Banach spaceIn mathematics, more specifically in functional analysis, a Banach space (pronounced ˈbanax) 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. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.
Lp spaceDISPLAYTITLE:Lp space 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 . Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces.
Fréchet spaceIn functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically Banach spaces.
Hilbert spaceIn mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Normed vector spaceIn 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: Non-negativity: for every ,. Positive definiteness: for every , if and only if is the zero vector.
Vector spaceIn mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.
Space (mathematics)In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. A space consists of selected mathematical objects that are treated as points, and selected relationships between these points. The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space.
EvidenceEvidence for a proposition is what supports the proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidence is what justifies beliefs or what makes it rational to hold a certain doxastic attitude. For example, a perceptual experience of a tree may act as evidence that justifies the belief that there is a tree. In this role, evidence is usually understood as a private mental state.
Evidence (law)The law of evidence, also known as the rules of evidence, encompasses the rules and legal principles that govern the proof of facts in a legal proceeding. These rules determine what evidence must or must not be considered by the trier of fact in reaching its decision. The trier of fact is a judge in bench trials, or the jury in any cases involving a jury. The law of evidence is also concerned with the quantum (amount), quality, and type of proof needed to prevail in litigation.
Hierarchy of evidenceA hierarchy of evidence, comprising levels of evidence (LOEs), that is, evidence levels (ELs), is a heuristic used to rank the relative strength of results obtained from experimental research, especially medical research. There is broad agreement on the relative strength of large-scale, epidemiological studies. More than 80 different hierarchies have been proposed for assessing medical evidence. The design of the study (such as a case report for an individual patient or a blinded randomized controlled trial) and the endpoints measured (such as survival or quality of life) affect the strength of the evidence.
Permutation groupIn mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by Sn, and may be called the symmetric group on n letters.
Evidence-based policyEvidence-based policy is a concept in public policy that advocates for policy decisions to be grounded on, or influenced by, rigorously established objective evidence. This concept presents a stark contrast to policymaking predicated on ideology, 'common sense,' anecdotes, or personal intuitions. The approach mirrors the effective altruism movement's philosophy within governmental circles. The methodology employed in evidence-based policy often includes comprehensive research methods such as randomized controlled trials (RCT).
Evidence-based practiceEvidence-based practice (EBP) is the idea that occupational practices ought to be based on scientific evidence. While seemingly obviously desirable, the proposal has been controversial, with some arguing that results may not specialize to individuals as well as traditional practices. Evidence-based practices have been gaining ground since the formal introduction of evidence-based medicine in 1992 and have spread to the allied health professions, education, management, law, public policy, architecture, and other fields.
Circumstantial evidenceCircumstantial evidence is evidence that relies on an inference to connect it to a conclusion of fact—such as a fingerprint at the scene of a crime. By contrast, direct evidence supports the truth of an assertion directly—i.e., without need for any additional evidence or inference. On its own, circumstantial evidence allows for more than one explanation. Different pieces of circumstantial evidence may be required, so that each corroborates the conclusions drawn from the others.
Group actionIn mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it.
Group (mathematics)In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
Covering spaceA covering of a topological space is a continuous map with special properties. Let be a topological space. A covering of is a continuous map such that there exists a discrete space and for every an open neighborhood , such that and is a homeomorphism for every . Often, the notion of a covering is used for the covering space as well as for the map . The open sets are called sheets, which are uniquely determined up to a homeomorphism if is connected. For each the discrete subset is called the fiber of .
Lie group actionIn differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. TOC Let be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism . A smooth manifold endowed with a Lie group action is also called a G-manifold.
Automorphism groupIn mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group.
Outer automorphism groupIn mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is trivial and G has a trivial center, then G is said to be complete. An automorphism of a group that is not inner is called an outer automorphism. The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups.
