# The modulus of coninuity of the set-valued metric projection

The modulus of coninuity of the set-valued metric projection Acta Mathematica Academiae Scientiarum Hungaricae Tomus 37 (4), (1981), 355~372. THE MODULUS OF CONINUITY OF THE SET-VALUED METRIC PROJECTION By G. GODINI (Bucharest) Introduction Let E and E~ be normed linear spaces, T: E~E~ a continuous mapping and xEE. The modulus of continuity of T at x (see e.g. ) is the function defined by (1) Or(X, 8) = sup {[]Zx-Tyl[: yEE, []Y-Xll <- 8} (8 >= 0). For McE, the (strong) modulus of continuity of T on M (see e.g. ) is defined by (2) E2r(M; 8) = sup {aT(x, 8): xEM} (8 >- 0). If G is a linear subspace of the normed linear space E and xEE, we denote by Po(x) the set of all best approximations of x out of G, i.e., Po(x) = {g0EG: [[X-goll = inf [lx-g[[}. oEG The (set-valued) mapping x~Po(x) is called the (set-valued) metric projection of E onto G. Let us also denote (3) Do = {xEE: Pa(x) #~} (4) F~ = {xEDo: PG(x) is a singleton}. G is called a proximinal (respectively Chebyshev) subspace of E if Da=E (re- spectively FG=E). If Po(x)#~, we shall sometimes denote an element of Po(x) by pa(x). S. B. STE~KIN posed the problem http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

# The modulus of coninuity of the set-valued metric projection

, Volume 37 (4) – Jun 18, 2005
18 pages      /lp/springer-journals/the-modulus-of-coninuity-of-the-set-valued-metric-projection-UwDGHSJjWf
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895135
Publisher site
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### Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 37 (4), (1981), 355~372. THE MODULUS OF CONINUITY OF THE SET-VALUED METRIC PROJECTION By G. GODINI (Bucharest) Introduction Let E and E~ be normed linear spaces, T: E~E~ a continuous mapping and xEE. The modulus of continuity of T at x (see e.g. ) is the function defined by (1) Or(X, 8) = sup {[]Zx-Tyl[: yEE, []Y-Xll <- 8} (8 >= 0). For McE, the (strong) modulus of continuity of T on M (see e.g. ) is defined by (2) E2r(M; 8) = sup {aT(x, 8): xEM} (8 >- 0). If G is a linear subspace of the normed linear space E and xEE, we denote by Po(x) the set of all best approximations of x out of G, i.e., Po(x) = {g0EG: [[X-goll = inf [lx-g[[}. oEG The (set-valued) mapping x~Po(x) is called the (set-valued) metric projection of E onto G. Let us also denote (3) Do = {xEE: Pa(x) #~} (4) F~ = {xEDo: PG(x) is a singleton}. G is called a proximinal (respectively Chebyshev) subspace of E if Da=E (re- spectively FG=E). If Po(x)#~, we shall sometimes denote an element of Po(x) by pa(x). S. B. STE~KIN posed the problem

### Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Jun 18, 2005

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