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Restrictions of convex subsets of 
Author:
Per Hag
Journal:
Trans. Amer. Math. Soc. 233 (1977), 283-294
MSC:
Primary 46E10
MathSciNet review:
0467264
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Abstract: The main result of this paper is a theorem giving a measure-theoretic condition which is necessary and sufficient for a closed convex subset S of  to have the so-called bounded extension property with respect to a closed subset F of X. This theorem generalizes well-known results on closed subspaces by Bishop, Gamelin and Semadeni.
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E. M. Alfsen and B. Hirsberg, On dominated extensions in linear subspaces of
 , Preprint series, Inst. of Math., Univ. of Oslo, 1970.
- [2]
The closedness of the image of a convex set, Zap. Naučn.
Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22
(1971), 171–174 (Russian). MR 0291780
(45 #871)
- [3]
Errett
Bishop, A general Rudin-Carleson
theorem, Proc. Amer. Math. Soc. 13 (1962), 140–143. MR 0133462
(24 #A3293), http://dx.doi.org/10.1090/S0002-9939-1962-0133462-4
- [4]
N. Dunford and J. T. Schwartz, Linear operators. Vol. I, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302.
- [5]
T.
W. Gamelin, Restrictions of subspaces of
𝐶(𝑋), Trans. Amer. Math.
Soc. 112 (1964),
278–286. MR 0162132
(28 #5331), http://dx.doi.org/10.1090/S0002-9947-1964-0162132-8
- [6]
Theodore
W. Gamelin, Uniform algebras, Prentice-Hall Inc., Englewood
Cliffs, N. J., 1969. MR 0410387
(53 #14137)
- [7]
Irving
Glicksberg, Measures orthogonal to algebras and
sets of antisymmetry, Trans. Amer. Math.
Soc. 105 (1962),
415–435. MR 0173957
(30 #4164), http://dx.doi.org/10.1090/S0002-9947-1962-0173957-5
- [8]
P. Hag, Restrictions of convex subsets of
 , Ph. D. dissertation, Univ. of Michigan, 1972.
- [9]
Per
Hag, A generalization of the Rudin-Carleson
theorem, Proc. Amer. Math. Soc. 43 (1974), 341–344. MR 0338755
(49 #3519), http://dx.doi.org/10.1090/S0002-9939-1974-0338755-1
- [10]
Per
Hag, Erratum to: “A generalization of
the Hudin-Carleson theorem” (Proc. Amer. Math. Soc. 43 (1974),
341–344), Proc. Amer. Math. Soc. 60 (1976), 377. MR 0415325
(54 #3414), http://dx.doi.org/10.1090/S0002-9939-1976-0415325-X
- [11]
E.
Michael and A.
Pełczyński, A linear extension theorem, Illinois
J. Math. 11 (1967), 563–579. MR 0217582
(36 #671)
- [12]
H.
L. Royden, Real analysis, The Macmillan Co., New York, 1963.
MR
0151555 (27 #1540)
- [13]
Z. Semadeni, Simultaneous extensions and projections in spaces of continuous functions Lecture Notes, Univ. of Aarhus, 1965.
- [1]
- E. M. Alfsen and B. Hirsberg, On dominated extensions in linear subspaces of , Preprint series, Inst. of Math., Univ. of Oslo, 1970.
- [2]
- E. L. Arenson, On the closure of the image of a convex set, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 22 (1971), 171-174. (Russian) MR 45 #871. MR 0291780 (45:871)
- [3]
- E. Bishop, A general Rudin-Carleson theorem, Proc. Amer. Math. Soc. 13 (1962), 140-143. MR 24 #A3293. MR 0133462 (24:A3293)
- [4]
- N. Dunford and J. T. Schwartz, Linear operators. Vol. I, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302.
- [5]
- T. W. Gamelin, Restrictions of subspaces of
 , Trans. Amer. Math. Soc. 112 (1964), 278-286. MR 28 #5331. MR 0162132 (28:5331)
- [6]
- -, Uniform algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969. MR 0410387 (53:14137)
- [7]
- I. Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415-435. MR 30 #4164. MR 0173957 (30:4164)
- [8]
- P. Hag, Restrictions of convex subsets of , Ph. D. dissertation, Univ. of Michigan, 1972.
- [9]
- -, A generalization of the Rudin-Carleson theorem, Proc. Amer. Math. Soc. 43 (1974), 341-344. MR 49 #3519. MR 0338755 (49:3519)
- [10]
- -, Erratum to ``A generalization of the Rudin-Carleson theorem", Proc. Amer. Math. Soc. 60 (1976), 377. MR 0415325 (54:3414)
- [11]
- E. Michael and A. Pełczyński, A linear extension theorem, Illinois J. Math. 11 (1967), 563-579. MR 36 #671. MR 0217582 (36:671)
- [12]
- H. L. Royden, Real analysis, Macmillan, New York, 1968. MR 0151555 (27:1540)
- [13]
- Z. Semadeni, Simultaneous extensions and projections in spaces of continuous functions Lecture Notes, Univ. of Aarhus, 1965.
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DOI:
http://dx.doi.org/10.1090/S0002-9947-1977-0467264-1
PII:
S 0002-9947(1977)0467264-1
Article copyright:
© Copyright 1977 American Mathematical Society
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