Topological vector space

Summary

In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) 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 (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a 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 (although this article does not). 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. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting on topolo

Official source

https://en.wikipedia.org/wiki/Topological_vector_space

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A Looping-Delooping Adjunction For Topological Spaces

Martina Rovelli

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.

Int Press Boston, Inc2017

Generic representations of orthogonal groups: the mixed functors

Christine Vespa

In previous work, we defined the category of functors F-quad, associated to F-2-vector spaces equipped with a nondegenerate quadratic form. In this paper, we define a special family of objects in the category F-quad, named the mixed functors. We give the complete decompositions of two elements of this family that give rise to two new infinite families of simple objects in the category F-quad.

2007

φ For all finite n ∈ N, there is a well-known isomorphism between the standard braid group Bn and the mapping class group π0Hn. This isomorphism has been exhaustively studied in literature, and generalized in many ways. For some basic topological reason, this strong link between finite braid groups and finite mapping class groups can-not be extended to the infinite case in a straightforward way, and, in particular, is not yet well studied in literature. In our work, we define the infinite braid group B∞ to be the group of braids with infinitely many strands, all of which can be possibly nontrivial, i.e., not straight. In particular, this definition does not correspond to the group of finitary infinite braids, which is just the union of all finite braid groups. Similar to the maps π0φn for finite n, we introduce a map that, in particular, turns out not to be an isomorphism. However, we prove its injectivity, and identify its image in B∞. The study of the link between mapping class groups and braid groups in the infinite case is motivated by the study of homeomorphisms in H∞ that give rise to a homoclinic tangle. In fact, the map π0φ∞ attributes to each isotopy class of such a homeomorphism an element of the infinite braid group B∞, and so, allows us to describe the isotopy classes of these homeomorphisms in terms of their image in B∞. Using the fact that the map π0φ∞ is injective, we prove a result that can be applied to the study of the topological structure of homoclinic tangles.

EPFL2008