Topological vector space (original) (raw)
فضاء هيلبرت و باناخ مثالان معروفان للفضاءات المتجهية الطوبولوجية.
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| dbo:abstract | En matemàtiques, un espai vectorial topològic és una estructura bàsica que combina l'estructura algebraica d'un espai vectorial amb una estructura topològica.El cos subjacent d'un espai vectorial topològic és un cos topològic, que en les aplicacions acostuma a ser el cos dels nombres reals R o el dels nombres complexos C. Els espais vectorials topològics són eines fonamentals en anàlisi funcional. En aquest camp els elements dels espais vectorials topològics són típicament funcions definides en certs espais topològics, o operadors lineals entre altres espais vectorials topològics, i la topologia de l'espai és definida sovint per tal de captar una noció particular de convergència de successions de funcions. Alguns tipus particulars molt importants d'espais vectorials topològics són els espais de Banach i els espais de Hilbert. (ca) فضاء هيلبرت و باناخ مثالان معروفان للفضاءات المتجهية الطوبولوجية. (ar) Ein topologischer Vektorraum ist ein Vektorraum, auf dem neben seiner algebraischen auch noch eine damit verträgliche topologische Struktur definiert ist. (de) En analitiko, topologia vektora spaco estas vektora spaco ekipita per topologio kiu kongruas kun la vektorspaca strukturo (t.e. adicio kaj skalara multipliko estas kontinuaj). (eo) Un espacio vectorial topológico es un espacio de puntos que aúna la estructura típica de un espacio vectorial convencional y un espacio topológico, es decir, es un espacio vectorial sobre el que se ha definido una estructura topológica. Probablemente los ejemplos más sencillos son el plano euclídeo y el espacio euclídeo en los que la topología se define mediante la distancia euclídea. El conjunto de bolas abiertas consistentes en el conjunto de puntos que equidistan de uno dado menos de una cierta distancia son una colección de conjuntos que permite construir la base de la topología. Además de este ejemplo los espacios normados como los espacios de Hilbert o los espacios de Sobolev son otros ejemplos de espacios topológicos más complicados (estos últimos suelen tener dimensión infinita y se usan en análisis funcional). (es) En mathématiques, les espaces vectoriels topologiques sont une des structures de base de l'analyse fonctionnelle. Ce sont des espaces munis d'une structure topologique associée à une structure d'espace vectoriel, avec des relations de compatibilité entre les deux structures. Les exemples les plus simples d'espaces vectoriels topologiques sont les espaces vectoriels normés, parmi lesquels figurent les espaces de Banach, en particulier les espaces de Hilbert. (fr) Dalam matematika, suatu ruang vektor topologis (juga disebut ruang topologis linear) adalah suatu ruang vektor yang mana suatu topologi yang serasi didefinisikan sebagai suatu tambahan pada struktur aljabarnya, sedemikian sehingga operasi pada ruang vektor menjadi fungsi kontinu. Lebih khusus lagi, ruang topologisnya memiliki , memungkinkan gagasan tentang . Ruang vektor topologis adalah salah satu struktur dasar yang diteliti dalam analisis fungsional. Elemen ruang vektor topologis biasanya fungsi atau operator linear yang bekerja pada ruang vektor topologis, dan topologi sering didefinisikan untuk menangkap gagasan tertentu dari konvergensi dari urutan fungsi. Ruang Banach, Ruang Hilbert dan adalah contoh yang terkenal. Kecuali dinyatakan lain, lapangan yang mendasari ruang vektor topologis diasumsikan sebagai bilangan kompleks ℂ atau bilangan riil ℝ. (in) In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are not necessarily locally convex. Banach spaces, Hilbert spaces and Sobolev spaces are other well-known examples of TVSs. Many topological vector spaces are spaces of functions, or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions. In this article, the scalar field of a topological vector space will be assumed to be either the complex numbers or the real numbers unless clearly stated otherwise. (en) 수학에서 위상 벡터 공간(位相vector空間, 영어: topological vector space, 약자 TVS)은 호환되는 위상이 주어진 벡터 공간이다. (ko) Topologische vectorruimten zijn het centrale studieobject van een tak van de wiskunde die functionaalanalyse heet. (nl) In matematica, uno spazio vettoriale topologico (a volte spazio topologico lineare) è uno spazio su cui sono definite sia una struttura topologica sia una struttura lineare, in modo che esse siano compatibili tra loro. Gli spazi topologici lineari sono tra gli oggetti più studiati dell'analisi funzionale. La ricerca riguardante gli spazi vettoriali topologici è stata iniziata da Stefan Banach negli anni trenta, come generalizzazione, appunto, degli spazi di Banach. (it) 数学における線型位相空間（せんけいいそうくうかん、英語: linear topological space）とは、ベクトル空間の構造（線型演算）とその構造に両立する位相構造を持ったもののことである。