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rdfs:label
Espai de Hilbert Χώρος Χίλμπερτ فضاء هيلبرت 希尔伯特空间 힐베르트 공간 Hilbert space Espace de Hilbert Hilbertraum Espacio de Hilbert ヒルベルト空間 Гільбертів простір Hilbertrum Espaço de Hilbert Hilbertruimte Гильбертово пространство Hilberta spaco Spazio di Hilbert Przestrzeń Hilberta Hilbertův prostor Ruang Hilbert
rdfs:comment
En matemáticas, el concepto de espacio de Hilbert es una generalización del concepto de espacio euclídeo. Esta generalización extiende los métodos del álgebra lineal y el cálculo aplicados en el espacio euclídeo de dos dimensiones y tres dimensiones a los espacios de dimensión arbitraria, incluyendo los espacios de dimensión infinita. Ejemplos de tales nociones y técnicas son la de ángulo entre vectores, ortogonalidad de vectores, el teorema de Pitágoras, proyección ortogonal, distancia entre vectores y convergencia de una sucesión. El nombre dado a estos espacios es en honor al matemático David Hilbert quien los utilizó en su estudio de las ecuaciones integrales. Ett hilbertrum (efter David Hilbert) är inom matematiken ett inre produktrum som är fullständigt med avseende på den norm som definieras av den inre produkten. Hilbertrum generaliserar och klargör begrepp såsom vissa linjära transformationer (till exempel fouriertransformer) och är absolut nödvändiga i formuleringen av kvantmekaniken. Hilbertrum studeras inom funktionalanalys. En mathématiques, un espace de Hilbert est un espace vectoriel réel (resp. complexe) muni d'un produit scalaire euclidien (resp. hermitien), qui permet de mesurer des longueurs et des angles et de définir une orthogonalité. De plus, un espace de Hilbert est complet, ce qui permet d'y appliquer des techniques d'analyse. Ces espaces doivent leur nom au mathématicien allemand David Hilbert. En matemàtiques, el concepte d'espai de Hilbert és una generalització del concepte d'espai euclidià. Aquesta generalització permet que nocions i tècniques algebraiques i geomètriques aplicables a espais de dimensió dos i tres s'estenguin a espais de dimensió arbitrària, incloent espais de dimensió infinita. Exemples de tals nocions i tècniques són la d'angle entre vectors, ortogonalitat de vectors, el teorema de Pitàgores, projecció ortogonal, distància entre vectors i convergència d'una successió. El nom donat a aquests espais és en honor del matemàtic David Hilbert qui els va utilitzar en el seu estudi de les equacions integrals. En matematiko, hilberta spaco (nomata laŭ David Hilbert) estas ĝeneraligo de eŭklida spaco kiu estas ne limigita per finia kvanto de dimensioj. Tial ĝi estas spaco, kio signifas ke ĝi havas nociojn de distanco kaj angulo (aparte la nocio de orteco). Ankaŭ, ĝi kontentigas pli teknikan kompletecon kiu certiĝas ke limigoj ekzistas kiam oni ilin atendas, kiu faciligas diversajn difinojn de kalkulo. Hilbertaj spacoj provizas ĉirkaŭtekston kun por formaligi kaj ĝeneraligi la konceptojn de la en terminoj de ajnaj kaj de la , kiu estas centra koncepto de . Hilbertaj spacoj estas gravaj en matematika formulaĵo de kvantummekaniko. 함수해석학에서 힐베르트 공간(Hilbert空間, 영어: Hilbert space)은 모든 코시 열의 극한이 존재하는 내적 공간이다. 유클리드 공간을 일반화한 개념이다. Η μαθηματική έννοια του χώρου Χίλμπερτ, που πήρε το όνομα του από τον Ντάβιντ Χίλμπερτ (David Hilbert), γενικεύει την έννοια του ευκλείδειου χώρου. Επεκτείνει τις μεθόδους της γραμμικής άλγεβρας και του λογισμού, τόσο από το δισδιάστατο ευκλείδειο χώρο, όσο και τον τριδιάστατο χώρο σε χώρους κάθε πεπερασμένου ή άπειρου αριθμού διαστάσεων. Ένας χώρος Χίλμπερτ είναι ένας αφηρημένος διανυσματικός χώρος εφοδιασμένος με εσωτερικό γινόμενο, το οποίο τον καθιστά μετρήσιμο. Επιπλέον, οι χώροι Χίλμπερτ είναι πλήρεις: υπάρχουν δηλαδή αρκετά όρια στον χώρο για να επιτρέψουν τις τεχνικές του λογισμού, που πρέπει να χρησιμοποιηθούν. Na matemática, um espaço de Hilbert é uma generalização do espaço euclidiano que não precisa estar restrita a um número finito de dimensões. É um espaço vetorial dotado de produto interno, ou seja, com noções de distância e ângulos. Esse espaço obedece uma relação de completude, que garante que os limites existem quando esperados, o que permite e facilita diversas definições da Análise. Os espaços de Hilbert permitem que, de certa maneira, noções intuitivas sejam aplicadas em espaços funcionais. Por exemplo, com eles podemos generalizar os conceitos de séries de Fourier em termos de . Os espaços de Hilbert são de importância crucial para a Mecânica Quântica. Гі́льбертів про́стір (на честь Давида Гільберта) — це узагальнення поняття евклідового простору на нескінченновимірний випадок. Є лінійним простором над полем дійсних або комплексних чисел (прийменник «над» означає, що у такому просторі дозволені операції множення на скаляри із відповідних полів), із визначеним скалярним добутком. Останній дозволяє вводити поняття, аналогічні звичним поняттям ортогональності і кута. 