## Abstract and Applied Analysis

### On projection constant problems and the existence of metric projections in normed spaces

#### Abstract

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spaces $l_p,1\leq p < \infty$ and $c_0$. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator from $l_p,1\leq p < \infty$ or $c_0$ onto anyone of their maximal proper subspaces.

#### Article information

Source
Abstr. Appl. Anal., Volume 6, Number 7 (2001), 401-411.

Dates
First available in Project Euclid: 13 April 2003

https://projecteuclid.org/euclid.aaa/1050266950

Digital Object Identifier
doi:10.1155/S1085337501000732

Mathematical Reviews number (MathSciNet)
MR1879573

Zentralblatt MATH identifier
1060.41025

#### Citation

El-Shobaky, Entisarat; Ali, Sahar Mohammed; Takahashi, Wataru. On projection constant problems and the existence of metric projections in normed spaces. Abstr. Appl. Anal. 6 (2001), no. 7, 401--411. doi:10.1155/S1085337501000732. https://projecteuclid.org/euclid.aaa/1050266950

#### References

• M. M. Day, Normed Linear Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. Heft, vol. 21, Springer-Verlag, Berlin, 1958.
• F. S. De Blasi and J. Myjak, On a generalized best approximation problem, J. Approx. Theory 94 (1998), no. 1, 54–72.
• F. S. De Blasi, J. Myjak, and P. L. Papini, Porous sets in best approximation theory, J. London Math. Soc. (2) 44 (1991), no. 1, 135–142.
• N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, vol. 6, Interscience, New York, 1958.
• E. M. El-Shobaky, S. M. Ali, and W. Takahashi, On the projection constants of some topological spaces and some applications, Abstr. Appl. Anal. 6 (2001), no. 5, 299–308.
• D. J. H. Garling and Y. Gordon, Relations between some constants associated with finite dimensional Banach spaces, Israel J. Math. 9 (1971), 346–361.
• Y. Gordon, On the projection and Macphail constants of $l^{n}_{p}$ spaces, Israel J. Math. 6 (1968), 295–302.
• B. Grünbaum, Projection constants, Trans. Amer. Math. Soc. 95 (1960), 451–465.
• H. K önig, C. Schütt, and N. Tomczak-Jaegermann, Projection constants of symmetric spaces and variants of Khintchine's inequality, J. Reine Angew. Math. 511 (1999), 1–42.
• H. K önig and N. Tomczak-Jaegermann, Norms of minimal projections, J. Funct. Anal. 119 (1994), no. 2, 253–280.
• R. R. Phelps, Čebyšev subspaces of finite codimension in ${C}({X})$, Pacific J. Math. 13 (1963), 647–655.
• D. Rutovitz, Some parameters associated with finite-dimensional Banach spaces, J. London Math. Soc. 40 (1965), 241–255.
• I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Die Grundlehren der mathematischen Wissenschaften, vol. 171, Springer-Verlag, New York, 1970.
• W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, 2000.