係数体は実数体 R や複素数体 C などの位相体であり、ベクトルの加法やスカラー倍などの演算が連続写像になっていることが要請される。線型位相空間においては、通常のベクトル空間におけるような代数的な操作に加えて、興味のあるベクトルを他のベクトルで近似することが可能になり、関数解析学における基本的な枠組みが与えられる。 ベクトル空間の代数的な構造はその次元のみによって完全に分類されるが、特に無限次元のベクトル空間に対してその上に考えられる位相には様々なものがある。有限次元の実・複素ベクトル空間上の、意義のある位相はそれぞれの空間に対して一意的に決まってしまうことから、この多様性は無限次元に特徴的なものといえる。 (ja) Em matemática, e em especial em análise funcional, um espaço vectorial topológico combina as noções de espaço vectorial e espaço topológico, de forma que as operações usuais definidas no espaço vectorial sejam funções contínuas. O conceito de espaço vectorial topológico ou espaço linear topológico (ELT) generaliza as técnicas em espaços normados para espaços vectoriais onde pode não ser possível definir uma norma. Observe também que embora os ELT sejam estruturas bastante gerais, nem sempre um espaço métrico linear é um ELT. (pt) Przestrzeń liniowo-topologiczna – przestrzeń liniowa z określoną w niej topologią, dla której działania dodawania wektorów i mnożenia przez skalar są ciągłe. O topologii dodatkowo zakłada się, że każdy punkt tej przestrzeni jest zbiorem domkniętym, innymi słowy przestrzeń spełnia pierwszy aksjomat oddzielania. Można udowodnić, że każda przestrzeń liniowo-topologiczna jest przestrzenią Hausdorffa, a nawet jest przestrzenią regularną. Grupa addytywna przestrzeni liniowo-topologicznej jest grupą topologiczną. Każda przestrzeń unormowana (a więc np. dowolna przestrzeń Banacha czy Hilberta) jest przestrzenią liniowo-topologiczną. Przestrzenie liniowo-topologiczne są głównym obiektem badań analizy funkcjonalnej. Najczęściej rozważane są przestrzenie liniowo-topologiczne będące przestrzeniami funkcyjnymi. (pl) Ett topologiskt vektorrum är ett vektorrum utrustat med en topologi som gör vektoraddition och skalärmultiplikation till kontinuerliga funktioner. I vissa sammanhang ingår även i definitionen att topologin ska vara Hausdorff, vilket för en vektorrumstopologi är ekvivalent med att varje punkt är en sluten mängd. Ett topologiskt vektorrum är lokalt konvext om det har en bas för sin topologi bestående av konvexa mängder. Speciellt är varje normerat rum ett lokalt konvext topologiskt vektorrum. (sv) Топологічний векторний простір над топологічним полем — векторний простір над , наділений топологією, що узгоджується зі структурою векторного простору, тобто задовольняє наступним аксіомам: 1. * відображення є неперервним; 2. * відображення є неперервним В цих означеннях добутки і наділені добутками відповідних топологій). Цілком аналогічно можна визначити топологічний лівий і правий векторний простори над (не обов'язково комутативним) топологічним тілом. Для позначення топологічного векторного простору з топологією іноді використовується символ . Топологічні векторні простори і над одним і тим же топологічним полем називаються ізоморфними, якщо існує неперервне лінійне взаємно однозначне відображення на , обернене до якого також є неперервним. Розмірністю топологічного векторного простору називається розмірність векторного простору . (uk) 拓撲向量空間是泛函分析研究中的一個基本結構。顧名思義就是要研究具有拓撲結構的向量空間。 拓撲向量空間主要都是函數空間，在上面定義的拓撲結構就是函數列收歛的條件。 希爾伯特空間及巴拿赫空間是典型的例子。 (zh) Топологи́ческое ве́кторное простра́нство, или топологи́ческое лине́йное простра́нство, — векторное пространство, наделённое топологией, относительно которой операции сложения и умножения на число непрерывны. Термин используется в основном в функциональном анализе. (ru) |
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| dbp:mathStatement | If is a group , is a topology on and is endowed with the product topology, then the addition map is continuous at the origin of if and only if the set of neighborhoods of the origin in is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood." (en) Suppose that is a real or complex vector space. If is a non-empty additive collection of balanced and absorbing subsets of then is a neighborhood base at for a vector topology on That is, the assumptions are that is a filter base that satisfies the following conditions: # Every is balanced and absorbing, # is additive: For every there exists a such that If satisfies the above two conditions but is a filter base then it will form a neighborhood basis at for a vector topology on (en) If is a topological vector space then the following three conditions are equivalent: # The origin is closed in and there is a countable basis of neighborhoods at the origin in # is metrizable . # There is a translation-invariant metric on that induces on the topology which is the given topology on # is a metrizable topological vector