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| In [[mathematics]], the '''category of topological vector spaces''' is the [[category (category theory)|category]] whose [[object (category theory)|objects]] are [[topological vector space]]s and whose [[morphism]]s are [[continuous linear map]]s between them. This is a category because the [[function composition|composition]] of two continuous linear maps is again continuous. The category is often denoted '''TVect''' or '''TVS'''.
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| Fixing a [[topological field]] ''K'', one can also consider the (sub-)category '''TVect'''<sub>''K''</sub> of topological vector spaces over ''K'' with continuous ''K''-linear maps as the morphisms.
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| ==TVect is a concrete category==
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| Like many categories, the category '''TVect''' is a [[concrete category]], meaning its objects are [[Set (mathematics)|sets]] with additional structure (i.e. a vector space structure and a topology) and its morphisms are [[function (mathematics)|functions]] preserving this structure. There are obvious [[forgetful functor]]s into the [[category of topological spaces]], the [[category of vector spaces]] and the [[category of sets]].
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| ==<math>\textbf{TVect}_K</math> is a topological category==
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| The category is topological, which means loosely speaken that it relates to its "underlying category" the category of vector spaces in the same way that '''Top''' relates to '''Set'''. Formally:
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| For every single K-vector space <math>V</math> and every family <math>( (V_i,\tau_i),f_i)_{i\in I}</math> of topological K-vector spaces <math>(V_i,\tau_i)</math> and K-linear maps <math>f_i: V\to V_i</math>, there exists a vector space topology <math>\tau</math> on <math>V</math> so that the following property is fulfilled:
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| Whenever <math>g: Z\to V</math> is a K-linear map from a topological K-vector space <math>(Z,\sigma)</math> it holds:
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| :<math>g: (Z,\sigma)\to (V,\tau)</math> is continuous <math>\iff</math> <math>\forall i\in I: f_i\circ g: (Z,\sigma)\to(V_i,\tau_i)</math> is continuous.
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| The topological vector space <math>(V,\tau)</math> is called "initial object" or "initial structure" with respect to the given data.
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| 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".
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| There are numerous consequences of this property. For example:
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| * "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 in fact that doesn't matter: Initial structures with respect to all classes exists iff they exists with respect to all sets)
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| * 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 <math>V</math> with respect to <math>(\tau_i,f_i)_{i\in I}</math>, the final structures don't 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 <math>\textbf{TVect}_K</math> do not carry the discrete topology.
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| * Since the following diagram of forgetful functors commutes
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| ::<math>\begin{array}{ccc}
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| \textbf{Vect}_K & \rightarrow & \textbf{Set} \\
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| \uparrow & & \uparrow \\
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| \textbf{TVect}_K & \rightarrow & \textbf{Top}
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| \end{array}</math>
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| :and the forgetful functor from <math>\textbf{Vect}_K</math> to '''Set''' is right adjoint, the forgetful functor from <math>\textbf{TVect}_K</math> to '''Top''' is right adjoint too (and the correspondig 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.
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| * Since{{clarify|date=October 2013}} <math>\textbf{Vect}_K</math> is (co)complete, <math>\textbf{TVect}_K</math> is (co)complete too.
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| ==References==
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| * {{cite book | last=Lang | first=Serge | authorlink = Serge Lang | title=Differential manifolds | publisher=Addison-Wesley Publishing Co., Inc. | location=Reading, Mass.–London–Don Mills, Ont. | year=1972 }}
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| [[Category:Category-theoretic categories|Topological vector spaces]]
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| [[Category:Topological vector spaces]]
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Hi there! :) My name is Kristen, I'm a student studying Environmental Management from Borsano, Italy.
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