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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

Attributes	Values
rdf:type	Thing
rdfs:label	Projektives Tensorprodukt (de) Projective tensor product (en)
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:seeAlso	Auxiliary normed spaces
name	Theorem (en)
dcterms:subject	Functional analysis
Wikipage page ID	63566617 (xsd:integer)
Wikipage revision ID	1114476549 (xsd:integer)
Link from a Wikipage to another Wikipage	Topological embedding DF-space Integral operator Null sequence Nuclear space Equicontinuous Fréchet space Minkowski functional Simple tensor Totally bounded space Auxiliary normed spaces Topological vector space Topological vector spaces Topology of uniform convergence Weak-* topology Absolutely convergent Alexander Grothendieck Dual system F-space Normable space Directed set Hilbert space Tensor product Functional analysis Bilinear map Injective function Injective tensor product Metrizable topological vector space Net (mathematics) Open map Reflexive space Locally convex Uncountable Strong topology Algebraic dual Balanced Banach disk Coarsest topology Semi-metrizable Semi-normable
Link from a Wikipage to an external page	https://ncatlab.org/nlab/show/nuclear+space
sameAs	Projective tensor product Projective tensor product Projective tensor product
dbp:wikiPageUsesTemplate	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
note	Grothendieck (en)
has 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)
math statement	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)
prov:wasDerivedFrom	wikipedia-en:Projective_tensor_product?oldid=1114476549&ns=0
page length (characters) of wiki page	50921 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Projective_tensor_product
is Link from a Wikipage to another Wikipage of	Schwartz kernel theorem DF-space Nuclear operators between Banach spaces Complete topological vector space Nuclear C*-algebra Nuclear operator Nuclear space Alexander Grothendieck Distribution (mathematics) Spaces of test functions and distributions Injective tensor product Integral linear operator List of topologies
is foaf:primaryTopic of	wikipedia-en:Projective_tensor_product

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