Dynamic Programming Related Questions

AI Thread Summary
The discussion addresses key concepts in dynamic programming, focusing on the distinctions between compact valued and single valued correspondences. It explores the implications of continuity on compact sets, particularly regarding boundedness and the attainment of maximum values. The difference between supremum and maximum is clarified, noting that some sets, like the open interval (0,1), have a least upper bound but no maximum. The conversation emphasizes that the maximum cannot be defined as (1-G) where G is the smallest real number greater than zero, as no such G exists. Overall, the thread provides insights into fundamental mathematical principles relevant to dynamic programming.
sampahmel
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Hi all,

(1.) Can someone tell me the difference between a compact valued and single valued correspondence?

(2.) I have been seeing repeating themes of "continuity on a compact set". Does that imply boundedness and thus possible to attain maximum?

(3.) What's the difference between Supremum (I know it is the l.u.b.) and Maximum?

Thank you for taking time out to read and to answer.
 
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sampahmel said:
(3.) What's the difference between Supremum (I know it is the l.u.b.) and Maximum?

Thank you for taking time out to read and to answer.

Some sets such as the open interval (0,1) have no maximum, but it has a least upper bound, namely 1. This is not the maximum value since it is not contained in the set.
 
Jarle said:
Some sets such as the open interval (0,1) have no maximum, but it has a least upper bound, namely 1. This is not the maximum value since it is not contained in the set.

Does that mean the maximum is (1-G) where G is the smallest real >0
 
sampahmel said:
Does that mean the maximum is (1-G) where G is the smallest real >0

No such G exists.
 

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