Previous |  Up |  Next


linear operator; extension of minimal norm; element of best approximation; strongly unique best approximation
Let $Y \subset l^{(n)}_{\infty }$ be a hyperplane and let $A \in {\Cal L}(Y)$ be given. Denote $$ \align {\Cal A} = & \{L\in {\Cal L}(l^{(n)}_{\infty },Y):L\mid Y = A\} \text{ and} \ & \lambda_{A} = \inf \{\parallel L \parallel : L\in {\Cal A}\}. \endalign $$ In this paper the problem of calculating of the constant $\lambda_{A}$ is studied. We present a complete characterization of those $A \in {\Cal L}(Y)$ for which $\lambda_{A} = \parallel A \parallel $. Next we consider the case $\lambda_{A} > \parallel A \parallel $. Finally some computer examples will be presented.
[1] Baronti M., Papini P.L.: Norm one projections onto subspaces of $l^p$. Ann. Mat. Pura Appl. IV (1988), 53-61. MR 0980971
[2] Blatter J., Cheney E.W.: Minimal projections onto hyperplanes in sequence spaces. Ann. Mat. Pura Appl. 101 (1974), 215-227. MR 0358179
[3] Collins H.S., Ruess W.: Weak compactness in spaces of compact operators and vector valued functions. Pacific J. Math. 106 (1983), 45-71. MR 0694671
[4] Odyniec Wl., Lewicki G.: Minimal Projections in Banach Spaces. Lecture Notes in Math. 1449, Springer-Verlag. MR 1079547 | Zbl 1062.46500
[5] Singer I.: On the extension of continuous linear functionals... Math. Ann. 159 (1965), 344-355. MR 0188758 | Zbl 0141.12002
[6] Singer I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer-Verlag, Berlin, Heidelberg, New York, 1970. MR 0270044 | Zbl 0197.38601
[7] Sudolski J., Wojcik A.: Some remarks on strong uniqueness of best approximation. Approximation Theory and its Applications 6 (1990), 44-78. MR 1078687 | Zbl 0704.41016
Partner of
EuDML logo