Closed linear operatorIn functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a partially definedlinear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator. The closed graph theorem states that a linear operator f:X→Y{\displaystyle f:X\to Y} between Banach spaces with full domain X{\displaystyle X} is a closed operator if and only if it is a bounded operator.
Hausdorff spaceIn topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjointneighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology.
Closed setThe boundary of the unit disc (pink) is the unit circle (blue). The open disc ‖x‖<1{\displaystyle \|x\|<1} does not include the boundary ‖x‖=1{\displaystyle \|x\|=1}, while the closed unit disc ‖x‖≤1{\displaystyle \|x\|\leq 1} does include the boundary and is therefore a closed set.
Compact spaceIn mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value infinitely often, by the pigeonhole principle.
Product topologyIn topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces.
Topological spaceIn mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.
Closed graph propertyIn mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A real function y=f(x){\displaystyle y=f(x)} is closed if the graph is closed, meaning that it contains all of its limit points. Every such continuous function has a closed graph, but the converse is not necessarily true.
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n{\displaystyle n}-dimensional Euclidean spaceRn{\displaystyle \mathbb {R} ^{n}} by the Euclidean metric.
Graph of a functionIn mathematics, the graph of a functionf{\displaystyle f} is the set of ordered pairs(x,y){\displaystyle (x,y)}, where f(x)=y.{\displaystyle f(x)=y.} In the common case where x{\displaystyle x} and f(x){\displaystyle f(x)} are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve. The graphical representation of the graph of a function is also known as a plot.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Continuous functionMathematical function with no sudden changes
Terence TaoTerence Chi-Shen Tao (Chinese: 陶哲轩; pinyin: Táo Zhéxuān; born 17 July 1975) is an Australian-American mathematician and professor of mathematics at the University of California, Los Angeles (UCLA). He was awarded the Fields Medal in 2006 for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory.*As an Amazon Associate I earn from qualifying purchases.