space. By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant. (en) If is a topological vector space then there exists a set of neighborhood strings in that is directed downward and such that the set of all knots of all strings in is a neighborhood basis at the origin for Such a collection of strings is said to be . Conversely, if is a vector space and if is a collection of strings in that is directed downward, then the set of all knots of all strings in forms a neighborhood basis at the origin for a vector topology on In this case, this topology is denoted by and it is called the topology generated by (en) Let be a collection of subsets of a vector space such that and for all For all let Define by if and otherwise let Then is subadditive and on so in particular, If all are symmetric sets then and if all are balanced then for all scalars such that and all If is a topological vector space and if all are neighborhoods of the origin then is continuous, where if in addition is Hausdorff and forms a basis of balanced neighborhoods of the origin in then is a metric defining the vector topology on (en) |
| dbp:name | dbr:Birkhoff–Kakutani_theorem Theorem (en) Characterization of continuity of addition at (en) |
| dbp:note | -valued function induced by a string (en) Neighborhood filter of the origin (en) Topology induced by strings (en) |
| dbp:proof | The proof of this dichotomy is straightforward so only an outline with the important observations is given. As usual, is assumed have the Euclidean topology. Let for all Let be a -dimensional vector space over If and is a ball centered at then whenever contains an "unbounded sequence", by which it is meant a sequence of the form where and is unbounded in normed space . Any vector topology on will be translation invariant and invariant under non-zero scalar multiplication, and for every the map given by is a continuous linear bijection. Because for any such every subset of can be written as for some unique subset And if this vector topology on has a neighborhood of the origin that is not equal to all of then the continuity of scalar multiplication at the origin guarantees the existence of an open ball centered at and an open neighborhood of the origin in such that which implies that does contain any "unbounded sequence". This implies that for every there exists some positive integer such that From this, it can be deduced that if does not carry the trivial topology and if then for any ball center at 0 in contains an open neighborhood of the origin in which then proves that is a linear homeomorphism. Q.E.D. (en) |
| dbp:title | Proof outline (en) |
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| rdfs:comment | فضاء هيلبرت و باناخ مثالان معروفان للفضاءات المتجهية الطوبولوجية. (ar) Ein topologischer Vektorraum ist ein Vektorraum, auf dem neben seiner algebraischen auch noch eine damit verträgliche topologische Struktur definiert ist. (de) En analitiko, topologia vektora spaco estas vektora spaco ekipita per topologio kiu kongruas kun la vektorspaca strukturo (t.e. adicio kaj skalara multipliko estas kontinuaj). (eo) En mathématiques, les espaces vectoriels topologiques sont une des structures de base de l'analyse fonctionnelle. Ce sont des espaces munis d'une structure topologique associée à une structure d'espace vectoriel, avec des relations de compatibilité entre les deux structures. Les exemples les plus simples d'espaces vectoriels topologiques sont les espaces vectoriels normés, parmi lesquels figurent les espaces de Banach, en particulier les espaces de Hilbert. (fr) 수학에서 위상 벡터 공간(位相vector空間, 영어: topological vector space, 약자 TVS)은 호환되는 위상이 주어진 벡터 공간이다. (ko) Topologische vectorruimten zijn het centrale studieobject van een tak van de wiskunde die functionaalanalyse heet. (nl) In matematica, uno spazio vettoriale topologico (a volte spazio topologico lineare) è uno spazio su cui sono definite sia una struttura topologica sia una struttura lineare, in modo che esse siano compatibili tra loro. Gli spazi topologici lineari sono tra gli oggetti più studiati dell'analisi funzionale. La ricerca riguardante gli spazi vettoriali topologici è stata iniziata da Stefan Banach negli anni trenta, come generalizzazione, appunto, degli spazi di Banach. (it) 数学における線型位相空間（せんけいいそうくうかん、英語: linear topological space）とは、ベクトル空間の構造（線型演算）とその構造に両立する位相構造を持ったもののことである。係数体は実数体 R や複素数体 C などの位相体であり、ベクトルの加法やスカラー倍などの演算が連続写像になっていることが要請される。線型位相空間においては、通常のベクトル空間におけるような代数的な操作に加えて、興味のあるベクトルを他のベクトルで近似することが可能になり、関数解析学における基本的な枠組みが与えられる。 ベクトル空間の代数的な構造はその次元のみによって完全に分類されるが、特に無限次元のベクトル空間に対してその上に考えられる位相には様々なものがある。有限次元の実・複素ベクトル空間上の、意義のある位相はそれぞれの空間に対して一意的に決まってしまうことから、この多様性は無限次元に特徴的なものといえる。 (ja) Em matemática, e em especial em análise funcional, um espaço vectorial topológico combina as noções de espaço vectorial e espaço topológico, de forma que as operações usuais definidas no espaço vectorial sejam funções contínuas. O conceito de espaço vectorial topológico ou espaço linear topológico (ELT) generaliza as técnicas em espaços normados para espaços vectoriais onde pode não ser possível definir uma norma. Observe também que embora os ELT sejam estruturas bastante gerais, nem sempre um espaço métrico linear é um ELT. (pt) Ett topologiskt vektorrum är ett vektorrum utrustat med en topologi som gör vektoraddition och skalärmultiplikation till kontinuerliga funktioner. I vissa sammanhang ingår även i definitionen att topologin ska vara Hausdorff, vilket för en vektorrumstopologi är ekvivalent med att varje punkt är en sluten mängd. Ett topologiskt vektorrum är lokalt konvext om det har en bas för sin topologi bestående av konvexa mängder. Speciellt är varje normerat rum ett lokalt konvext topologiskt vektorrum. (sv) 拓撲向量空間是泛函分析研究中的一個基本結構。顧名思義就是要研究具有拓撲結構的向量空間。 拓撲向量空間主要都是函數空間，在上面定義的拓撲結構就是函數列收歛的條件。 希爾伯特空間及巴拿赫空間是典型的例子。 (zh) Топологи́ческое ве́кторное простра́нство, или топологи́ческое лине́йное простра́нство, — векторное пространство, наделённое топологией, относительно которой операции сложения и умножения на число непрерывны. Термин используется в основном в функциональном анализе. (ru) En matemàtiques, un espai vectorial topològic és una estructura bàsica que combina l'estructura algebraica d'un espai vectorial amb una estructura topològica.El cos subjacent d'un espai vectorial topològic és un cos topològic, que en les aplicacions acostuma a ser el cos dels nombres reals R o el dels nombres complexos C. Alguns tipus particulars molt importants d'espais vectorials topològics són els espais de Banach i els espais de Hilbert. (ca) Un espacio vectorial topológico es un espacio de puntos que aúna la estructura típica de un espacio vectorial convencional y un espacio topológico, es decir, es un espacio vectorial sobre el que se ha definido una estructura topológica. (es) Dalam matematika, suatu ruang vektor topologis (juga disebut ruang topologis linear) adalah suatu ruang vektor yang mana suatu topologi yang serasi didefinisikan sebagai suatu tambahan pada struktur aljabarnya, sedemikian sehingga operasi pada ruang vektor menjadi fungsi kontinu. Lebih khusus lagi, ruang topologisnya memiliki , memungkinkan gagasan tentang . Ruang vektor topologis adalah salah satu struktur dasar yang diteliti dalam analisis fungsional. Ruang Banach, Ruang Hilbert dan adalah contoh yang terkenal. (in) In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a vector topology and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness. Some authors also require that the space is a Hausdorff space (although this article does not). One of the most widely studied categories of TVSs are locally convex topological vector spaces. This article focuses on TVSs that are n (en) Przestrzeń liniowo-topologiczna – przestrzeń liniowa z określoną w niej topologią, dla której działania dodawania wektorów i mnożenia przez skalar są ciągłe. O topologii dodatkowo zakłada się, że każdy punkt tej przestrzeni jest zbiorem domkniętym, innymi słowy przestrzeń spełnia pierwszy aksjomat oddzielania. Przestrzenie liniowo-topologiczne są głównym obiektem badań analizy funkcjonalnej. Najczęściej rozważane są przestrzenie liniowo-topologiczne będące przestrzeniami funkcyjnymi. (pl) Топологічний векторний простір над топологічним полем — векторний простір над , наділений топологією, що узгоджується зі структурою векторного простору, тобто задовольняє наступним аксіомам: 1. * відображення є неперервним; 2. * відображення є неперервним В цих означеннях добутки і наділені добутками відповідних топологій). Цілком аналогічно можна визначити топологічний лівий і правий векторний простори над (не обов'язково комутативним) топологічним тілом. Для позначення топологічного векторного простору з топологією іноді використовується символ . (uk) |
| rdfs:label | فضاء متجهي طوبولوجي (ar) Espai vectorial topològic (ca) Topologický vektorový prostor (cs) Topologischer Vektorraum (de) Topologia vektora spaco (eo) Espacio vectorial topológico (es) Espace vectoriel topologique (fr) Ruang vektor topologis (in) Spazio vettoriale topologico (it) 위상 벡터 공간 (ko) 線型位相空間 (ja) Topologische vectorruimte (nl) Przestrzeń liniowo-topologiczna (pl) Espaço vectorial topológico (pt) Topological vector space (en) Топологическое векторное пространство (ru) Topologiskt vektorrum (sv) Топологічний векторний простір (uk) 拓撲向量空間 (zh) |
| rdfs:seeAlso | dbr:Locally_convex_topological_vector_space |
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