Banach space (original) (raw)
فضاء باناخ هو فضاء معياري كامل، مما يعني ان كل متتالية كوشي من عناصر هذا الفضاء تنتهي داخل الفضاء نفسه وهذا ما يجعل منه . سمي هذا الفضاء هكذا نسبة إلى عالم الرياضيات البولندي ستيفن باناخ.
| Property | Value | |
|---|---|---|
| dbo:abstract | فضاء باناخ هو فضاء معياري كامل، مما يعني ان كل متتالية كوشي من عناصر هذا الفضاء تنتهي داخل الفضاء نفسه وهذا ما يجعل منه . سمي هذا الفضاء هكذا نسبة إلى عالم الرياضيات البولندي ستيفن باناخ. (ar) En matemàtiques, un espai de Banach és un espai vectorial normat i complet. Pren el seu nom en el matemàtic Stefan Banach. (ca) Banachovy prostory jsou normované lineární prostory, které jsou navíc úplné. Jsou to jedny z ústředních objektů zkoumání funkcionální analýzy. Jsou pojmenovány podle Stefana Banacha, který je studoval. (cs) Στα μαθηματικά, και πιο συγκεκριμένα στη συναρτησιακή ανάλυση, ένας χώρος Μπάναχ (προφορά: /ˈbanax/,) είναι ένας πλήρης . Έτσι, ένας χώρος Μπάναχ είναι ένας διανυσματικός χώρος με μια μετρική που επιτρέπει τον υπολογισμό του μήκος του διανύσματος και της απόστασης μεταξύ των διανυσμάτων και είναι πλήρης, με την έννοια ότι μια Κωσύ ακολουθία διανυσμάτων πάντα συγκλίνει σε ένα καλά ορισμένο όριο που είναι μέσα στο χώρο. Οι χώροι Μπάναχ πήραν το όνομά τους από τον Πολωνό μαθηματικό Στέφαν Μπάναχ, ο οποίος εισήγαγε αυτή την έννοια και την μελέτησε συστηματικά από το 1920 έως το 1922, μαζί με τον Χανς Χαν και τον Έντουαρντ Χέλλυ. Οι χώροι Μπάναχ αρχικά αναπτύχθηκαν από την μελέτη των συναρτησιακών χώρων από τον Χίλμπερτ, τον Φρεσέτ, και τον Ριτζ νωρίτερα τον ίδιο αιώνα. Οι χώροι Μπάναχ παίζουν κεντρικό ρόλο στη συναρτησιακή ανάλυση. Σε άλλους τομείς της ανάλυσης, οι χώροι υπό μελέτη είναι συχνά χώροι Μπάναχ. (el) En analitiko, banaĥa spaco estas vektora spaco kun kompleta normo. (eo) In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space."Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. (en) Ein Banachraum (auch Banach-Raum, Banachscher Raum) ist in der Mathematik ein vollständiger normierter Vektorraum. Banachräume gehören zu den zentralen Studienobjekten der Funktionalanalysis. Insbesondere sind viele unendlichdimensionale Funktionenräume Banachräume. Sie sind nach dem Mathematiker Stefan Banach benannt, der sie 1920–1922 gemeinsam mit Hans Hahn und Eduard Helly vorstellte. (de) En mathématiques, plus particulièrement en analyse fonctionnelle, on appelle espace de Banach un espace vectoriel normé sur un sous-corps K de ℂ (en général, K = ℝ ou ℂ), complet pour la distance issue de sa norme.Comme la topologie induite par sa distance est compatible avec sa structure d’espace vectoriel, c’est un espace vectoriel topologique. Les espaces de Banach possèdent de nombreuses propriétés qui font d'eux un outil essentiel pour l'analyse fonctionnelle. Ils doivent leur nom au mathématicien polonais Stefan Banach. (fr) En matemáticas, un espacio de Banach, llamado así en honor del matemático polaco, Stefan Banach, es uno de los objetos de estudio más importantes en análisis funcional. Los espacios de Banach son un concepto importante en el análisis matemático y se utilizan en una amplia variedad de aplicaciones, como la teoría de operadores lineales y la teoría de funciones de variable compleja. Un espacio de Banach es típicamente un espacio de funciones de dimensión infinita. (es) Dalam matematika, lebih khusus lagi dalam analisis fungsional, Ruang Banach adalah ruang vektor bernorma lengkap. Jadi, ruang Banach adalah ruang vektor dengan metrik yang memungkinkan penghitungan panjang vektor dan jarak antara vektor dan lengkap dalam arti bahwa vektor selalu konvergen ke limit yang didefinisikan dengan baik yang ada di dalam ruang. Spasi Banach dinamai menurut ahli matematika Polandia Stefan Banach, yang memperkenalkan konsep ini dan mempelajarinya secara sistematis pada 1920–1922 bersama dengan dan . adalah orang pertama yang menggunakan istilah "ruang Banach" dan Banach pada gilirannya kemudian menciptakan istilah "."Ruang Banach awalnya tumbuh dari studi oleh Hilbert, , and di awal abad ini. Ruang Banach memainkan peran sentral dalam analisis fungsional. Di bidang lain , ruang yang diteliti sering kali merupakan ruang Banach. (in) 数学におけるバナッハ空間（バナッハくうかん、英: Banach space; バナハ空間）は、完備なノルム空間、即ちノルム付けられた線型空間であって、そのノルムが定めるが完備であるものを言う。 