在数学裡,希尔伯特空间(英語:Hilbert space)即完备的内积空间,也就是一個帶有內積的完備向量空間。希尔伯特空间是有限维欧几里得空间的一个推广,使之不局限于實數的情形和有限的维数,但又不失完备性(而不像一般的非欧几里得空间那样破坏了完备性)。与欧几里得空间相仿,希尔伯特空间也是一个内积空间,其上有距离和角的概念(及由此引申而来的正交性与垂直性的概念)。此外,希尔伯特空间还是一个完备的空间,其上所有的柯西序列會收敛到此空間裡的一點,从而微积分中的大部分概念都可以无障碍地推广到希尔伯特空间中。 希尔伯特空间为基于任意上的多项式表示的傅立叶级数和傅立叶变换提供了一种有效的表述方式,而这也是泛函分析的核心概念之一。希尔伯特空间是公設化数学和量子力学的关键性概念之一。 Im mathematischen Teilgebiet der Funktionalanalysis ist ein Hilbertraum (Hilbert‧raum, auch Hilbert-Raum, Hilbertscher Raum), benannt nach dem deutschen Mathematiker David Hilbert, ein Vektorraum über dem Körper der reellen oder komplexen Zahlen, versehen mit einem Skalarprodukt – und damit Winkel- und Längenbegriffen –, der vollständig bezüglich der vom Skalarprodukt induzierten Norm (des Längenbegriffs) ist. Ein Hilbertraum ist ein Banachraum, dessen Norm durch ein Skalarprodukt induziert ist. Lässt man die Bedingung der Vollständigkeit fallen, spricht man von einem Prähilbertraum. Przestrzeń Hilberta – przestrzeń unitarna zupełna. Oznacza to, że jest to przestrzeń liniowa nad ciałem liczb rzeczywistych lub zespolonych, która * ma zdefiniowany iloczyn skalarny, * traktowana jako przestrzeń metryczna z metryką indukowaną przez iloczyn skalarny (poprzez normę) jest zupełna, tzn. każdy ciąg Cauchy’ego ma granicę. Każda przestrzeń Hilberta jest przestrzenią Banacha (z normą indukowaną przez iloczyn skalarny), przestrzenią Frécheta oraz lokalnie wypukłą przestrzenią liniowo-topologiczną – ze względu na unormowanie i zupełność. Ги́льбертово простра́нство — обобщение евклидова пространства, допускающее бесконечную размерность и полное по метрике, порождённой скалярным произведением.Названо в честь Давида Гильберта. Важнейшим объектом исследования в гильбертовом пространстве являются линейные операторы. Само понятие гильбертова пространства сформировалось в работах Гильберта и Шмидта по теории интегральных уравнений, а абстрактное определение было дано в работах фон Неймана, Риса и Стоуна по теории эрмитовых операторов. In matematica uno spazio di Hilbert è uno spazio vettoriale completo secondo la norma indotta da un certo prodotto scalare. La nozione di spazio di Hilbert è stata introdotta dal celebre matematico David Hilbert all'inizio del XX secolo e ha fornito un enorme contributo allo sviluppo dell'analisi funzionale e armonica. Il suo interesse principale risiede nella conservazione di alcune proprietà degli spazi euclidei in spazi di funzioni infinito-dimensionali. Grazie alla definizione di spazio di Hilbert è possibile formalizzare la teoria delle serie di Fourier e generalizzarla a basi arbitrarie. المفهوم الرياضياتي لفضاء هيلبرت (بالإنجليزية: Hilbert space)‏ يعمم مفهوم الفضاء الإقليدي. فهو فضاء معياري معرف عليه دالة الجداء الداخلي بشرط أن يكون المعيار المعرف عليه هو بدلالة دالة الجداء الداخلي هذه، بالإضافة إلى وجوب كونه فضاء معياريا كاملا أو ما يدعى ب فضاء باناخ. وهذا يعني أن أي فضاء هيلبرت هو فضاء باناخ ولكن العكس غير صحيح.مثال على ذلك، الفضاء Q هو فضاء منتظم تحت النظيم العادي ولكنه ليس بفضاء بانخ. Konsep matematika dari ruang Hilbert, dinamai David Hilbert, menggeneralisasi gagasan ruang Euklides. Maka memperluas metode aljabar vektor dan kalkulus dari dua dimensi bidang Euclidean dan ruang tiga dimensi ke ruang dengan berhingga atau tak hingga. Ruang Hilbert adalah ruang vektor yang dilengkapi dengan , sebuah operasi yang memungkinkan untuk menentukan panjang dan sudut. Lebih lanjut, ruang Hilbert adalah lengkap, yang berarti bahwa ada cukup limit di ruang untuk memungkinkan teknik kalkulus digunakan. Hilbertovým prostorem je v matematice a fyzice označován vektorový prostor, v kterém je možné měřit úhly a velikosti vektorů a konstruovat ortogonální projekce vektorů na podprostory. In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. 数学におけるヒルベルト空間(ヒルベルトくうかん、英: Hilbert space)は、ダフィット・ヒル��ルトにその名を因む、ユークリッド空間の概念を一般化したものである。これにより、二次元のユークリッド平面や三次元のユークリッド空間における線型代数学や微分積分学の方法論を、任意の有限または無限次元の空間へ拡張して持ち込むことができる。ヒルベルト空間は、内積の構造を備えた抽象ベクトル空間(内積空間)になっており、そこでは角度や長さを測るということが可能である。ヒルベルト空間は、さらに完備距離空間の構造を備えている(極限が十分に存在することが保証されている)ので、その中で微分積分学がきちんと展開できる。 In de functionaalanalyse, een deelgebied van de wiskunde, is een hilbertruimte, vernoemd naar de Duitse wiskundige David Hilbert, een abstracte reële of complexe vectorruimte die voorzien is van de extra structuur van een inwendig product. Een hilbertruimte is een algemene vorm van het begrip euclidische ruimte en breidt de methoden van de lineaire algebra en de analyse van het tweedimensionale euclidische vlak en de driedimensionale ruimte uit naar ruimten met een eindig of oneindig aantal dimensies.
