In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.
The Euclidean norm on is the non-negative function defined by
Like all norms, it induces a canonical metric defined by The metric induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points and is
In any metric space, the open balls form a base for a topology on that space. [1] The Euclidean topology on is the topology generated by these balls. In other words, the open sets of the Euclidean topology on are given by (arbitrary) unions of the open balls defined as for all real and all where is the Euclidean metric.
When endowed with this topology, the real line is a T5 space. Given two subsets say and of with where denotes the closure of there exist open sets and with and such that [2]
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms:
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
In mathematical analysis and in probability theory, a σ-algebra on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space.
In mathematics, an open set is a generalization of an open interval in the real line.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right.
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space. It is used in the proof of results in many areas of analysis and geometry, including some of the fundamental theorems of functional analysis.
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include and .
In differential geometry, a Riemannian manifold or Riemannian space(M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p.
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball or an open ball.
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based only on given values of the distances between pairs of points. More abstractly, it is the study of semimetric spaces and the isometric transformations between them. In this view, it can be considered as a subject within general topology.
In mathematics, a bump function is a function on a Euclidean space which is both smooth and compactly supported. The set of all bump functions with domain forms a vector space, denoted or The dual space of this space endowed with a suitable topology is the space of distributions.
In 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. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
In mathematics, the real coordinate space or real coordinate n-space, of dimension n, denoted Rn or , is the set of all ordered n-tuples of real numbers, that is the set of all sequences of n real numbers, also known as coordinate vectors. Special cases are called the real lineR1, the real coordinate planeR2, and the real coordinate three-dimensional spaceR3. With component-wise addition and scalar multiplication, it is a real vector space.
In mathematics, an LB-space, also written (LB)-space, is a topological vector space that is a locally convex inductive limit of a countable inductive system of Banach spaces. This means that is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Banach space.
