Almost Chebyshev subspaces, lower semicontinuity, and Hahn-Banach extensions
HTML articles powered by AMS MathViewer
- by Edward Rozema PDF
- Proc. Amer. Math. Soc. 39 (1973), 117-121 Request permission
Abstract:
A subset $M$ of a Banach space is called almost Chebyshev iff the set of elements with more than one best approximation from $M$ is the first category. It is first shown that if the metric projection onto a proximinal almost Chebyshev subset $M$ is lower semicontinuous, then $M$ is Chebyshev. Next, let $M$ be a subspace of a separable Banach space. Then ${M^ \bot }$ is almost Chebyshev iff the set of elements in ${M^\ast }$ which fail to have a unique Hahn-Banach extension is the first category.References
- Jörg Blatter, Zur Stetigkeit von mengenwertigen metrischen Projektionen, Forschungsberichte des Landes Nordrhein-Westfalen, Nr. 1870, Westdeutscher Verlag, Cologne, 1967, pp. 17–38. MR 0219963 B. Brosowski et al., Stetigkeitssätz für metrischen Projektion. II. $(P)$-Räume und ${\mathcal {T}_s}$-Stetigkeit der metrische Projektion, Max-Planck-Institut für Physik and Astrophysik 19, München, 1969.
- A. L. Brown, Best $n$-dimensional approximation to sets of functions, Proc. London Math. Soc. (3) 14 (1964), 577–594. MR 167761, DOI 10.1112/plms/s3-14.4.577
- A. L. Brown, On continuous selections for metric projections in spaces of continuous functions, J. Functional Analysis 8 (1971), 431–449. MR 0296666, DOI 10.1016/0022-1236(71)90005-x
- A. L. Garkavi, On Chebyshev and almost-Chebyshev subspaces, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 799–818 (Russian). MR 0165351
- A. L. Garkavi, Almost-Čebyšev systems of continuous functions, Izv. Vysš. Učebn. Zaved. Matematika 1965 (1965), no. 2 (45), 36–44 (Russian). MR 0184076
- Ernest Michael, Continuous selections. I, Ann. of Math. (2) 63 (1956), 361–382. MR 77107, DOI 10.2307/1969615
- R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238–255. MR 113125, DOI 10.1090/S0002-9947-1960-0113125-4
- S. B. Stečkin, Approximation properties of sets in normed linear spaces, Rev. Math. Pures Appl. 8 (1963), 5–18 (Russian). MR 155168
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 117-121
- MSC: Primary 41A65; Secondary 46B05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0312131-9
- MathSciNet review: 0312131