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Hilbert space Abstract linear spaces
dbo:abstract
함수해석학에서 힐베르트 공간(Hilbert空間, 영어: Hilbert space)은 모든 코시 열의 극한이 존재하는 내적 공간이다. 유클리드 공간을 일반화한 개념이다. المفهوم الرياضياتي لفضاء هيلبرت (بالإنجليزية: Hilbert space)‏ يعمم مفهوم الفضاء الإقليدي. فهو فضاء معياري معرف عليه دالة الجداء الداخلي بشرط أن يكون المعيار المعرف عليه هو بدلالة دالة الجداء الداخلي هذه، بالإضافة إلى وجوب كونه فضاء معياريا كاملا أو ما يدعى ب فضاء باناخ. وهذا يعني أن أي فضاء هيلبرت هو فضاء باناخ ولكن العكس غير صحيح.مثال على ذلك، الفضاء Q هو فضاء منتظم تحت النظيم العادي ولكنه ليس بفضاء بانخ. يمكننا فضاء هلبرت من تعميم أساليب الجبر الخطي والتفاضل والتكامل المستخدمة في الفضاءات الإقليدية ثنائية الأبعاد وثلاثية الأبعاد إلى فضاءات قد تكون لانهائية البُعد. فضاء هلبرت هو فضاء متجهي مزود بجداء داخلي، وبذا يسمح بتعريف دالة مسافة وتعامد. وإضافة لذلك، فإن فضاء هلبرت هو فضاء متري كامل مع دالة المسافة المعرفة فيه (في هذه الحالة هي دالة المعيار)، مما يعني توفر النهايات التي تسمح باستخدام التفاضل والتكامل. فضاءات هلبرت تظهر بشكل طبيعي كثيرًا في الرياضيات والفيزياء، عادةً كفضاء دالي لا نهائي الأبعاد. أقدم فضاءات هلبرت دراسةً كانت على يد ديفيد هيلبرت وإرهارد شميدت وفريجيس ريش "Frigyes Riesz" في العقد الأول من القرن العشرين. وتعتبر من الأدوات الهامة في المعادلات التفاضلية الجزئية، وميكانيكا الكم، وتحليل فورييه (الذي يتضمن تطبيقات لمعالجة الإشارات ونقل الحرارة)، ونظرية إرجوديك (التي تشكل الأساس الرياضي للديناميكا الحرارية). صاغ جون فون نيومان مصطلح فضاء هيلبرت للمفهوم التجريدي المستخدم في العديد من هذه التطبيقات المتنوعة. أدى نجاح أساليب فضاء هيلبرت إلى ازدهار التحليل الدالي. بعيدًا عن الفضاءات الإقليدية الكلاسيكية، فإن من أمثلة فضاءات هيلبرت فضاءات الدوال الكمولة تربيعيًا "Lp space"، وفضاءات المتسلسلات "Sequence space"، وفضاءات سوبوليف "Sobolev space" المكونة من دوال معممة، وفضاءات هاردي "Hardy space" للدوال تامة الشكل. يلعب الحدس الهندسي دورًا مهمًا في العديد من جوانب نظرية فضاء هلبرت. توجد نظائر مطابقة لنظرية فيثاغورس وقانون متوازي الأضلاع في فضاء هلبرت. على مستوى أعمق، يلعب الإسقاط العمودي على فضاء جزئي (مناظر لـ «إسقاط ارتفاع» المثلث) دورًا مهمًا في مسائل التحسين وجوانب أخرى في النظرية. يمكن تحديد بشكل فريد عنصر في فضاء هيلبرت عن طريق إحداثياته المنسوبة لزمرة من محاور الإحداثيات (متعامدة القاعدة في فضاء هلبرت)، على غرار الإحداثيات الديكارتية في المستوى. عندما تكون هذه المجموعة من المحاور معدودة غير منتهية، في هذه الحالة يمكن النظر لفضاء هيلبرت على أنه فضاء متسلسلات لانهائية التي يمكن جمعها تربيعيًا. فضاء المتسلسلات اللانهائية هذا هو ما يشار إليه في كثير من الأحيان في الأدبيات الرياضية القديمة بفضاء هلبرت. المؤثرات الخطية في فضاء هلبرت أيضًا لها حضور كبير: في الحالات الجيدة، هي تحويلات تساهم في تمديد الفضاء بواسطة عوامل مختلفة في اتجاهات متعامدة تبادليًا بصورة دقيقة تظهر من خلال دراسة طيفها. 数学におけるヒルベルト空間(ヒルベルトくうかん、英: Hilbert space)は、ダフィット・ヒルベルトにその名を因む、ユークリッド空間の概念を一般化したものである。これにより、二次元のユークリッド平面や三次元のユークリッド空間における線型代数学や微分積分学の方法論を、任意の有限または無限次元の空間へ拡張して持ち込むことができる。ヒルベルト空間は、内積の構造を備えた抽象ベクトル空間(内積空間)になっており、そこでは角度や長さを測るということが可能である。ヒルベルト空間は、さらに完備距離空間の構造を備えている(極限が十分に存在することが保証されている)ので、その中で微分積分学がきちんと展開できる。 ヒルベルト空間は、典型的には無限次元の関数空間として、数学、物理学、工学などの各所に自然に現れる。そういった意味でのヒルベルト空間の研究は、20世紀冒頭10年の間にヒルベルト、シュミット、リースらによって始められた。ヒルベルト空間の概念は、偏微分方程式論、量子力学、フーリエ解析(信号処理や熱伝導などへの応用も含む)、熱力学の研究の数学的基礎を成すエルゴード理論などの理論において欠くべからざる道具になっている。これら種々の応用の多くの根底にある抽象概念を「ヒルベルト空間」と名付けたのは、フォン・ノイマンである。ヒルベルト空間を用いる方法の成功は、関数解析学の実りある時代のさきがけとなった。古典的なユークリッド空間はさておき、ヒルベルト空間の例としては、自乗可積分関数の空間 L2、自乗総和可能数列の空間 、超関数からなるソボレフ空間 、正則関数の成すハーディ空間 などが挙げられる。 ヒルベルト空間論の多くの場面で、幾何学的直観は重要である。例えば、三平方の定理や中線定理(の厳密な類似対応物)は、ヒルベルト空間においても成り立つ。より深いところでは、部分空間への直交射影(例えば、三角形に対してその「高さを潰す」操作の類似対応物)は、ヒルベルト空間論における最適化問題やその周辺で重要である。ヒルベルト空間の各元は、平面上の点がそのデカルト座標(直交座標)によって特定できるのと同様に、座標軸の集合(正規直交基底)に関する座標によって一意的に特定することができる。このことは、座標軸の集合が可算無限であるときには、ヒルベルト空間を自乗総和可能な無限列の集合と看做すことも有用であることを意味する。ヒルベルト空間上の線型作用素は、ほぼ具体的な対象として扱うことができる。条件がよければ、空間を互いに直交するいくつかの異なる要素に分解してやると、線型作用素はそれぞれの要素の上では単に拡大縮小するだけの変換になる(これはまさに線型作用素のスペクトルを調べるということである)。 