解析学に現れる多くの函数空間、例えば連続函数の空間（の空間）、 Lp-空間と呼ばれるの空間、ハーディ空間と呼ばれる正則函数の空間などはバナッハ空間を成す。これらはもっとも広く用いられる位相線型空間であり、これらの位相はノルムから規定されるものになっている。 バナッハ空間の名称は、この概念をハーンとヘリーらと共に1920-1922年に導入したポーランドの数学者ステファン・バナフに因む。 (ja) In de functionaalanalyse, een deelgebied van de wiskunde, is een banachruimte een reële of complexe vectorruimte die voorzien is van een norm en die ten aanzien van die norm volledig is. Banachruimten zijn de meestgebruikte topologische vectorruimten; hun topologie wordt gegeven door een norm. Veel van de oneindigdimensionale functieruimten die in de analyse worden bestudeerd, zijn banachruimten. Daaronder zijn ook ruimten van continue functies (continue functies op een compacte hausdorff-ruimte), ruimten van lebesgue-integreerbare functies, die bekendstaan als Lp-ruimtes en ruimten van holomorfe functies, die bekendstaan als hardy-ruimten. Banachruimten zijn genoemd naar de Poolse wiskundige Stefan Banach, die zo rond 1920-1922 dit begrip introduceerde, samen met Hans Hahn en Eduard Helly. (nl) In matematica uno spazio di Banach è uno spazio normato completo rispetto alla metrica indotta dalla norma. Gli spazi di Banach furono studiati inizialmente da Stefan Banach, da cui hanno preso il nome, e costituiscono un oggetto di studio molto importante dell'analisi funzionale: molti spazi di funzioni sono, infatti, spazi di Banach. (it) 함수해석학에서 바나흐 공간(Banach空間, 영어: Banach space)은 완비 노름 공간이다. 함수해석학의 주요 연구 대상 가운데 하나다. 스테판 바나흐의 이름을 땄다. (ko) Przestrzeń Banacha – przestrzeń unormowana (z normą | | · |
| dbo:thumbnail | wiki-commons:Special:FilePath/Tensor-diagramB.jpg?width=300 | |
| dbo:wikiPageExternalLink | https://books.google.com/books%3Fid=7n9sAAAAMAAJ https://archive.org/details/sequencesseriesi0000dies/page/ | |
| dbo:wikiPageID | 3989 (xsd:integer) | |
| dbo:wikiPageLength | 102064 (xsd:nonNegativeInteger) | |
| dbo:wikiPageRevisionID | 1122629643 (xsd:integer) | |
| dbo:wikiPageWikiLink | dbr:Product_topology dbr:Robert_Phelps dbr:Metrizable_space dbr:Montel_space dbr:Norm_induced_metric dbr:Bifunctor dbr:Bounded_set_(topological_vector_space) dbr:David_Hilbert dbr:Algebra_over_a_field dbr:Antilinear_map dbr:Approximation_property dbc:Banach_spaces dbr:Holomorphic_function dbr:Homeomorphic dbr:Per_Enflo dbr:Riesz_representation_theorem dbr:Uniform_boundedness_principle dbr:Uniform_space dbr:Vector_space dbr:Dvoretzky's_theorem dbr:Infinite-dimensional_optimization dbr:L-semi-inner_product dbr:Limit_of_a_sequence dbr:James'_theorem dbc:Science_and_technology_in_Poland dbr:Compact_space dbr:Complemented_subspace dbr:Complete_topological_vector_space dbr:Completion_(metric_space) dbr:Complex_number dbr:Continuous_dual_space dbr:Continuous_function dbr:Continuous_linear_functional dbr:Continuous_linear_operator dbr:Convex_set dbr:Convolution dbr:Countable_set dbr:Anderson–Kadec_theorem dbr:Mathematics dbr:Maurice_René_Fréchet dbr:Gelfand_representation dbr:Gelfand–Mazur_theorem dbr:Gelfand–Naimark_theorem dbr:Norm_(mathematics) dbr:Operator_norm dbr:Order_topology dbr:Separable_space dbr:Quasi-derivative dbr:Closed_graph_theorem dbr:Closure_(topology) dbr:Frigyes_Riesz dbr:Fréchet_space dbr:Fréchet–Urysohn_space dbr:Gateaux_derivative dbr:Gilles_Pisier dbr:Bounded_operator dbr:Convex_function dbr:Convex_hull dbr:Coproduct dbr:Total_derivative dbr:Milman–Pettis_theorem dbr:Hom_space