Operator normIn mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm of a linear map is the maximum factor by which it "lengthens" vectors. Given two normed vector spaces and (over the same base field, either the real numbers or the complex numbers ), a linear map is continuous if and only if there exists a real number such that The norm on the left is the one in and the norm on the right is the one in .
Separable extensionIn field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
Field extensionIn mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
Galois extensionIn mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
Group extensionIn mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence If is an extension of by , then is a group, is a normal subgroup of and the quotient group is isomorphic to the group . Group extensions arise in the context of the extension problem, where the groups and are known and the properties of are to be determined.
Algebraic extensionIn mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic. The algebraic extensions of the field of the rational numbers are called algebraic number fields and are the main objects of study of algebraic number theory.
Matrix normIn mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Given a field of either real or complex numbers, let be the K-vector space of matrices with rows and columns and entries in the field . A matrix norm is a norm on . This article will always write such norms with double vertical bars (like so: ).
Amenable groupIn mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
Pontryagin dualityIn mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a p-adic field.
Peer supportPeer support occurs when people provide knowledge, experience, emotional, social or practical help to each other. It commonly refers to an initiative consisting of trained supporters (although it can be provided by peers without training), and can take a number of forms such as peer mentoring, reflective listening (reflecting content and/or feelings), or counseling. Peer support is also used to refer to initiatives where colleagues, members of self-help organizations and others meet, in person or online, as equals to give each other connection and support on a reciprocal basis.