Hilbertovým prostorem je v matematice a fyzice označován vektorový prostor, v kterém je možné měřit úhly a velikosti vektorů a konstruovat ortogonální projekce vektorů na podprostory. En mathématiques, un espace de Hilbert est un espace vectoriel réel (resp. complexe) muni d'un produit scalaire euclidien (resp. hermitien), qui permet de mesurer des longueurs et des angles et de définir une orthogonalité. De plus, un espace de Hilbert est complet, ce qui permet d'y appliquer des techniques d'analyse. Ces espaces doivent leur nom au mathématicien allemand David Hilbert. Le concept d'espace de Hilbert étend les méthodes de l'algèbre linéaire en généralisant les notions d'espace euclidien (comme le plan euclidien ou l'espace usuel de dimension 3) et d'espace hermitien à des espaces de dimension quelconque (finie ou infinie). Des espaces de Hilbert apparaissent fréquemment en mathématiques et en physique, essentiellement en tant qu'espaces fonctionnels de dimension infinie. Les premiers espaces de Hilbert ont été étudiés sous cet aspect pendant la première décennie du XXe siècle par David Hilbert, Erhard Schmidt et Frigyes Riesz. Ils sont des outils indispensables dans les théories des équations aux dérivées partielles, mécanique quantique, analyse de Fourier (ce qui inclut des applications au traitement du signal et le transfert thermique) et la théorie ergodique qui forme le fondement mathématique de la thermodynamique. John von Neumann forgea l'expression espace de Hilbert pour désigner le concept abstrait qui sous-tend nombre de ces applications. Les succès des méthodes apportées par les espaces de Hilbert menèrent à une époque très prolifique pour l'analyse fonctionnelle. En plus des espaces euclidiens classiques, les exemples les plus courants d'espaces de Hilbert sont les espaces de fonctions de carré intégrable, les espaces de Sobolev qui sont constitués de fonctions généralisées, et les espaces de Hardy de fonctions holomorphes. L'intuition géométrique intervient dans de nombreux aspects de la théorie des espaces de Hilbert. Ces espaces possèdent des théorèmes analogues au théorème de Pythagore et à la règle du parallélogramme. En mathématiques appliquées, les projections orthogonales sur un sous-espace (ce qui correspond à aplatir l'espace de quelques dimensions) jouent un rôle important dans des problèmes d'optimisation entre autres aspects de la théorie. Un élément d'un espace de Hilbert peut être défini de manière unique par ses coordonnées relativement à une base de Hilbert, de façon analogue aux coordonnées cartésiennes dans une base orthonormale du plan. Quand cet ensemble d'axes est dénombrable, l'espace de Hilbert peut être vu comme un ensemble de suites de carré sommable. Les opérateurs linéaires sur un espace de Hilbert sont semblables à des objets concrets : dans les « bons » cas, ce sont simplement des transformations qui étirent l'espace suivant différents coefficients dans des directions deux à deux perpendiculaires, en un sens qui est précisé par l'étude de leur spectre. Przestrzeń Hilberta – przestrzeń unitarna zupełna. Oznacza to, że jest to przestrzeń liniowa nad ciałem liczb rzeczywistych lub zespolonych, która * ma zdefiniowany iloczyn skalarny, * traktowana jako przestrzeń metryczna z metryką indukowaną przez iloczyn skalarny (poprzez normę) jest zupełna, tzn. każdy ciąg Cauchy’ego ma granicę. Każda przestrzeń Hilberta jest przestrzenią Banacha (z normą indukowaną przez iloczyn skalarny), przestrzenią Frécheta oraz lokalnie wypukłą przestrzenią liniowo-topologiczną – ze względu na unormowanie i zupełność. Nazwa przestrzeni pochodzi od nazwiska Davida Hilberta, który wprowadził je pod koniec XIX wieku. Przestrzenie Hilberta są wykorzystywane w wielu dziedzinach fizyki, m.in. w mechanice kwantowej i kwantowej teorii pola (np. przestrzeń Foka nad przestrzenią Hilberta). Гі́льбертів про́стір (на честь Давида Гільберта) — це узагальнення поняття евклідового простору на нескінченновимірний випадок. Є лінійним простором над полем дійсних або комплексних чисел (прийменник «над» означає, що у такому просторі дозволені операції множення на скаляри із відповідних полів), із визначеним скалярним добутком. Останній дозволяє вводити поняття, аналогічні звичним поняттям ортогональності і кута. En matemáticas, el concepto de espacio de Hilbert es una generalización del concepto de espacio euclídeo. Esta generalización extiende los métodos del álgebra lineal y el cálculo aplicados en el espacio euclídeo de dos