Uniform Topology (other)
In mathematics, the uniform topology on a space may mean: * In functional analysis, it sometimes refers to a polar topology on a topological vector space. * In general topology, it is the topology carried by a uniform space. * In real analysis, it is the topology of uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitra ...

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ...
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Polar Topology
In functional analysis and related areas of mathematics a polar topology, topology of \mathcal-convergence or topology of uniform convergence on the sets of \mathcal is a method to define locally convex topologies on the vector spaces of a pairing. Preliminaries A pairing is a triple (X, Y, b) consisting of two vector spaces over a field \mathbb (either the real numbers or complex numbers) and a bilinear map b : X \times Y \to \mathbb. A dual pair or dual system is a pairing (X, Y, b) satisfying the following two separation axioms: # Y separates/distinguishes points of X: for all non-zero x \in X, there exists y \in Y such that b(x, y) \neq 0, and # X separates/distinguishes points of Y: for all non-zero y \in Y, there exists x \in X such that b(x, y) \neq 0. Polars The polar or absolute polar of a subset A \subseteq X is the set :A^ := \left\. Dually, the polar or absolute polar of a subset B \subseteq Y is denoted by B^, and defined by :B^ := \left\. In this case, ...
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Uniform Space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closene ...
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