dbr:Linear_operator dbr:Locally_convex_topological_vector_space dbr:Lp_space dbr:Simple_tensor dbr:Sobolev_space dbr:Stefan_Banach dbr:Sublinear_function dbr:Closed_set dbr:Compact_operator dbr:Comparison_of_topologies dbr:Complete_metric_space dbr:Complex_conjugate dbr:Dense_set dbr:Fréchet_derivative dbr:Function_space dbr:Functional_analysis dbr:Hardy_space dbr:Harmonic_analysis dbr:Schauder_basis dbr:Spectrum_of_a_C*-algebra dbr:Unit_sphere dbr:Mazur–Ulam_theorem dbr:Balanced_set dbr:Ball_(mathematics) dbr:Banach_space dbr:Banach–Alaoglu_theorem dbr:Banach–Mazur_theorem dbr:Timothy_Gowers dbr:Topological_vector_space dbr:Topology dbr:Weak_topology dbr:Disk_algebra dbr:Dual_norm dbr:Haar_wavelet dbr:Hausdorff_space dbr:Linear_map dbr:Linear_span dbr:Linear_subspace dbr:Locally_compact_space dbr:Uniformly_convex_space dbr:Absolute_value dbr:Affine_space dbr:Alexander_Grothendieck dbr:Dual_space dbr:Eduard_Helly dbr:Equivalent_norm dbr:Errett_Bishop dbr:Extreme_point dbr:F-space dbr:Field_(mathematics) dbr:Banach_algebra dbr:Banach–Mazur_compactum dbr:Banach–Stone_theorem dbc:Normed_spaces dbr:Nicole_Tomczak-Jaegermann dbr:Normable_space dbr:Normed_space dbr:Parseval's_theorem dbr:Partial_differential_equation dbr:Cauchy_sequence dbr:Directional_derivative dbr:Forgetful_functor dbr:Goldstine_theorem dbr:Kolmogorov's_normability_criterion dbr:Pointwise_convergence dbr:Linear_functional dbr:Product_(category_theory) dbr:Projection_(linear_algebra) dbr:Radon_measure dbr:Range_of_a_function dbr:Hahn–Banach_theorem dbr:Hans_Hahn_(mathematician) dbr:Hilbert_space dbr:Involution_(mathematics) dbr:Isometry dbr:Israel_Gelfand dbr:Baire_category_theorem dbr:Baire_function dbr:Baire_space dbr:Tensor_product dbr:Hyperplane dbr:Hamel_basis dbr:Surjective dbr:Absolute_convergence dbr:L2-space dbr:LF-space dbr:Eberlein–Šmulian_theorem dbr:Homeomorphism dbr:Topological_tensor_product dbr:Dirac_measure dbr:Distribution_(mathematics) dbr:C*-algebra dbr:Polarization_identity dbr:Spaces_of_test_functions_and_distributions dbr:Continuous_function_(topology) dbr:Ground_field dbr:Inner_product dbr:Inner_product_space dbr:Interior_(topology) dbr:Metric_space dbr:Metric_topology dbr:Metrizable_topological_vector_space dbr:Natural_number dbr:Neighbourhood_basis dbr:Neighbourhood_system dbr:Net_(mathematics) dbr:Open_mapping_theorem_(functional_analysis) dbr:Open_set dbr:Ordinal_number dbr:Real_number dbr:Real_part dbr:Reflexive_space dbr:Seminorm dbr:Sequence_space_(mathematics) dbr:Series_(mathematics) dbr:Tychonoff's_theorem dbr:Uniformity_(topology) dbr:Locally_convex dbr:Wiener_algebra dbr:Linear_transformation dbr:Metric_(mathematics) dbr:Universal_property dbr:Probability_measure dbr:Schur's_property dbr:Set-theoretic_topology dbr:Injective dbr:Normed_vector_space dbr:Uniformly_continuous dbr:Strong_dual_space dbr:Hahn–Banach_separation_theorem dbr:Translation_invariant dbr:Translation_invariant_topology dbr:Parallelogram_identity dbr:Analysis_(mathematics) dbr:Finer_topology dbr:Finite-dimensional_vector_space dbr:Hull-kernel_topology dbr:Continuity_at_a_point dbr:Hermitian_symmetry dbr:Product_space dbr:Real_and_imaginary_parts_of_a_linear_functional dbr:Short_map dbr:Banach–Mazur_distance dbr:Closed_convex_hull dbr:Complete_metric dbr:Bounded_linear_functional dbr:Bounded_linear_operator dbr:Space_of_real_sequences dbr:Neighbourhood_(topology) dbr:Pre-Hilbert_space dbr:Subadditive_function dbr:Surjection dbr:File:Tensor-diagramB.jpg | |
| dbp:id | p/b015190 (en) | |