dimensiones y tres dimensiones a los espacios de dimensión arbitraria, incluyendo los espacios de dimensión infinita. Ejemplos de tales nociones y técnicas son la de ángulo entre vectores, ortogonalidad de vectores, el teorema de Pitágoras, proyección ortogonal, distancia entre vectores y convergencia de una sucesión. El nombre dado a estos espacios es en honor al matemático David Hilbert quien los utilizó en su estudio de las ecuaciones integrales. Más formalmente, se define como un espacio de producto interior que es completo con respecto a la norma vectorial definida por el producto interno. Los espacios de Hilbert sirven para clarificar y para generalizar el concepto de series de Fourier y ciertas transformaciones lineales tales como la transformación de Fourier, además son de importancia crucial en la formulación matemática de la mecánica cuántica. Los espacios de Hilbert y sus propiedades se estudian dentro del análisis funcional. Ги́льбертово простра́нство — обобщение евклидова пространства, допускающее бесконечную размерность и полное по метрике, порождённой скалярным произведением.Названо в честь Давида Гильберта. Важнейшим объектом исследования в гильбертовом пространстве являются линейные операторы. Само понятие гильбертова пространства сформировалось в работах Гильберта и Шмидта по теории интегральных уравнений, а абстрактное определение было дано в работах фон Неймана, Риса и Стоуна по теории эрмитовых операторов. Η μαθηματική έννοια του χώρου Χίλμπερτ, που πήρε το όνομα του από τον Ντάβιντ Χίλμπερτ (David Hilbert), γενικεύει την έννοια του ευκλείδειου χώρου. Επεκτείνει τις μεθόδους της γραμμικής άλγεβρας και του λογισμού, τόσο από το δισδιάστατο ευκλείδειο χώρο, όσο και τον τριδιάστατο χώρο σε χώρους κάθε πεπερασμένου ή άπειρου αριθμού διαστάσεων. Ένας χώρος Χίλμπερτ είναι ένας αφηρημένος διανυσματικός χώρος εφοδιασμένος με εσωτερικό γινόμενο, το οποίο τον καθιστά μετρήσιμο. Επιπλέον, οι χώροι Χίλμπερτ είναι πλήρεις: υπάρχουν δηλαδή αρκετά όρια στον χώρο για να επιτρέψουν τις τεχνικές του λογισμού, που πρέπει να χρησιμοποιηθούν. Χώροι Χίλμπερτ προκύπτουν φυσικά και συχνά στα μαθηματικά και τη φυσική, συνήθως ως απειροδιάστατες λειτουργίες χώρων. Οι πρώτοι χώροι Χίλμπερτ μελετήθηκαν από την άποψη αυτή κατά την πρώτη δεκαετία του 20ού αιώνα από τους Ντέιβιντ Χίλμπερτ, και Φρίντες Ρης. Είναι απαραίτητα εργαλεία στις θεωρίες των μερικών διαφορικών εξισώσεων, την κβαντομηχανική, την ανάλυση Φουριέ (η οποία περιλαμβάνει εφαρμογές για την επεξεργασία σήματος και τη μεταφορά θερμότητας) και την εργοδική θεωρία, η οποία αποτελεί τη μαθηματική υποστήριξη της θερμοδυναμικής. Ο Τζον φον Νόιμαν επινόησε τον όρο χώρο Χίλμπερτ για την αφηρημένη έννοια, που κρύβεται πίσω από πολλές από αυτές τις ποικίλες εφαρμογές. Η επιτυχία των μεθόδων του χώρου Χίλμπερτ μπαίνει σε μια πολύ γόνιμη εποχή για τη . Εκτός από τους κλασσικούς Ευκλείδειους χώρους, παραδείγματα των χώρων Χίλμπερτ περιλαμβάνονται στους χώρους των τετραγωνικά ολοκληρώσιμων συναρτήσεων, τους , τους που αποτελούνται από και τους των αναλυτικών συναρτήσεων. Η γεωμετρική διαίσθηση παίζει σημαντικό ρόλο σε πολλές πτυχές της θεωρίας του χώρου Hilbert. Ακριβή ανάλογα του πυθαγορείου θεωρήματος και του νόμου παραλληλογράμμου συγκρατούν ένα χώρο Hilbert. Σε ένα βαθύτερο επίπεδο, η κάθετη προβολή επί ενός υπόχωρου (το ανάλογο της «» ενός τριγώνου) παίζει σημαντικό ρόλο στη βελτιστοποίηση των προβλημάτων και σε άλλες πτυχές της θεωρίας. Ένα στοιχείο του χώρου Hilbert μπορεί να προσδιοριστεί με μοναδικό τρόπο από τις συντεταγμένες του σε σχέση με ένα σύνολο από τους άξονες συντεταγμένων, σε αναλογία με καρτεσιανές συντεταγμένες στο επίπεδο. Όταν αυτό το σύνολο των αξόνων είναι , αυτό σημαίνει ότι ο χώρος Χίλμπερτ μπορεί επίσης χρησίμως να θεωρηθεί από την άποψη των που είναι τετράγωνο-αθροίσιμων τετραγωνικών. Οι γραμμικοί φορείς σε ένα χώρο Χίλμπερτ είναι επίσης αρκετά συγκεκριμένα αντικείμενα: σε καλές περιπτώσεις, είναι απλοί μετασχηματισμοί που τεντώνουν το χώρο από διάφορους παράγοντες σε αμοιβαίες κατακόρυφες κατευθύνσεις, με την έννοια ότι γίνονται ακριβείς από τη μελέτη του τους. In matematica uno spazio di Hilbert è uno spazio vettoriale completo secondo la norma