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In mathematics, a topological space is, roughly speaking, a geometrical 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. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate.
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space. One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces.
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.
In mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps: where carries the product topology. That means is an additive topological group and a multiplicative topological semigroup.
In functional analysis, an F-space is a vector space over the real or complex numbers together with a metric such that
- Scalar multiplication in is continuous with respect to and the standard metric on or
- Addition in is continuous with respect to
- The metric is translation-invariant; that is, for all
- The metric space is complete.
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
In 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 finite abelian groups, and the additive group of the integers, the real numbers, and every finite-dimensional vector space over the reals or a p-adic field.
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers, or the real numbers or the circle are locally compact abelian groups.
In 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 on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebraA, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum  is also naturally a topological space; this is similar to the notion of the spectrum of a ring.
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and spaces can be generalized.
In mathematics, a classification theorem answers the classification problem: "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the same as Borel measures that are finite on compact sets. In general Baire sets and Borel sets need not be the same. In spaces with non-Baire Borel sets, Baire measures are used because they connect to the properties of continuous functions more directly.
In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields.
In algebra, a locally compact field is a topological field whose topology forms a locally compact Hausdorff space. These kinds of fields were originally introduced in p-adic analysis since the fields are locally compact topological spaces constructed from the norm on . The topology is essential because it allows one to construct analogues of algebraic number fields in the p-adic context.
In algebra, a linear topology on a left -module is a topology on that is invariant under translations and admits a fundamental system of neighborhood of that consists of submodules of If there is such a topology, is said to be linearly topologized. If is given a discrete topology, then becomes a topological -module with respect to a linear topology.
References
- Banaszczyk, Wojciech (1991). Additive subgroups of topological vector spaces. Lecture Notes in Mathematics. Vol. 1466. Berlin: Springer-Verlag. pp. viii+178. ISBN 3-540-53917-4. MR 1119302.
- Fourier analysis on Groups, by Walter Rudin.
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