| dbp:mathStatement | Let be a linear mapping between Banach spaces. The graph of is closed in if and only if is continuous. (en) If a Banach space is the internal direct sum of closed subspaces then is isomorphic to (en) Let be a normed vector space. Then the closed unit ball of the dual space is compact in the weak* topology. (en) If and are compact Hausdorff spaces and if and are isometrically isomorphic, then the topological spaces and are homeomorphic. (en) For every measure the space is weakly sequentially complete. (en) Let be a separable Banach space. The following are equivalent: *The space does not contain a closed subspace isomorphic to *Every element of the bidual is the weak*-limit of a sequence in (en) A Banach space isomorphic to all its infinite-dimensional closed subspaces is isomorphic to a separable Hilbert space. (en) for all (en) Let be a normed space. If is separable, then is separable. (en) Suppose that and are Banach spaces and that Suppose further that the range of is closed in Then is isomorphic to (en) Let be a bounded sequence in a Banach space. Either has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of (en) Let be a Banach space and be a normed vector space. Suppose that is a collection of continuous linear operators from to The uniform boundedness principle states that if for all in we have then (en) Let be an uncountable compact metric space. Then is isomorphic to (en) Let be a vector space over the field Let further * be a linear subspace, * be a sublinear function and * be a linear functional so that for all Then, there exists a linear functional so that (en) For a Banach space the following two properties are equivalent: * is reflexive. * for all in there exists with so that (en) A set in a Banach space is relatively weakly compact if and only if every sequence in has a weakly convergent subsequence. (en) Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism. (en) Let and be Banach spaces and be a surjective continuous linear operator, then is an open map. (en) Let be a reflexive Banach space. Then is separable if and only if is separable. (en) | |
| dbp:name | dbr:Uniform_boundedness_principle dbr:Closed_graph_theorem dbr:Banach–Alaoglu_theorem dbr:Banach–Stone_theorem dbr:Eberlein–Šmulian_theorem dbr:Open_mapping_theorem_(functional_analysis) Theorem (en) Corollary (en) Hahn–Banach theorem (en) James' Theorem (en) Parallelogram identity (en) The First Isomorphism Theorem for Banach spaces (en) | |
| dbp:title | Banach space (en) | |
| dbp:wikiPageUsesTemplate | dbt:Springer dbt:Banach_spaces dbt:Annotated_link dbt:Authority_control dbt:Citation dbt:Clear dbt:Commons_category dbt:Em dbt:Harvtxt dbt:Main dbt:Math dbt:MathWorld dbt:Reflist dbt:See_also dbt:Sfn dbt:Short_description dbt:Visible_anchor dbt:IPA-pl dbt:Bourbaki_Topological_Vector_Spaces dbt:Conway_A_Course_in_Functional_Analysis dbt:Edwards_Functional_Analysis_Theory_and_Applications dbt:Functional_Analysis dbt:Grothendieck_Topological_Vector_Spaces dbt:Khaleelulla_Counterexamples_in_Topological_Vector_Spaces dbt:Math_theorem dbt:Narici_Beckenstein_Topological_Vector_Spaces dbt:Robertson_Topological_Vector_Spaces dbt:Rudin_Walter_Functional_Analysis dbt:Schaefer_Wolff_Topological_Vector_Spaces dbt:Swartz_An_Introduction_to_Functional_Analysis dbt:Trèves_François_Topological_vector_spaces,_distributions_and_kernels dbt:Wilansky_Modern_Methods_in_Topological_Vector_Spaces dbt:Bachman_Narici_Functional_Analysis_2nd_Edition dbt:Banach_Théorie_des_Opérations_Linéaires dbt:ListOfBanachSpaces dbt:Aliprantis_Border_Infinite_Dimensional...is_A_Hitchhiker's_Guide_Third_Edition | |
| dcterms:subject | dbc:Banach_spaces dbc:Science_and_technology_in_Poland dbc:Normed_spaces | |
| gold:hypernym | dbr:Space | |
| rdf:type | owl:Thing yago:WikicatBanachSpaces yago:WikicatNormedSpaces yago:Abstraction100002137 yago:Attribute100024264 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 yago:Space100028651 yago:WikicatPropertiesOfTopologicalSpaces | |