indotta da un certo prodotto scalare. La nozione di spazio di Hilbert è stata introdotta dal celebre matematico David Hilbert all'inizio del XX secolo e ha fornito un enorme contributo allo sviluppo dell'analisi funzionale e armonica. Il suo interesse principale risiede nella conservazione di alcune proprietà degli spazi euclidei in spazi di funzioni infinito-dimensionali. Grazie alla definizione di spazio di Hilbert è possibile formalizzare la teoria delle serie di Fourier e generalizzarla a basi arbitrarie. Esso generalizza la nozione di spazio euclideo ed euristicamente uno spazio di Hilbert è un insieme con una struttura lineare (spazio vettoriale) su cui è definito un prodotto scalare (quindi è possibile parlare di norma, distanze, angoli, ortogonalità) e tale che sia garantita la completezza, ossia che qualunque successione di Cauchy ammetta come limite un elemento dello spazio stesso. Nelle applicazioni i vettori elementi di uno spazio di Hilbert sono frequentemente successioni di numeri complessi o funzioni. È cruciale nella formalizzazione matematica della meccanica quantistica (si veda la relativa sezione sottostante per maggiori dettagli). Im mathematischen Teilgebiet der Funktionalanalysis ist ein Hilbertraum (Hilbert‧raum, auch Hilbert-Raum, Hilbertscher Raum), benannt nach dem deutschen Mathematiker David Hilbert, ein Vektorraum über dem Körper der reellen oder komplexen Zahlen, versehen mit einem Skalarprodukt – und damit Winkel- und Längenbegriffen –, der vollständig bezüglich der vom Skalarprodukt induzierten Norm (des Längenbegriffs) ist. Ein Hilbertraum ist ein Banachraum, dessen Norm durch ein Skalarprodukt induziert ist. Lässt man die Bedingung der Vollständigkeit fallen, spricht man von einem Prähilbertraum. Die Struktur eines Hilbertraums ist eindeutig festgelegt durch seine Hilbertraumdimension. Diese kann eine beliebige Kardinalzahl sein. Ist die Dimension endlich und betrachtet man als Körper die reellen Zahlen, so handelt es sich um einen euklidischen Raum. In vielen Gebieten, etwa in der mathematischen Beschreibung der Quantenmechanik, ist „der“ Hilbertraum mit abzählbarer Dimension, d. h. mit der kleinstmöglichen unendlichen Dimension, von besonderer Bedeutung. Ein Element eines Hilbertraums kann als eine Familie einer der Dimension entsprechenden Anzahl reeller bzw. komplexer Werte (im Endlichdimensionalen kartesische Koordinaten genannt) aufgefasst werden. Analog zu Vektorräumen, deren Elemente stets nur in endlich vielen Koordinaten einer Hamelbasis ungleich null sind, ist jedes Element eines Hilbertraums nur in abzählbar vielen Koordinaten einer Orthonormalbasis ungleich null und die Koordinatenfamilie ist quadratsummabel. Hilberträume tragen durch ihr Skalarprodukt eine topologische Struktur. Dadurch sind hier im Gegensatz zu allgemeinen Vektorräumen Grenzwertprozesse möglich. Hilberträume sind abgeschlossen unter abzählbaren Summen von orthogonalen Elementen mit einer quadratsummablen Folge von Normen bzw. von parallelen Elementen mit einer absolutsummablen Folge von Normen. En matemàtiques, el concepte d'espai de Hilbert és una generalització del concepte d'espai euclidià. Aquesta generalització permet que nocions i tècniques algebraiques i geomètriques aplicables a espais de dimensió dos i tres s'estenguin a espais de dimensió arbitrària, incloent espais de dimensió infinita. Exemples de tals nocions i tècniques són la d'angle entre vectors, ortogonalitat de vectors, el teorema de Pitàgores, projecció ortogonal, distància entre vectors i convergència d'una successió. El nom donat a aquests espais és en honor del matemàtic David Hilbert qui els va utilitzar en el seu estudi de les equacions integrals. Més formalment, es defineix com un espai de producte interior que és complet respecte a la norma vectorial definida pel producte interior. Els espais de Hilbert serveixen per aclarir i generalitzar el concepte de sèries de Fourier, certes transformacions lineals tals com la transformació de Fourier hi són d'importància crucial en la formulació matemàtica de la mecànica quàntica. Els espais de Hilbert i les seves propietats s'estudien dins de l'anàlisi funcional. Konsep matematika dari