About: Injective tensor product

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In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -valued fun

Attributes	Values
rdf:type	Thing
rdfs:label	Injektives Tensorprodukt (de) Injective tensor product (en)
rdfs:comment	Das injektive Tensorprodukt ist eine Erweiterung der in der Mathematik betrachteten Tensorprodukte von Vektorräumen auf den Fall, dass zusätzlich Topologien auf den Vektorräumen vorhanden sind. In dieser Situation liegt es nahe, auch auf dem Tensorprodukt der Räume eine Topologie erklären zu wollen. Unter den vielen Möglichkeiten, dies zu tun, sind das projektive Tensorprodukt und das hier zu behandelnde injektive Tensorprodukt natürliche Wahlen. (de) In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -valued fun (en)
rdfs:seeAlso	Schwartz space Series (mathematics)
name	Theorem (en) Theorem: (en)
dcterms:subject	Functional analysis
Wikipage page ID	63566619 (xsd:integer)
Wikipage revision ID	1118598630 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective tensor product Schwartz space Barrelled space Inductive tensor product Complete topological vector space Continuous dual space Nuclear operator Nuclear space Separable space Equicontinuous Fréchet space Minkowski functional Locally convex topological vector space Strongest Comparison of topologies Polar set Mackey topology Banach space Topological homomorphism Topological vector space Topology of uniform convergence Transpose Weak-* topology Alexander Grothendieck Nuclear spaces Directed set Radon measure Hilbert space Isometry Tensor product Functional analysis Bipolar theorem Differentiable vector-valued functions from Euclidean space Distribution (mathematics) Net (mathematics) Open map Series (mathematics) Uncountable Strong dual space Coarsest topology TVS-embedding
Link from a Wikipage to an external page	https://ncatlab.org/nlab/show/nuclear+space
sameAs	Injective tensor product Injective tensor product Injective tensor product
dbp:wikiPageUsesTemplate	dbt:Annotated_link dbt:Cite_book dbt:Clarify dbt:Em dbt:Main dbt:Reflist dbt:See_also dbt:Sfn dbt:Khaleelulla_Counterexamples_in_Topological_Vector_Spaces dbt:Math_theorem dbt:Narici_Beckenstein_Topological_Vector_Spaces dbt:Schaefer_Wolff_Topological_Vector_Spaces dbt:Trèves_François_Topological_vector_spaces,_distributions_and_kernels dbt:TopologicalTensorProductsAndNuclearSpaces
date	October 2022 (en)
reason	should probably link to Comparison of topologies or Strong dual space (en)
has abstract	Das injektive Tensorprodukt ist eine Erweiterung der in der Mathematik betrachteten Tensorprodukte von Vektorräumen auf den Fall, dass zusätzlich Topologien auf den Vektorräumen vorhanden sind. In dieser Situation liegt es nahe, auch auf dem Tensorprodukt der Räume eine Topologie erklären zu wollen. Unter den vielen Möglichkeiten, dies zu tun, sind das projektive Tensorprodukt und das hier zu behandelnde injektive Tensorprodukt natürliche Wahlen. Zunächst wird der leichter zugängliche Fall der normierten Räume und Banachräume besprochen, anschließend wird auf die Verallgemeinerungen in der Theorie der lokalkonvexen Räume eingegangen. Die Konstruktion für normierte Räume und Banachräume geht auf Robert Schatten zurück, die Verallgemeinerungen auf lokalkonvexe Räume wurden von Alexander Grothendieck erzielt. (de) In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to -valued functions. (en)
math statement	If is a complete Hausdorff locally convex space, then is canonically isomorphic to the injective tensor product (en) If is a complete locally convex space, then is canonically isomorphic to (en) The dual of consists of exactly those continuous bilinear forms v on that can be represented in the form of a map where and are some closed, equicontinuous subsets of and respectively, and is a positive Radon measure on the compact set with total mass Furthermore, if is an equicontinuous subset of then the elements can be represented with fixed and running through a norm bounded subset of the space of Radon measures on (en) The canonical embedding becomes an embedding of topological vector spaces when is given the injective topology and furthermore, its range is dense in its codomain. If is a completion of then the continuous extension of this embedding is an isomorphism of TVSs. So in particular, if is complete then is canonically isomorphic to (en)
prov:wasDerivedFrom	wikipedia-en:Injective_tensor_product?oldid=1118598630&ns=0
page length (characters) of wiki page	46138 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Injective_tensor_product
is differentFrom of	Inductive tensor product
is rdfs:seeAlso of	Topological tensor product Integral linear operator
is Link from a Wikipage to another Wikipage of	Projective tensor product Schwartz kernel theorem Complete topological vector space Nuclear C*-algebra Nuclear space Differentiable vector–valued functions from Euclidean space Distribution (mathematics) Spaces of test functions and distributions Integral linear operator List of topologies
is foaf:primaryTopic of	wikipedia-en:Injective_tensor_product

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