| rdfs:comment | فضاء باناخ هو فضاء معياري كامل، مما يعني ان كل متتالية كوشي من عناصر هذا الفضاء تنتهي داخل الفضاء نفسه وهذا ما يجعل منه . سمي هذا الفضاء هكذا نسبة إلى عالم الرياضيات البولندي ستيفن باناخ. (ar) En matemàtiques, un espai de Banach és un espai vectorial normat i complet. Pren el seu nom en el matemàtic Stefan Banach. (ca) Banachovy prostory jsou normované lineární prostory, které jsou navíc úplné. Jsou to jedny z ústředních objektů zkoumání funkcionální analýzy. Jsou pojmenovány podle Stefana Banacha, který je studoval. (cs) En analitiko, banaĥa spaco estas vektora spaco kun kompleta normo. (eo) Ein Banachraum (auch Banach-Raum, Banachscher Raum) ist in der Mathematik ein vollständiger normierter Vektorraum. Banachräume gehören zu den zentralen Studienobjekten der Funktionalanalysis. Insbesondere sind viele unendlichdimensionale Funktionenräume Banachräume. Sie sind nach dem Mathematiker Stefan Banach benannt, der sie 1920–1922 gemeinsam mit Hans Hahn und Eduard Helly vorstellte. (de) En mathématiques, plus particulièrement en analyse fonctionnelle, on appelle espace de Banach un espace vectoriel normé sur un sous-corps K de ℂ (en général, K = ℝ ou ℂ), complet pour la distance issue de sa norme.Comme la topologie induite par sa distance est compatible avec sa structure d’espace vectoriel, c’est un espace vectoriel topologique. Les espaces de Banach possèdent de nombreuses propriétés qui font d'eux un outil essentiel pour l'analyse fonctionnelle. Ils doivent leur nom au mathématicien polonais Stefan Banach. (fr) En matemáticas, un espacio de Banach, llamado así en honor del matemático polaco, Stefan Banach, es uno de los objetos de estudio más importantes en análisis funcional. Los espacios de Banach son un concepto importante en el análisis matemático y se utilizan en una amplia variedad de aplicaciones, como la teoría de operadores lineales y la teoría de funciones de variable compleja. Un espacio de Banach es típicamente un espacio de funciones de dimensión infinita. (es) 数学におけるバナッハ空間（バナッハくうかん、英: Banach space; バナハ空間）は、完備なノルム空間、即ちノルム付けられた線型空間であって、そのノルムが定めるが完備であるものを言う。 解析学に現れる多くの函数空間、例えば連続函数の空間（の空間）、 Lp-空間と呼ばれるの空間、ハーディ空間と呼ばれる正則函数の空間などはバナッハ空間を成す。これらはもっとも広く用いられる位相線型空間であり、これらの位相はノルムから規定されるものになっている。 バナッハ空間の名称は、この概念をハーンとヘリーらと共に1920-1922年に導入したポーランドの数学者ステファン・バナフに因む。 (ja) In matematica uno spazio di Banach è uno spazio normato completo rispetto alla metrica indotta dalla norma. Gli spazi di Banach furono studiati inizialmente da Stefan Banach, da cui hanno preso il nome, e costituiscono un oggetto di studio molto importante dell'analisi funzionale: molti spazi di funzioni sono, infatti, spazi di Banach. (it) 함수해석학에서 바나흐 공간(Banach空間, 영어: Banach space)은 완비 노름 공간이다. 함수해석학의 주요 연구 대상 가운데 하나다. 스테판 바나흐의 이름을 땄다. (ko) Em matemática, um espaço de Banach, é um espaço vectorial normado completo. Deve seu nome ao matemático polaco Stefan Banach(1892-1945), o qual contribuiu para a teoria das séries ortogonais e inovações na teoria de medida e integração, sendo a sua contribuição mais importante na análise funcional. (pt) Banachrum är i matematiken i allmänhet oändligdimensionella rum av funktioner. Banachrum är uppkallat efter Stefan Banach som studerade dem, ett av de centrala objekten inom funktionalanalys. (sv) Ба́нахово пространство — нормированное векторное пространство, полное по метрике, порождённой нормой. Основной объект изучения функционального анализа. Названо по имени польского математика Стефана Банаха (1892—1945), который с 1922 года систематически изучал эти пространства. (ru) 在數學裡，尤其是在泛函分析之中，巴拿赫空間（英語：Banach space）是一個完備賦範向量空間。更精確地說，巴拿赫空間是一個具有範數並對此範數完備的向量空間。其完备性体现在，空间内任意向量的柯西序列总是收敛到一个良定义的位于空间内部的極限。 巴拿赫空間有兩種常見的類型：「實巴拿赫空間」及「複巴拿赫空間」，分別是指將巴拿赫空間的向量空間定義於由實數或複數組成的域之上。許多在數學分析中學到的無限維函數空間都是巴拿赫空間，包括由連續函數（）組成的空間、由勒貝格可積函數組成的Lp空間及由全純函數組成的哈代空間。上述空間是拓撲向量空間中最常見的類型，這些空間的拓撲都自來其範數。 巴拿赫空間是以波蘭數學家斯特凡·巴拿赫的名字來命名，他和漢斯·哈恩及爱德华·赫利於1920-1922年提出此空間。 (zh) Банахів простір — повний нормований векторний простір. Тобто векторний простір над полем дійсних або комплексних чисел з нормою такою, що кожна фундаментальна послідовність є збіжною до елементу з за метрикою Центральний об'єкт у функціональному аналізі. Названий на честь Стефана Банаха. (uk) Στα μαθηματικά, και πιο συγκεκριμένα στη συναρτησιακή ανάλυση, ένας χώρος Μπάναχ (προφορά: /ˈbanax/,) είναι ένας πλήρης . Έτσι, ένας χώρος Μπάναχ είναι ένας διανυσματικός χώρος με μια μετρική που επιτρέπει τον υπολογισμό του μήκος του διανύσματος και της απόστασης μεταξύ των διανυσμάτων και είναι πλήρης, με την έννοια ότι μια Κωσύ ακολουθία διανυσμάτων πάντα συγκλίνει σε ένα καλά ορισμένο όριο που είναι μέσα στο χώρο. (el) In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. (en) Dalam matematika, lebih khusus lagi dalam analisis fungsional, Ruang Banach adalah ruang vektor bernorma lengkap. Jadi, ruang Banach adalah ruang vektor dengan metrik yang memungkinkan penghitungan panjang vektor dan jarak antara vektor dan lengkap dalam arti bahwa vektor selalu konvergen ke limit yang didefinisikan dengan baik yang ada di dalam ruang. (in) In de functionaalanalyse, een deelgebied van de wiskunde, is een banachruimte een reële of complexe vectorruimte die voorzien is van een norm en die ten aanzien van die norm volledig is. Banachruimten zijn de meestgebruikte topologische vectorruimten; hun topologie wordt gegeven door een norm. Banachruimten zijn genoemd naar de Poolse wiskundige Stefan Banach, die zo rond 1920-1922 dit begrip introduceerde, samen met Hans Hahn en Eduard Helly. (nl) Przestrzeń Banacha – przestrzeń unormowana (z normą | | · |
| rdfs:label | Banach space (en) فضاء باناخ (ar) Espai de Banach (ca) Banachův prostor (cs) Banachraum (de) Χώρος Μπάναχ (el) Banaĥa spaco (eo) Espacio de Banach (es) Espace de Banach (fr) Ruang Banach (in) Spazio di Banach (it) 바나흐 공간 (ko) バナッハ空間 (ja) Banachruimte (nl) Przestrzeń Banacha (pl) Espaço de Banach (pt) Банахово пространство (ru) Банахів простір (uk) Banachrum (sv) 巴拿赫空间 (zh) | |
| rdfs:seeAlso | dbr:Bidual | |
| owl:sameAs | freebase:Banach space yago-res:Banach space wikidata:Banach space dbpedia-ar:Banach space http://azb.dbpedia.org/resource/باناخ_اۇزایی dbpedia-bg:Banach space dbpedia-ca:Banach space http://ckb.dbpedia.org/resource/بۆشاییی_باناخ dbpedia-cs:Banach space dbpedia-de:Banach space dbpedia-el:Banach space dbpedia-eo:Banach space dbpedia-es:Banach space dbpedia-et:Banach space dbpedia-fa:Banach space dbpedia-fi:Banach space dbpedia-fr:Banach space dbpedia-he:Banach space dbpedia-hu:Banach space dbpedia-id:Banach space dbpedia-is:Banach space dbpedia-it:Banach space dbpedia-ja:Banach space dbpedia-kk:Banach space dbpedia-ko:Banach space dbpedia-nl:Banach space dbpedia-nn:Banach space dbpedia-no:Banach space dbpedia-pl:Banach space dbpedia-pms:Banach space dbpedia-pt:Banach space dbpedia-ro:Banach space dbpedia-ru:Banach space dbpedia-sk:Banach space dbpedia-sv:Banach space dbpedia-uk:Banach space dbpedia-vi:Banach space dbpedia-zh:Banach space https://global.dbpedia.org/id/rd6W | |
| prov:wasDerivedFrom | wikipedia-en:Banach_space?oldid=1122629643&ns=0 | |
| foaf:depiction | wiki-commons:Special:FilePath/Tensor-diagramB.jpg | |
| foaf:isPrimaryTopicOf | wikipedia-en:Banach_space | |
| is dbo:academicDiscipline of | dbr:Stevo_Todorčević dbr:Mikhail_Kadets | |
| is dbo:knownFor of | dbr:Robert_Phelps dbr:Stefan_Banach dbr:Jean_Bourgain dbr:Peter_Orno | |
| is dbo:wikiPageDisambiguates of | dbr:Banach_(disambiguation) | |
| is dbo:wikiPageRedirects of | dbr:Banach_spaces dbr:Dual_banach_space dbr:Banach_Space dbr:Linear_Algebra/Banach_Spaces dbr:Banach_Spaces dbr:Banach_function_space dbr:Banach_norm dbr:Complete_norm dbr:Complete_normed_space dbr:Complete_normed_vector_space | |
