Some remarks concerning norm preserving extension and best approximation


Let \(X,Y\) be two normed spaces, \(X_{1}\) a subspace of \(X\) and \(A:X\rightarrowY\) 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.


Costica Mustata
“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.


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Revue d’Analyse Numer.Theor. Approx.

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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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