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SMOOTH POINTS OF UNIT BALLS OF OPERATOR AND FUNCTION SPACES

SMOOTH POINTS OF UNIT BALLS OF OPERATOR AND FUNCTION SPACES DEMONSTRATIO MATHEMATICAVol. XXIXNo 41996Roshdi KhalilS M O O T H P O I N T S OF U N I T BALLSOF O P E R A T O R A N D F U N C T I O N SPACES0. IntroductionLet X be a Banach space and i?i(X) be the unit ball of X. A pointx G B\(X) is called a smooth point if there exists a unique / G X*, thedual of X, such that f(x) = ||/|| = 1.In [4], Holub characterized the smooth points of Bi(N(l2))andB\(K(l2)), where N(X)K(X))denotes the nuclear (compact) operators onX. Smooth points of B\(K{lp)) and B\{L{lp))l < p < oo were studied in[1], where L(X) is the space of bounded linear operators on X.The structure of the Hilbert space I 2 , was heavily used in the study ofsmooth points of B\{N(l2)), in [4], So the proofs can't be generalized to thecase of lp, p ^ 2. Another difference between the cases of I2 and lp,p ^ 2, isthe fact that the extreme points of B\{L{12)) are completely characterized(in [5]), while the description of the extreme points of B\(L(lp)) is a verydifficult (and an open) problem, [6].In Section II of this http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

SMOOTH POINTS OF UNIT BALLS OF OPERATOR AND FUNCTION SPACES

Demonstratio Mathematica , Volume 29 (4): 10 – Oct 1, 1996

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Publisher
de Gruyter
Copyright
© by Roshdi Khalil
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1996-0406
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXIXNo 41996Roshdi KhalilS M O O T H P O I N T S OF U N I T BALLSOF O P E R A T O R A N D F U N C T I O N SPACES0. IntroductionLet X be a Banach space and i?i(X) be the unit ball of X. A pointx G B\(X) is called a smooth point if there exists a unique / G X*, thedual of X, such that f(x) = ||/|| = 1.In [4], Holub characterized the smooth points of Bi(N(l2))andB\(K(l2)), where N(X)K(X))denotes the nuclear (compact) operators onX. Smooth points of B\(K{lp)) and B\{L{lp))l < p < oo were studied in[1], where L(X) is the space of bounded linear operators on X.The structure of the Hilbert space I 2 , was heavily used in the study ofsmooth points of B\{N(l2)), in [4], So the proofs can't be generalized to thecase of lp, p ^ 2. Another difference between the cases of I2 and lp,p ^ 2, isthe fact that the extreme points of B\{L{12)) are completely characterized(in [5]), while the description of the extreme points of B\(L(lp)) is a verydifficult (and an open) problem, [6].In Section II of this

Journal

Demonstratio Mathematicade Gruyter

Published: Oct 1, 1996

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