| is dbo:wikiPageWikiLink of | dbr:Product_rule dbr:Robert_Phelps dbr:Robert_Schatten dbr:Schilder's_theorem dbr:Entanglement_witness dbr:Envelope_(category_theory) dbr:Enveloping_von_Neumann_algebra dbr:List_of_functional_analysis_topics dbr:Modes_of_convergence dbr:Pauline_Mellon dbr:Mellin_inversion_theorem dbr:Method_of_continuity dbr:Metric_differential dbr:Monotonic_function dbr:Montel_space dbr:Strict_differentiability dbr:Variational_inequality dbr:Partial_trace dbr:Representation_theory dbr:Smith_space dbr:Barrelled_space dbr:Basis_(linear_algebra) dbr:Basis_function dbr:Bergman_space dbr:Besov_space dbr:Boris_Tsirelson dbr:Bornological_space dbr:Bounded_inverse_theorem dbr:Bounded_variation dbr:Bra–ket_notation dbr:Derivative dbr:Algebra_over_a_field dbr:Algebraic_structure dbr:Approximation_property dbr:Holomorphic_function dbr:Joram_Lindenstrauss dbr:Jordan_matrix dbr:Joseph_Diestel dbr:Per_Enflo dbr:Renato_Caccioppoli dbr:Resolvent_formalism dbr:Richard_S._Hamilton dbr:Riemann–Liouville_integral dbr:Riesz's_lemma dbr:Riesz_representation_theorem dbr:Riesz–Fischer_theorem dbr:Charles_Read_(mathematician) dbr:Cyclic_vector dbr:Cylinder_set_measure dbr:Cylindrical_σ-algebra dbr:Unbounded_operator dbr:Uniform_boundedness_principle dbr:Valuation_(geometry) dbr:Vector-valued_Hahn–Banach_theorems dbr:Vector_measure dbr:Vector_space dbr:Vitali_Milman dbr:Decomposition_of_spectrum_(functional_analysis) dbr:Dominated_convergence_theorem dbr:Doris_Stockton dbr:Dunford–Pettis_property dbr:Dynamical_system dbr:Earle–Hamilton_fixed-point_theorem dbr:Inertial_manifold dbr:Infinite-dimensional_Lebesgue_measure dbr:Infinite-dimensional_holomorphy dbr:Information-based_complexity dbr:Integral_transform dbr:Integration_by_parts_operator dbr:Interpolation dbr:Invariance_of_domain dbr:Invariant_subspace dbr:Invariant_subspace_problem dbr:Inverse_function_theorem dbr:Inverse_problem dbr:James'_space dbr:James's_theorem dbr:Kuratowski_embedding dbr:Kōmura's_theorem dbr:LB-space dbr:Operator_algebra dbr:Pre-abelian_category dbr:Real_analysis dbr:Lethargy_theorem dbr:Lie's_third_theorem dbr:Lie_group dbr:Limiting_absorption_principle dbr:List_of_important_publications_in_mathematics dbr:Weak_formulation dbr:Nuclear_operators_between_Banach_spaces dbr:Reduced_derivative dbr:Pseudo-monotone_operator dbr:Time-scale_calculus dbr:Transversality_(mathematics) dbr:Timeline_of_Polish_science_and_technology dbr:Compact-open_topology dbr:Compact_space dbr:Complemented_subspace dbr:Complete_topological_vector_space dbr:Continuous_linear_operator dbr:Convenient_vector_space dbr:Convolution dbr:Cornelia_Druțu dbr:Analytic_function dbr:Analytic_semigroup dbr:Anderson–Kadec_theorem dbr:Essential_spectrum dbr:Gaussian_measure dbr:Gauss–Kuzmin–Wirsing_operator dbr:Generalizations_of_the_derivative dbr:Generalized_Fourier_series dbr:Generic_programming dbr:Geodesic_bicombing dbr:Geometry_of_numbers dbr:Lp_sum dbr:Norm_(mathematics) dbr:Nuclear_space dbr:Operator_(mathematics) dbr:Ornstein–Uhlenbeck_operator dbr:Modulus_and_characteristic_of_convexity dbr:Morera's_theorem dbr:Vitali–Hahn–Saks_theorem dbr:Signed_measure dbr:Uniform_absolute-convergence dbr:Uniformly_smooth_space dbr:Separable_space dbr:Quasi-abelian_category dbr:Quasi-derivative dbr:Quasi-invariant_measure dbr:Claude_Ambrose_Rogers dbr:Closed_graph_property dbr:Closed_graph_theorem dbr:Closed_graph_theorem_(functional_analysis) dbr:Edward_George_Effros dbr:Ehrling's_lemma dbr:Eigenvalues_and_eigenvectors dbr:Equicontinuity dbr:Fredholm_determinant dbr:Fredholm_solvability dbr:Frobenius_theorem_(differential_topology) dbr:Fréchet_space dbr:Gateaux_derivative dbr:Gilles_Pisier dbr:Gradient dbr:Mollifier dbr:Multidimensional_Chebyshev's_inequality dbr:Concentration_dimension dbr:Concentration_of_measure dbr:Condition_number dbr:Connected_space