Metric spaceIn mathematics, a metric space is a set together with a notion of distance between its points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance.
Uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n ) {\displaystyle (f_{n})} converges uniformly to a limiting function f {\displaystyle f} if, roughly speaking, they uniformly approximate the function f {\displaystyle f} over the whole domain, meaning that all but finitely many of the functions of the sequence lie in a uniform error bar of the original function.
Bounded functionIn mathematics, a functionf{\displaystyle f} defined on some setX{\displaystyle X} with real or complex values is called bounded if the set of its values (its image) is bounded. In other words, there exists a real number M{\displaystyle M} such that |f(x)|≤M{\displaystyle |f(x)|\leq M}for allx{\displaystyle x} in X{\displaystyle X}. A function that is not bounded is said to be unbounded.
Continuous functionMathematical function with no sudden changes
Infimum and supremumGreatest lower bound and least upper boundA set P{\displaystyle P} of real numbers (hollow and filled circles), a subset S{\displaystyle S} of P{\displaystyle P} (filled circles), and the infimum of S.{\displaystyle S.} Note that for totally ordered finite sets, the infimum and the minimum are equal.A set A{\displaystyle A} of real numbers (blue circles), a set of upper bounds of A{\displaystyle A} (red diamond and circles), and the smallest such upper bound, that is, the supremum of A{\displaystyle A} (red diamond).In mathematics, the infimum (abbreviated inf; pl.: infima) of a subsetS{\displaystyle S} of a partially ordered setP{\displaystyle P} is the greatest element in P{\displaystyle P} that is less than or equal to each element of S,{\displaystyle S,} if such an element exists.
Norm (mathematics)Length in a vector spaceIn mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what constitutes a "set" or "collection", because such a definition would have to be in terms of something else previously defined.
Mathematical analysisMathematical analysis is the branch of mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives and integrals to study variable quantities, and in the 19th century its foundations were reformulated with greater rigor. Basic objects of study in mathematical analysis include the real numbers, functions, sequences, series, and limits.
Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
Complex numberNumber with a real and an imaginary partA complex number z = a + bi can be visually represented as a pair of numbers (a, b) forming a position vector (blue) or a point (red) on an Argand diagram, representing the complex plane.
Interval (mathematics)In mathematics, an interval is the set of all real numbers lying between two fixed lower or upper bounds ("endpoints") with no "gaps". For example, the set of real numbers consisting of 0, 1, and all numbers in between is an interval, denoted [0, 1] and called the unit interval. An interval may contain neither endpoint (called an open interval), both endpoints (called a closed interval), or either endpoint (called a semi-open or semi-closed interval).
If and only ifIn logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false.*As an Amazon Associate I earn from qualifying purchases.