ruang Hilbert, dinamai David Hilbert, menggeneralisasi gagasan ruang Euklides. Maka memperluas metode aljabar vektor dan kalkulus dari dua dimensi bidang Euclidean dan ruang tiga dimensi ke ruang dengan berhingga atau tak hingga. Ruang Hilbert adalah ruang vektor yang dilengkapi dengan , sebuah operasi yang memungkinkan untuk menentukan panjang dan sudut. Lebih lanjut, ruang Hilbert adalah lengkap, yang berarti bahwa ada cukup limit di ruang untuk memungkinkan teknik kalkulus digunakan. Ruang Hilbert muncul secara alami dan sering dalam matematika dan fisika, biasanya sebagai . Ruang Hilbert paling awal dipelajari dari sudut pandang ini pada dekade pertama abad ke-20 oleh David Hilbert, , dan . Mereka adalah alat yang sangat diperlukan dalam teori persamaan diferensial parsial s, , Analisis Fourier (yang mencakup aplikasi untuk pemrosesan sinyal dan perpindahan panas), dan (yang membentuk dasar matematika termodinamika). John von Neumann menciptakan istilah ruang Hilbert untuk konsep abstrak yang mendasari banyak aplikasi yang beragam ini. Keberhasilan metode ruang Hilbert mengantarkan era yang sangat bermanfaat bagi analisis fungsional. Terlepas dari ruang Euclidean klasik, contoh ruang Hilbert meliputi , , terdiri dari , dan dari . Intuisi geometris memainkan peran penting dalam banyak aspek teori ruang Hilbert. Analog tepat dari Teorema Pythagoras dan hukum jajaran genjang berlaku di ruang Hilbert. Pada tingkat yang lebih dalam, proyeksi tegak lurus ke subruang (analog dari "" dari segitiga) memainkan peran penting dalam masalah pengoptimalan dan lainnya sebagai. Sebuah elemen ruang Hilbert dapat secara unik ditentukan oleh koordinatnya sehubungan dengan satu set (sebuah ), dalam analogi dengan koordinat Kartesius pada bidang. Ketika himpunan sumbu itu , ruang Hilbert juga dapat dianggap berguna dalam hal ruang yang . Ruang terakhir sering dalam literatur yang lebih tua disebut sebagai ruang Hilbert. pada ruang Hilbert juga merupakan objek yang cukup konkret: dalam kasus yang baik, mereka hanyalah transformasi yang meregangkan ruang oleh faktor-faktor berbeda dalam arah yang saling tegak lurus dalam arti yang dibuat tepat oleh studi . In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. En matematiko, hilberta spaco (nomata laŭ David Hilbert) estas ĝeneraligo de eŭklida spaco kiu estas ne limigita per finia kvanto de dimensioj. Tial ĝi estas spaco, kio signifas ke ĝi havas nociojn de distanco kaj angulo (aparte la nocio de orteco). Ankaŭ, ĝi kontentigas pli teknikan kompletecon kiu certiĝas ke limigoj ekzistas kiam oni ilin atendas, kiu faciligas diversajn difinojn de kalkulo. Hilbertaj spacoj provizas ĉirkaŭtekston kun por formaligi kaj ĝeneraligi la konceptojn de la en terminoj de ajnaj kaj de la , kiu estas centra koncepto de . Hilbertaj spacoj estas gravaj en matematika formulaĵo de kvantummekaniko. 在数学裡,希尔伯特空间(英語:Hilbert space)即完备的内积空间,也就是一個帶有內積的完備向量空間。希尔伯特空间是有限维欧几里得空间的一个推广,使之不局限于實數的情形和有限的维数,但又不失完备性(而不像一般的非欧几里得空间那样破坏了完备性)。与欧几里得空间相仿,希尔伯特空间也是一个内积空间,其上有距离和角的概念(及由此引申而来的正交性与垂直性的概念)。此外,希尔伯特空间还是一个完备的空间,其上所有的柯西序列會收敛到此空間裡的一點,从而微积分中的大部分概念都可以无障碍地推广到希尔伯特空间中。 希尔伯特空间为基于任意上的多项式表示的傅立叶级数和傅立叶变换提供了一种有效的表述方式,而这也是泛函分析的核心概念之一。希尔伯特空间是公設化数学和量子力学的关键性概念之一。 Na matemática, um espaço de Hilbert é uma generalização do espaço euclidiano que não precisa estar restrita a um número finito de dimensões. É um espaço vetorial dotado de produto interno, ou seja, com noções de distância e ângulos. Esse espaço obedece uma relação de completude, que garante que os limites existem quando esperados, o que permite e facilita diversas definições da Análise. Os espaços de Hilbert permitem que, de certa maneira, noções intuitivas sejam aplicadas em espaços funcionais. Por exemplo, com eles podemos generalizar os conceitos de séries de Fourier em termos de . Os espaços de Hilbert são de importância crucial para a Mecânica Quântica. Espaços de Hilbert foram criados por David Hilbert, que os estudou no contexto de equações integrais. John von Neumann criou a nomenclatura "der abstrakte Hilbertsche Raum" em seu famoso trabalho em operadores Hermitianos não limitados, publicado em 1929. Talvez, John Von Neumann seja o matemático que melhor reconheceu a