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Similarly, if the supremum of belongs to , it is a maximum or greatest element of . For example, consider the set of negative real numbers (excluding zero). This set has no greatest element, since for every element of the set, there is another, larger, element.
The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
Here, the supremum is taken over balls B in R n which contain the point x and |B| denotes the measure of B (in this case a multiple of the radius of the ball raised to the power n). One can also study the centred maximal function, where the supremum is taken just over balls B which have centre x. In practice there is little difference between ...
The supremum of this function (largest value) is 5, and the infimum (smallest value) is −4. However, the function takes these values only on the sets {} and {}, respectively, which are of measure zero. Everywhere else, the function takes the value 2.
Supremum. For a poset P and a subset X of P, the least element in the set of upper bounds of X (if it exists, which it may not) is called the supremum, join, or least upper bound of X. It is denoted by sup X or X. The supremum of two elements may be written as sup{x,y} or x ∨ y.
An element of is the join (or least upper bound or supremum) of in if the following two conditions are satisfied: x ≤ j and y ≤ j {\displaystyle x\leq j{\text{ and }}y\leq j} (that is, j {\displaystyle j} is an upper bound of x and y {\displaystyle x{\text{ and }}y} ).
If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm.