Category of topological vector spaces
- Category of topological vector spaces
In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again continuous. The category is often denoted TVect or TVS.
Fixing a topological field "K", one can also consider the (sub-)category TVect"K" of topological vector spaces over "K" with continuous "K"-linear maps as the morphisms.
TVect is a concrete category
Like many categories, the category TVect is a concrete category, meaning its objects are sets with additional structure (i.e. a vector space structure and a topology) and its morphisms are functions preserving this structure. There are obvious forgetful functors into the category of topological spaces, the category of vector spaces and the category of sets.
References
*
Wikimedia Foundation.
2010.
Look at other dictionaries:
Topological vector space — In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a… … Wikipedia
Locally convex topological vector space — In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) which generalize normed spaces. They can be defined as topological vector… … Wikipedia
Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… … Wikipedia
Topological group — Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product … Wikipedia
Topological space — Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The… … Wikipedia
Vector bundle — The Möbius strip is a line bundle over the 1 sphere S1. Locally around every point in S1, it looks like U × R, but the total bundle is different from S1 × R (which is a cylinder instead). In mathematics, a vector bundle is a… … Wikipedia
Topological quantum field theory — A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.Although TQFTs were invented by physicists (notably Edward Witten), they are primarily of mathematical… … Wikipedia
Category (mathematics) — In mathematics, a category is an algebraic structure that comprises objects that are linked by arrows . A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A … Wikipedia
*-autonomous category — In mathematics, a * autonomous (read star autonomous ) category C is a symmetric monoidal closed category equipped with a dualizing object . Contents 1 Definition 2 Properties 3 Examples … Wikipedia
List of category theory topics — This is a list of category theory topics, by Wikipedia page. Specific categories *Category of sets **Concrete category *Category of vector spaces **Category of graded vector spaces *Category of finite dimensional Hilbert spaces *Category of sets… … Wikipedia