importância desse trabalho original. Os elementos de espaço de Hilbert abstratos são chamados vetores. Em aplicações, eles são tipicamente sequências de números complexos ou funções. Em Mecânica Quântica, por exemplo, um sistema físico é descrito por um espaço de Hilbert complexo que contém os , que contém todas as informações do sistema e complexidades multifocais. Ett hilbertrum (efter David Hilbert) är inom matematiken ett inre produktrum som är fullständigt med avseende på den norm som definieras av den inre produkten. Hilbertrum generaliserar och klargör begrepp såsom vissa linjära transformationer (till exempel fouriertransformer) och är absolut nödvändiga i formuleringen av kvantmekaniken. Hilbertrum studeras inom funktionalanalys. In de functionaalanalyse, een deelgebied van de wiskunde, is een hilbertruimte, vernoemd naar de Duitse wiskundige David Hilbert, een abstracte reële of complexe vectorruimte die voorzien is van de extra structuur van een inwendig product. Een hilbertruimte is een algemene vorm van het begrip euclidische ruimte en breidt de methoden van de lineaire algebra en de analyse van het tweedimensionale euclidische vlak en de driedimensionale ruimte uit naar ruimten met een eindig of oneindig aantal dimensies. Hierdoor zijn de begrippen lengte en hoek in een hilbertruimte gedefinieerd en kunnen lengten en hoeken in een hilbertruimte altijd gemeten worden. In aanvulling hierop vereist men verder dat hilbertruimten volledig met betrekking tot de daardoor gedefinieerde norm zijn. Volledigheid houdt in dat een hilbertruimte een voldoende aantal limieten kent, zodanig dat de technieken van de analyse kunnen worden gebruikt in een hilbertruimte. Hilbertruimten blijken van nature voor te komen in de wiskunde, natuurkunde en de techniek, meestal als oneindigdimensionale functieruimten. Vanuit dit gezichtspunt werden de vroegste hilbertruimten in het eerste decennium van de 20e eeuw ook bestudeerd door David Hilbert, Erhard Schmidt en Frigyes Riesz. Hilbertruimten zijn onmisbare hulpmiddelen in de theorieën van partiële differentiaalvergelijkingen, de kwantummechanica, de fourier-analyse (met inbegrip van toepassingen in de signaalverwerking) en de ergodische theorie, die de wiskundige onderbouwing vormt voor de studie van de thermodynamica. John von Neumann bedacht 'hilbertruimte' voor het abstracte begrip dat 'ten grondslag ligt aan veel van deze uiteenlopende toepassingen'. Het succes van de methoden van de hilbertruimte luidde het begin in van een zeer vruchtbare periode voor de functionaalanalyse. Afgezien van de klassieke euclidische ruimten zijn voorbeelden van hilbertruimten ruimtes van kwadratisch integreerbare functies, rijruimten, sobolev-ruimten, bestaande uit veralgemeende functies en hardy-ruimten van holomorfe functies. Meetkundige intuïtie speelt een belangrijke rol in veel aspecten van de hilbertruimtetheorie. Analoga van de stelling van Pythagoras en de parallellogramwet zijn geldig in een hilbertruimte. Op een dieper niveau spelen loodrechte projecties op een deelruimte (het analogon van het bepalen van de hoogtelijn in een driehoek) een belangrijke rol in optimalisatieproblemen en andere aspecten van de theorie. Een element van een hilbertruimte kan uniek worden bepaald dor zijn coördinaten met betrekking tot een verzameling van coördinaatassen (een orthonormale basis), in analogie met cartesiaanse coördinaten in het vlak. Als deze verzameling van coördinatenassen aftelbaar oneindig is, betekent dit dat een hilbertruimte ook beschreven kan worden in termen van oneindige rijen die kwadratisch optelbaar zijn. Lineaire afbeeldingen op een hilbertruimte zijn eveneens vrij concrete objecten: in veel gevallen zijn het transformaties die de ruimte met verschillende factoren in onderling loodrechte richtingen oprekken of inkrimpen in een betekenis die gepreciseerd wordt door het bestuderen van hun spectrum. Elke hilbertruimte is een banachruimte, maar niet alle banachruimten zijn hilbertruimten. Als een banachruimte een hilbertruimte is, kan men het inwendig product van de hilbertruimte op eenduidige wijze reconstrueren uit de normfunctie.
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