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| Volume 4, Number 3, September 2008, pp. 641-649 | |||||||||
| H. Mohebi and E. Naraghirad | |||||||||
| Key words: | |||||||||
| best approximation, Banach lattice space, continuous along diagonal lines function, Downward set | |||||||||
| Mathematices Subject Classification: 46N10, 54D65 | |||||||||
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| Volume 4, Number 3, September 2008, pp. 641-649 | |||||||||
| H. Mohebi and E. Naraghirad | |||||||||
| Key words: | |||||||||
| best approximation, Banach lattice space, continuous along diagonal lines function, Downward set | |||||||||
| Mathematices Subject Classification: 46N10, 54D65 | |||||||||
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| Abstract: | |||
| Let X be a Banach lattice space with a strong unit 1. We obtain a characterization of an arbitrary downward set W in X as the level set of an increasing continuous along diagonal lines function. The main result of this paper indicates that the distance from a point x ∈ X to a closed downward set W in X can be expressed as the upper envelope of a certain family of functions ψ (x , y ) = sup { λ ∈ lR: x - y≥ λ 1 }. |
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| Distance from a point to a downward set in a Banach lattice | ||