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the supremum is also referred to as the least upper bound (or LUB). The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema...

concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis...

for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are...

directed supremum. Dually, if S {\displaystyle S} is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum. Let A...

is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A conditionally complete lattice satisfies at least...

and greatest element (which are just the supremum and infimum of the empty subset), Lattices, in which every non-empty finite set has a supremum and infimum...

case (E is a grid and B is bounded), the supremum and infimum operators can be replaced by the maximum and minimum. Thus, dilation and erosion are particular...

I {\displaystyle I} in its domain is the difference between the supremum and infimum of f {\displaystyle f} : ω f ( I ) = sup x ∈ I f ( x ) − inf x ∈...

relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are...

infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of...

non-empty finite sets have both a supremum and an infimum is called a lattice. It suffices to require that all suprema and infima of two elements exist to...

of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural...

limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit...

contain a supremum and infimum of the empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence...

median filter and the gaussian filter. Complete lattices are partially ordered sets, where every subset has an infimum and a supremum. In particular...

an important role in PCF theory. Greatest element and least element Infimum and supremum Maximal and minimal elements Schaefer, Helmut H.; Wolff, Manfred...

closed and bounded for any compact set K {\displaystyle K} , which in turn implies that f {\displaystyle f} attains its supremum and infimum on any (nonempty)...

lattice, with infimum and supremum symbolized by ∧ {\displaystyle \wedge } and ∨ {\displaystyle \vee } , respectively. Its universe and least element...

denote the supremum and infimum, respectively. The constant 0 {\displaystyle 0} map is the origin of the vector space X ∗ {\displaystyle X^{*}} and it always...

|x-y|^{2}=(x-y)^{2}} and squaring is monotonic on the nonnegative reals. In any subset S of the real numbers which has an Infimum and a Supremum, the absolute...