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Fundamentals of Linear and Topological Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
(b) A directed set D is a partially ordered set which satisfies the following condition: for alla,b∈Dthree existsc∈Dsuch thata≺_candb≺_c.
Locally Pseudoconvex Spaces and Algebras
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
for each n ∈ N. Since Sμ ∈ S and Onμ⊆Onλ1∩Onλ2 for each n ∈ N0, then λ1 ≺ μ and λ2 ≺ μ. It means that (Λ, ≺) is a directed set.
Compact-Like Operators in Vector Lattices Normed by Locally Solid Lattices
Published in Hemen Dutta, Topics in Contemporary Mathematical Analysis and Applications, 2020
A partially ordered set I is called directed if, for each a1, a2 ∈ I, there is another a ∈ I such that a ≥ a1 and a ≥ a2 (or, a ≤ a1 and a ≤ a2). A function from a directed set I into a set E is called a net in E. A vector lattice is order complete if 0 ≤ xα ↑≤ x implies the existence of sup xα. A net (xα)α∈A in a vector lattice X is called order convergent (or shortly, o-convergent) to x ∈ X if there exists another net (yβ)β∈Bsatisfying yβ ↓ 0, and for any β ∈ B, there exists αβ ∈ A such that |xα − x| ≤ yβ for all α ≥ αβ. In this case, we write xα→ox; for more details, see, for example, [3,31,32]. In a vector lattice X, a net (xα) is unbounded order convergent to x ∈ X if |xα−x|∧u→o0 for every u ∈ X+; see, for example, [15,17–19,28].
Arc length of function graphs via Taylor's formula
Published in International Journal of Mathematical Education in Science and Technology, 2021
A net is a function defined on a directed set. Note that a sequence is a type of net, with D being the set of natural numbers directed as usual, namely that means . In Definition 2.1, D is defined to be the set of choices of finite sets of points along the curve C, with the first point being A and the last being B. If α and β are two such choices, then put if , that is if β is a refinement of α. In an analogous fashion, Riemann sums can be considered as limits of nets by defining refinements of partitions of intervals. So the approach using nets covers in a natural way many seemingly unrelated mathematical concepts in a beautiful way. It also has the advantage of making different types of limit proofs look like sequence limit proofs. However, it is highly abstract and it is also unsuitable for concrete calculations, for instance using computer simulations. Disregarding these difficulties, the typical calculus book (see e.g. Adams, 2006; Hass et al., 2017; Stewart, 2015) will then state some variant of the following result which is then used in exercises to calculate lengths of function graphs in particular cases.
Deterministic and random fixed points for maps on extension type spaces
Published in Applicable Analysis, 2018
For a subset of a topological space , we denote by the directed set of all coverings of by open sets of (usually we write ). Given a map and , a point is said to be an -fixed point of if there exists a member such that and .
Weaker forms of specification for maps on uniform spaces
Published in Dynamical Systems, 2023
Let f be a uniformly continuous self-map on a uniform space , and let denote the set of symmetric entourages. For a positive integer n and any symmetric entourage , a subset of X is said to be -separated with respect to f if for each pair of distinct points x, y in there exists j such that and . Let denote the set of all compact subsets of X. For , let denote the maximal cardinality of -separated sets contained in . Define Note that, the collection forms a directed set under set inclusion, and hence , forms a net in . Define The uniform entropy of f with respect to the uniformity is given by (for more details, see [9]).