closed subspaces of a Banach space X are e-weakly Chebyshev if and only if X is ... Key Words e-quasi Chebyshev subspace, e-weakly Chebyshev subspace.
Analysis in Theory and Applications
19: 2, 2003, 130--135
E-WEAKLY SUBSPACES
CHEBYSHEV
OF BANACH
SPACES
Sh. Rezapour
(Azarbaidjan University of Tartn'at , Iran)
Received May 14, 2003
Abstract
We will define and characterize t-weakly Chebyshev subspaces of Banach spaces. We will prove that all closed subspaces of a Banach space X are e-weakly Chebyshev if and only if X is reflexive. Key Words
e-quasi Chebyshev subspace, e-weakly Chebyshev subspace
AMS(2000)
subject classification
1
46B50, 41A65
Introduction and Preliminaries
Let X be a (complex or real) Banach space, e ~ 0 be given and let W be a subspace of X. A point y o E W is said to be an e-approximation (best approximation) for x E X if
IIx For x E X ,
-
yo
I1 ~
(b). SincePw,,(O)={gEW: Ilgll~ (a). It is obvious. Corollary 2.8.
(a) All closed subspaces of a Banach space X is e-weakly CHebyshev
in X i f and only i f X is reflexive. (b) Let W be a closed subspace of a Banach space X. Then, W is 3-weakly Chebyshev
in X for some ~ 0
i f and only i f W is e-weakly Chebyshev in X for all e>O.
Sh. Rezapour : t-Weakly Chebyshev Subspaces of Banach Spaces
135
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D e p a r t m e n t of M a t h e m a t i c s A z a r b a i d j a n U n i v e r s i t y of T a r b i a t Moallem, 51745--406,
Tabriz, Iran
e - m a i l : s h a h r a m r e z a p o u r @ y a h o o , ca ~ sh. r e z a p o u r @ a z a r u n i v , edu
