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 a continuous linear map. The category is often denoted **TVect** or **TVS**.

- TVect is a concrete category
- TVect K {\displaystyle K} is a topological category
- See also
- References

Fixing a topological field *K*, one can also consider the subcategory **TVect**_{K} of topological vector spaces over *K* with continuous *K*-linear maps as the morphisms.

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.

The category is topological, which means loosely speaking that it relates to its "underlying category", the category of vector spaces, in the same way that **Top** relates to **Set**. Formally, for every *K*-vector space and every family of topological *K*-vector spaces and *K*-linear maps there exists a vector space topology on so that the following property is fulfilled:

Whenever is a *K*-linear map from a topological *K*-vector space it holds that

- is continuous is continuous.

The topological vector space is called "initial object" or "initial structure" with respect to the given data.

If one replaces "vector space" by "set" and "linear map" by "map", one gets a characterisation of the usual initial topologies in **Top**. This is the reason why categories with this property are called "topological".

There are numerous consequences of this property. For example:

- "Discrete" and "indiscrete" objects exist. A topological vector space is indiscrete iff it is the initial structure with respect to the empty family. A topological vector space is discrete iff it is the initial structure with respect to the family of all possible linear maps into all topological vector spaces. (This family is a proper class, but that does not matter: Initial structures with respect to all classes exists iff they exists with respect to all sets)
- Final structures (the similar defined analogue to final topologies) exist. But there is a catch: While the initial structure of the above property is in fact the usual initial topology on with respect to , the final structures do not need to be final with respect to given maps in the sense of
**Top**. For example: The discrete objects (= final with respect to the empty family) in do not carry the discrete topology. - Since the following diagram of forgetful functors commutes

- and the forgetful functor from to
**Set**is right adjoint, the forgetful functor from to**Top**is right adjoint too (and the corresponding left adjoints fit in an analogue commutative diagram). This left adjoint defines "free topological vector spaces". Explicitly these are free*K*-vector spaces equipped with a certain initial topology.

- Since
^{[ clarification needed ]}is (co)complete, is (co)complete too.

- Category of groups
- Category of metric spaces
- Category of sets
- Category of topological spaces
- Category of topological spaces with base point

In mathematics, an **associative algebra** is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give *A* the structure of a ring; the addition and scalar multiplication operations together give *A* the structure of a vector space over *K*. In this article we will also use the term *K*-algebra to mean an associative algebra over the field *K*. A standard first example of a *K*-algebra is a ring of square matrices over a field *K*, with the usual matrix multiplication.

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In mathematics, a **function space** is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set `X` into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function *space*.

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In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

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In general topology and related areas of mathematics, the **initial topology** on a set , with respect to a family of functions on , is the coarsest topology on *X* that makes those functions continuous.

In mathematics, a **pointed space** is a topological space with a distinguished point, the **basepoint**. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as that remains unchanged during subsequent discussion, and is kept track of during all operations.

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In general topology and related areas of mathematics, the **final topology** on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

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In category theory, a branch of mathematics, a **dual object** is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a **dualizable object**. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space *V*^{∗} doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property.

In mathematics, the **category of manifolds**, often denoted **Man**^{p}, is the category whose objects are manifolds of smoothness class *C*^{p} and whose morphisms are *p*-times continuously differentiable maps. This is a category because the composition of two *C*^{p} maps is again continuous and of class *C*^{p}.

In mathematics, and in particular representation theory, **Frobenius reciprocity** is a theorem expressing a duality between the process of restricting and inducting. It can be used to leverage knowledge about representations of a subgroup to find and classify representations of "large" groups that contain them. It is named for Ferdinand Georg Frobenius, the inventor of the representation theory of finite groups.

- Lang, Serge (1972).
*Differential manifolds*. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.CS1 maint: discouraged parameter (link)

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