dbr:Convergence_of_Fourier_series dbr:Convex_hull dbr:Coorbit_theory dbr:Equivalent_definitions_of_mathematical_structures dbr:Ergodicity dbr:Lagrange_multipliers_on_Banach_spaces dbr:Milman–Pettis_theorem dbr:Pushforward_measure dbr:Strictly_singular_operator dbr:Structure_theorem_for_Gaussian_measures dbr:Operator_ideal dbr:Operator_space dbr:Operator_topologies dbr:Orlicz_sequence_space dbr:Orlicz_space dbr:Bernard_Maurey dbr:Linear_algebra dbr:Linear_approximation dbr:Linear_form dbr:Lipschitz_continuity dbr:Locally_convex_topological_vector_space dbr:Louis_Nirenberg dbr:Lp_space dbr:Bochner_integral dbr:Bochner_measurable_function dbr:Bochner_space dbr:Calabi_conjecture dbr:Calkin_algebra dbr:Cholesky_decomposition dbr:Choquet_theory dbr:Sobolev_space dbr:Stanisław_Mazur dbr:Stefan_Banach dbr:Steffensen's_method dbr:Stevo_Todorčević dbr:Stone–Weierstrass_theorem dbr:Stone–Čech_compactification dbr:Closed_range_theorem dbr:Compact_embedding dbr:Compact_operator_on_Hilbert_space dbr:Complete_metric_space dbr:Complex_geodesic dbr:Complex_measure dbr:Composition_operator dbr:Computability_in_Analysis_and_Physics dbr:Delta-convergence dbr:Densely_defined_operator dbr:Yuri_Petunin dbr:Faà_di_Bruno's_formula dbr:Feller_process dbr:Fraňková–Helly_selection_theorem dbr:Fréchet_derivative dbr:Function_space dbr:Functional_analysis dbr:Hardy_space dbr:Helly's_selection_theorem dbr:Koszul_complex dbr:Krein–Rutman_theorem dbr:Kuiper's_theorem dbr:Leray–Schauder_degree dbr:Schauder_basis dbr:Mackey_space dbr:Müntz–Szász_theorem dbr:Orbital_stability dbr:Pseudospectrum dbr:Space_(mathematics) dbr:Spectral_theory dbr:Spectrum_(functional_analysis) dbr:Spectrum_of_a_C*-algebra dbr:Stone's_theorem_on_one-parameter_unitary_groups dbr:Strictly_convex_space dbr:Zonal_spherical_function dbr:Markushevich_basis dbr:Matrix_completion dbr:Matroid dbr:Mazur's_lemma dbr:Measure_of_non-compactness dbr:Michael_selection_theorem dbr:Totally_bounded_space dbr:Auxiliary_normed_space dbr:Balanced_set dbr:Banach_lattice dbr:Banach_space dbr:Banach_spaces dbr:Banach–Alaoglu_theorem dbr:Banach–Mazur_theorem dbr:Cauchy–Schwarz_inequality dbr:Tight_span dbr:Topological_group dbr:Topological_vector_space dbr:Wald's_equation dbr:Weak_topology dbr:Dissipative_operator dbr:Distinguished_space dbr:Distortion_problem dbr:Dixmier-Ng_Theorem dbr:Dual_norm dbr:Girth_(functional_analysis) dbr:Haar_wavelet dbr:Hadamard_derivative dbr:Hadamard_three-lines_theorem dbr:Heine–Borel_theorem dbr:Juliusz_Schauder dbr:Józef_Schreier dbr:K-space_(functional_analysis) dbr:Landau–Kolmogorov_inequality dbr:Laplace–Stieltjes_transform dbr:Linear_span dbr:Lions–Magenes_lemma dbr:List_of_Banach_spaces dbr:Radon–Riesz_property dbr:Representation_theory_of_finite_groups dbr:Schur_decomposition dbr:Radonifying_function dbr:Singular_value dbr:Uniformly_convex_space dbr:Tsirelson_space dbr:Alexandra_Bellow dbr:Alexandru_Ghika dbr:Amenable_group dbr:Analytic_function_of_a_matrix dbr:Dagmar_R._Henney dbr:Dual_banach_space dbr:Dual_space dbr:Dual_system dbr:Errett_Bishop dbr:Euclidean_vector dbr:Extreme_point dbr:F-space dbr:F._Riesz's_theorem dbr:Fields_Medal dbr:Filters_in_topology dbr:Fourier_transform dbr:Frame_of_reference dbr:Banach_algebra dbr:Banach_bundle dbr:Banach_bundle_(non-commutative_geometry) dbr:Banach_limit dbr:Banach_measure dbr:Banach–Stone_theorem dbr:Barrier_cone dbr:Brouwer_fixed-point_theorem dbr:Browder_fixed-point_theorem dbr:Browder–Minty_theorem dbr:Nicole_Tomczak-Jaegermann dbr:Norbert_Wiener dbr:Null_set dbr:Carathéodory_metric dbr:Cauchy_sequence dbr:Daniell_integral | |
| is dbp:knownFor of | dbr:Stefan_Banach dbr:Nicole_Tomczak-Jaegermann | |
| is rdfs:seeAlso of | dbr:Geometry_of_numbers dbr:Gilles_Pisier dbr:Strong_dual_space | |
| is foaf:primaryTopic of | wikipedia-en:Banach_space |
