| dbo:abstract |
Das projektive 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 injektive Tensorprodukt und das hier zu behandelnde projektive Tensorprodukt natürliche Wahlen. Die Untersuchung des projektiven Tensorproduktes lokalkonvexer Räume geht auf Alexander Grothendieck zurück.Einige Resultate über Banachräume wurden zuvor von Robert Schatten erzielt. Zunächst wird der leichter zugängliche Fall der normierten Räume und Banachräume besprochen, anschließend wird auf die Verallgemeinerungen in der Theorie der lokalkonvexen Räume eingegangen. (de) The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) continuous is called the projective topology or the π-topology. When is endowed with this topology then it is denoted by and called the projective tensor product of and (en) |
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https://ncatlab.org/nlab/show/nuclear+space |
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dbr:Topological_embedding dbr:DF-space dbr:Integral_operator dbr:Null_sequence dbr:Nuclear_space dbr:Equicontinuous dbr:Fréchet_space dbr:Minkowski_functional dbr:Simple_tensor dbr:Totally_bounded_space dbr:Auxiliary_normed_spaces dbr:Topological_vector_space dbr:Topological_vector_spaces dbr:Topology_of_uniform_convergence dbr:Weak-*_topology dbr:Absolutely_convergent dbr:Alexander_Grothendieck dbr:Dual_system dbr:F-space dbr:Normable_space dbr:Directed_set dbr:Hilbert_space dbr:Tensor_product dbc:Functional_analysis dbr:Bilinear_map dbr:Injective_function dbr:Injective_tensor_product dbr:Metrizable_topological_vector_space dbr:Net_(mathematics) dbr:Open_map dbr:Reflexive_space dbr:Locally_convex dbr:Uncountable dbr:Strong_topology dbr:Algebraic_dual dbr:Balanced dbr:Banach_disk dbr:Coarsest_topology dbr:Semi-metrizable dbr:Semi-normable |
| dbp:mathStatement |
Let and be Hilbert spaces and endow with the trace-norm. When the space of compact linear operators is equipped with the operator norm then its dual is and its bidual is the space of all continuous linear operators (en) Let and be metrizable locally convex TVSs and let Then is the sum of an absolutely convergent series where and and are null sequences in and respectively. (en) The canonical embedding becomes an embedding of topological vector spaces when is given the projective 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) Let and be locally convex TVSs with nuclear. Assume that both and are Fréchet spaces or else that they are both DF-spaces. Then: # The strong dual of can be identified with ; # The bidual of can be identified with ; # If in addition is reflexive then is a reflexive space; # Every separately continuous bilinear form on is continuous; # The strong dual of can be identified with so in particular if is reflexive then so is (en) Let and be Fréchet spaces and let be a balanced open neighborhood of the origin in . Let be a compact subset of the convex balanced hull of There exists a compact subset of the unit ball in and sequences and contained in and respectively, converging to the origin such that for every there exists some such that (en) |
| dbp:name |
Theorem (en) |
| dbp:note |
Grothendieck (en) |
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dbt:Grothendieck_Produits_Tensoriels_Topologiques_et_Espaces_Nucléaires dbt:Annotated_link dbt:Cite_book dbt:Em dbt:Harv dbt:Main dbt:More_footnotes dbt:Reflist dbt:See_also dbt:Sfn dbt:Functional_Analysis 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 |
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dbc:Functional_analysis |
| rdf:type |
owl:Thing |
| rdfs:comment |
The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map (defined by sending to ) continuous is called the projective topology or the π-topology. When is endowed with this topology then it is denoted by and called the projective tensor product of and (en) Das projektive 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 injektive Tensorprodukt und das hier zu behandelnde projektive Tensorprodukt natürliche Wahlen. (de) |
| rdfs:label |
Projektives Tensorprodukt (de) Projective tensor product (en) |
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dbr:Auxiliary_normed_spaces |
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wikidata:Projective tensor product dbpedia-de:Projective tensor product https://global.dbpedia.org/id/7cEHd |
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wikipedia-en:Projective_tensor_product?oldid=1114476549&ns=0 |
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wikipedia-en:Projective_tensor_product |
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dbr:Schwartz_kernel_theorem dbr:DF-space dbr:Nuclear_operators_between_Banach_spaces dbr:Complete_topological_vector_space dbr:Nuclear_C*-algebra dbr:Nuclear_operator dbr:Nuclear_space dbr:Alexander_Grothendieck dbr:Distribution_(mathematics) dbr:Spaces_of_test_functions_and_distributions dbr:Injective_tensor_product dbr:Integral_linear_operator dbr:List_of_topologies |
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wikipedia-en:Projective_tensor_product |