zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings. (English) Zbl 0653.41029
Let X be a paracompact space, Y a normed vector space and $2\sp Y$ the collection of all non-empty subsets of Y. A function f: $X\to 2\sp Y$ is called a set-valued mapping. The relationships between lower semicontinuity, almost lower semicontinuity and the existence of various kinds of continuous selections for such a mapping are studied. For the cases $Y=C\sb 0(T)$ and $Y=L\sb 1$, the authors give intrinsic characterizations of the one-dimensional subspaces whose metric projections admit continuous selections.
Reviewer: H.R.Dowson

41A65Abstract approximation theory
Full Text: DOI
[1] Arens, R. F.; Kelley, J. L.: Characterizations of the space of continuous functions. Trans. amer. Math. soc. 62, 499-508 (1947) · Zbl 0032.03202
[2] Blatter, J.: Zur stetigkeit von mengenwertigen metrischen projektionen. Schriftenreihe rh.-westf. Inst. math. Univ. Bonn, ser. A 16, 19-38 (1967) · Zbl 0184.15201
[3] Brosowski, B.; Deutsch, F.: On some geometrical properties of suns. J. approx. Theory 10, 245-267 (1974) · Zbl 0272.41020
[4] Brown, A. L.: Best n-dimensional approximation to sets of functions. Proc. London math. Soc. 14, 577-594 (1964) · Zbl 0129.04702
[5] Deutsch, F.; Kenderov, P.: Continuous selections and approximate selections for set-valued mappings and applications to metric projections. SIAM J. Math. anal. 14, 185-194 (1983) · Zbl 0518.41031
[6] Dunford, N.; Schwartz, J. T.: Linear operators. (1958)
[7] Hahn, H.: Reele funktionen. (1932)
[8] Krüger, H.: A remark on the lower semicontinuity of the set-valued metric projection. J. approx. Theory 28, 83-86 (1980) · Zbl 0428.41025
[9] Lazar, A. J.: Spaces of affine continuous functions on simplexes. Trans. amer. Math. soc. 134, 503-525 (1968) · Zbl 0174.17102
[10] Lazar, A. J.; Morris, P. D.; Wulbert, D. E.: Continuous selections for metric projections. J. funct. Anal. 3, 193-216 (1969) · Zbl 0174.17101
[11] Michael, E.: Continuous selections, I. Ann. of math. 63, 361-382 (1956) · Zbl 0071.15902
[12] Nürnberger, G.; Sommer, M.: Continuous selections in Chebyshev approximation. Isnm 72 (1985) · Zbl 0585.41032