaces by elements of linear subspaces, Ed. Acad. RSR, Bucharest, 1967 (in Romanian); English Translation, Springer, Berlin, 1970.
Received by the editors: February 3, 2003.
- *"T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro, cmustata2001@yahoo.com.