Information AboutSupremum |
| CATEGORIES ABOUT SUPREMUM | |
| order theory | |
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In Mathematics , the supremum of an Ordered Set ''S'' is the Least Element that is greater than or equal to each element of ''S''. Consequently, it is also referred to as the '''least upper bound''' (also '''lub''' and '''LUB'''). The supremum may, or may not, belong to the set ''S''. If ''S'' contains a Greatest Element , then that element is the supremum; and if not, then the supremum does not belong to the set. Suprema are often considered for subsets of Real Number s, Rational Number s, or any other well-known mathematical structures for which it is immediately clear what it means for an element to be "greater-or-equal" than another element. Nonetheless, the definition generalizes easily to the more abstract setting of Order Theory where one considers arbitrary Partially Ordered Set s. In any case, suprema must not be confused with '' Minimal '' Upper Bound s, or with Maximal or Greatest Element s. Some notes on these issues follow below. SUPREMUM OF A SET OF REAL NUMBERS In set of real numbers that is bounded above has a supremum. If, in addition, we define sup(''S'') = −∞ when ''S'' is Empty and sup(''S'') = +∞ when ''S'' is not bounded above, then ''every'' set of real numbers has a supremum (see Extended Real Number Line ). Examples: : : : : : : The supremum of ''S'' may or may not belong to ''S''. In particular, note the third example where the supremum of a set of Rationals is Irrational (which means that the rationals are Incomplete ). However, if the supremum value belongs to the set then it is the Greatest element in the set. The term '' Maximal Element '' is also synonymous as long as one deals with real numbers or any other Totally Ordered Set . Since sup(''S'') is the ''least'' upper bound, to show that sup(''S'') ≤ ''a'', one only has to show that ''a'' itself is an upper bound for ''S'', i.e. one only has to show that ''x'' ≤ ''a'' for all ''x'' in ''S''. Showing that sup(''S'') ≥ ''a'' is a bit harder: for any ''b'' < ''a'', we must find an ''x'' in ''S'' with ''x'' ≥ ''b''. In Functional Analysis , one often considers the Supremum Norm (also sometimes referred to as the Uniform Norm ) of a bounded function ''f'' : ''X'' -> R (or '''C'''); it is defined as |
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