On a sufficient condition for proximity

Author:
Ka Sing Lau

Journal:
Trans. Amer. Math. Soc. **251** (1979), 343-356

MSC:
Primary 46B99; Secondary 41A65, 47D15

MathSciNet review:
531983

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Abstract | References | Similar Articles | Additional Information

Abstract: A closed subspace *M* in a Banach space *X* is called *U*-proximinal 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 *U*-proximinal, for example, the subspaces with the 2-ball property (semi *M*-ideals) 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****[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; Interscience Publishers, Ltd., London, 1958. MR**0117523****[3]**H. Fakhoury,*Sur les M-idéaux dans certains espaces d'opérateurs et l'approximation par des opérateurs compacts*(to appear).**[4]**Julien Hennefeld,*A decomposition for 𝐵(𝑋)* and unique Hahn-Banach extensions*, Pacific J. Math.**46**(1973), 197–199. MR**0370265****[5]**R. B. Holmes,*𝑀-ideals in approximation theory*, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 391–396. MR**0427927****[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**, 10.1090/S0002-9904-1965-11433-1**[7]**Richard Holmes and Bernard Kripke,*Smoothness of approximation*, Michigan Math. J.**15**(1968), 225–248. MR**0228904****[8]**Richard B. Holmes and Bernard R. Kripke,*Best approximation by compact operators*, Indiana Univ. Math. J.**21**(1971/72), 255–263. MR**0296659****[9]**Richard Holmes, Bruce Scranton, and Joseph Ward,*Approximation from the space of compact operators and other 𝑀-ideals*, Duke Math. J.**42**(1975), 259–269. MR**0394301****[10]**Ka Sing Lau,*Approximation by continuous vector-valued functions*, Studia Math.**68**(1980), no. 3, 291–298. MR**599151****[11]**-,*M-ideals 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**, 10.1090/S0002-9947-1977-0430747-4**[13]**Joram Lindenstrauss,*Extension of compact operators*, Mem. Amer. Math. Soc. No.**48**(1964), 112. MR**0179580****[14]**Joram Lindenstrauss,*On nonlinear projections in Banach spaces*, Michigan Math. J.**11**(1964), 263–287. MR**0167821****[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**, 10.1016/0021-9045(78)90116-8**[16]**R. R. Smith and J. D. Ward,*𝑀-ideal structure in Banach algebras*, J. Functional Analysis**27**(1978), no. 3, 337–349. MR**0467316**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1979-0531983-0

Keywords:
Compact operators,
measurable functions,
*M*-ideals,
proximity,
uniformly convex

Article copyright:
© Copyright 1979
American Mathematical Society