On a sufficient condition for proximity
Author:
Ka Sing Lau
Journal:
Trans. Amer. Math. Soc. 251 (1979), 343356
MSC:
Primary 46B99; Secondary 41A65, 47D15
MathSciNet review:
531983
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Abstract: A closed subspace M in a Banach space X is called Uproximinal if it satisfies: , for some positive valued function , , and as , where S is the closed unit ball of X. One of the important properties of this class of subspaces is that the metric projections are continuous. We show that many interesting subspaces are Uproximinal, for example, the subspaces with the 2ball property (semi Mideals) and certain subspaces of compact operators in the spaces of bounded linear operators.
 [1]
Erik
M. Alfsen and Edward
G. Effros, Structure in real Banach spaces. I, II, Ann. of
Math. (2) 96 (1972), 98–128; ibid. (2) 96 (1972),
129–173. MR 0352946
(50 #5432)
 [2]
Nelson
Dunford and Jacob
T. Schwartz, Linear Operators. I. General Theory, With the
assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics,
Vol. 7, Interscience Publishers, Inc., New York, 1958. MR 0117523
(22 #8302)
 [3]
H. Fakhoury, Sur les Midéaux dans certains espaces d'opérateurs et l'approximation par des opérateurs compacts (to appear).
 [4]
Julien
Hennefeld, A decomposition for 𝐵(𝑋)* and unique
HahnBanach extensions, Pacific J. Math. 46 (1973),
197–199. MR 0370265
(51 #6492)
 [5]
R.
B. Holmes, 𝑀ideals in approximation theory,
Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin,
Tex., 1976), Academic Press, New York, 1976, pp. 391–396. MR 0427927
(55 #957)
 [6]
B.
R. Kripke and R.
B. Holmes, Approximation of bounded functions by
continuous functions, Bull. Amer. Math.
Soc. 71 (1965),
896–897. MR 0182702
(32 #185), http://dx.doi.org/10.1090/S000299041965114331
 [7]
Richard
Holmes and Bernard
Kripke, Smoothness of approximation, Michigan Math. J.
15 (1968), 225–248. MR 0228904
(37 #4483)
 [8]
Richard
B. Holmes and Bernard
R. Kripke, Best approximation by compact operators, Indiana
Univ. Math. J. 21 (1971/72), 255–263. MR 0296659
(45 #5718)
 [9]
Richard
Holmes, Bruce
Scranton, and Joseph
Ward, Approximation from the space of compact operators and other
𝑀ideals, Duke Math. J. 42 (1975),
259–269. MR 0394301
(52 #15104)
 [10]
Ka
Sing Lau, Approximation by continuous vectorvalued functions,
Studia Math. 68 (1980), no. 3, 291–298. MR 599151
(82c:54016)
 [11]
, Mideals and approximation by compact operators (unpublished).
 [12]
Ȧsvald
Lima, Intersection properties of balls and
subspaces in Banach spaces, Trans. Amer. Math.
Soc. 227 (1977),
1–62. MR
0430747 (55 #3752), http://dx.doi.org/10.1090/S00029947197704307474
 [13]
Joram
Lindenstrauss, Extension of compact operators, Mem. Amer.
Math. Soc. No. 48 (1964), 112. MR 0179580
(31 #3828)
 [14]
Joram
Lindenstrauss, On nonlinear projections in Banach spaces,
Michigan Math. J. 11 (1964), 263–287. MR 0167821
(29 #5088)
 [15]
Jaroslav
Mach and Joseph
D. Ward, Approximation by compact operators on certain Banach
spaces, J. Approx. Theory 23 (1978), no. 3,
274–286. MR
505751 (80j:47054), http://dx.doi.org/10.1016/00219045(78)901168
 [16]
R.
R. Smith and J.
D. Ward, 𝑀ideal structure in Banach algebras, J.
Functional Analysis 27 (1978), no. 3, 337–349.
MR
0467316 (57 #7175)
 [1]
 E. Alfsen and E. Effros, Structure in real Banach spaces, Ann. of Math. (2) 96 (1972), 98173. MR 0352946 (50:5432)
 [2]
 N. Dunford and J. Schwartz, Linear operators. I, Interscience Publishers, New York, 1958. MR 0117523 (22:8302)
 [3]
 H. Fakhoury, Sur les Midéaux dans certains espaces d'opérateurs et l'approximation par des opérateurs compacts (to appear).
 [4]
 J. Hennefeld, A decomposition of and unique HahnBanach extensions, Pacific J. Math. 46 (1973), 197199. MR 0370265 (51:6492)
 [5]
 R. Holmes, Mideals in approximation theory, Approximation Theory. II, Academic Press, New York, 1976, pp. 391396. MR 0427927 (55:957)
 [6]
 R. Holmes and B. Kripke, Approximation of bounded functions by continuous functions, Bull. Amer. Math. Soc. 71 (1965), 896897. MR 0182702 (32:185)
 [7]
 , Smoothness of approximation, Michigan Math. J. 15 (1968), 225248. MR 0228904 (37:4483)
 [8]
 , Best approximation by compact operators, Indiana Univ. Math. J. 21 (1971), 255263. MR 0296659 (45:5718)
 [9]
 R. Holmes, B. Scranton and J. Ward, Approximation from the space of compact operators and other Mideals, Duke Math. J. 42 (1975), 259269. MR 0394301 (52:15104)
 [10]
 K. Lau, Approximation by continuous vector valued functions, Studia Math. (to appear). MR 599151 (82c:54016)
 [11]
 , Mideals and approximation by compact operators (unpublished).
 [12]
 A. Lima, Intersection of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 162. MR 0430747 (55:3752)
 [13]
 J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. No. 48, 1964. MR 0179580 (31:3828)
 [14]
 , On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263287. MR 0167821 (29:5088)
 [15]
 J. Mach and J. Ward, Approximation by compact operators on certain Banach spaces, J. Approximation Theory (to appear). MR 505751 (80j:47054)
 [16]
 R. Smith and J. Ward, Mideal structure in Banach algebra, J. Functional Analysis 27 (1978), 337349. MR 0467316 (57:7175)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905319830
PII:
S 00029947(1979)05319830
Keywords:
Compact operators,
measurable functions,
Mideals,
proximity,
uniformly convex
Article copyright:
© Copyright 1979 American Mathematical Society
