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In mathematics, the injective tensor produ … 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.lex-valued functions to -valued functions.
, Das injektive Tensorprodukt ist eine Erwei … 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.wurden von Alexander Grothendieck erzielt.
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https://ncatlab.org/nlab/show/nuclear%2Bspace +
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63566619
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46138
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1118598630
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http://dbpedia.org/resource/Banach_space +
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, http://dbpedia.org/resource/Fr%C3%A9chet_space +
, http://dbpedia.org/resource/Complete_topological_vector_space +
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, http://dbpedia.org/resource/Strongest +
, http://dbpedia.org/resource/Coarsest_topology +
, http://dbpedia.org/resource/Isometry +
, http://dbpedia.org/resource/Inductive_tensor_product +
, http://dbpedia.org/resource/Comparison_of_topologies +
, http://dbpedia.org/resource/Locally_convex_topological_vector_space +
, http://dbpedia.org/resource/Nuclear_space +
, http://dbpedia.org/resource/Separable_space +
, http://dbpedia.org/resource/Nuclear_spaces +
, http://dbpedia.org/resource/Schwartz_space +
, http://dbpedia.org/resource/Minkowski_functional +
, http://dbpedia.org/resource/TVS-embedding +
, http://dbpedia.org/resource/Polar_set +
, http://dbpedia.org/resource/Topological_homomorphism +
, http://dbpedia.org/resource/Alexander_Grothendieck +
, http://dbpedia.org/resource/Mackey_topology +
, http://dbpedia.org/resource/Differentiable_vector-valued_functions_from_Euclidean_space +
, http://dbpedia.org/resource/Open_map +
, http://dbpedia.org/resource/Strong_dual_space +
, http://dbpedia.org/resource/Topology_of_uniform_convergence +
, http://dbpedia.org/resource/Directed_set +
, http://dbpedia.org/resource/Continuous_dual_space +
, http://dbpedia.org/resource/Barrelled_space +
, http://dbpedia.org/resource/Topological_vector_space +
, http://dbpedia.org/resource/Equicontinuous +
, http://dbpedia.org/resource/Nuclear_operator +
, http://dbpedia.org/resource/Hilbert_space +
, http://dbpedia.org/resource/Radon_measure +
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, http://dbpedia.org/resource/Transpose +
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, http://dbpedia.org/resource/Projective_tensor_product +
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, http://dbpedia.org/resource/Uncountable +
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| http://dbpedia.org/property/date
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October 2022
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| http://dbpedia.org/property/mathStatement
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If is a complete Hausdorff locally convex space, then is canonically isomorphic to the injective tensor product
, The canonical embedding becomes an embed … 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 toomplete then is canonically isomorphic to
, The dual of consists of exactly those co … 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 ond subset of the space of Radon measures on
, If is a complete locally convex space, then is canonically isomorphic to
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Theorem
, Theorem:
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should probably link to Comparison of topologies or Strong dual space
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http://dbpedia.org/resource/Category:Functional_analysis +
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http://en.wikipedia.org/wiki/Injective_tensor_product?oldid=1118598630&ns=0 +
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http://en.wikipedia.org/wiki/Injective_tensor_product +
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| rdfs:comment |
In mathematics, the injective tensor produ … 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 funal/complex-valued functions to -valued fun
, Das injektive Tensorprodukt ist eine Erwei … 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.injektive Tensorprodukt natürliche Wahlen.
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| rdfs:label |
Injektives Tensorprodukt
, Injective tensor product
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http://dbpedia.org/resource/Series_%28mathematics%29 +
, http://dbpedia.org/resource/Schwartz_space +
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