Lp space (original) (raw)
En matemàtiques, els espais Lp són certs espais funcionals definits a partir de generalitzacions naturals de les p-normes dels espais vectorials de dimensió finita. S'anomenen a vegades espais de Lebesgue, en honor d'Henri Lebesgue, encara que potser van ser introduïts abans per Frigyes Riesz el 1910. Formen una classe important d'exemples d'espais de Banach dins l'anàlisi funcional.Els espais Lp tenen aplicacions en física, estadística, finances, enginyeria i altres disciplines.
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| dbo:abstract | En matemàtiques, els espais Lp són certs espais funcionals definits a partir de generalitzacions naturals de les p-normes dels espais vectorials de dimensió finita. S'anomenen a vegades espais de Lebesgue, en honor d'Henri Lebesgue, encara que potser van ser introduïts abans per Frigyes Riesz el 1910. Formen una classe important d'exemples d'espais de Banach dins l'anàlisi funcional.Els espais Lp tenen aplicacions en física, estadística, finances, enginyeria i altres disciplines. (ca) Lp prostor je v matematické analýze funkcí integrovatelných s p-tou mocninou. (cs) Die -Räume, auch Lebesgue-Räume, sind in der Mathematik spezielle Räume, die aus allen p-fach integrierbaren Funktionen bestehen. Das in der Bezeichnung geht auf den französischen Mathematiker Henri Léon Lebesgue zurück, da diese Räume über das Lebesgue-Integral definiert werden. Im Fall Banachraum-wertiger Funktionen (wie im Folgenden allgemein für Vektorräume dargestellt) bezeichnet man sie auch als Bochner-Lebesgue-Räume. Das in der Bezeichnung ist ein reeller Parameter: Für jede Zahl ist ein -Raum definiert. Die Konvergenz in diesen Räumen wird als Konvergenz im p-ten Mittel bezeichnet. (de) Los espacios son los espacios vectoriales normados más importantes en el contexto de la teoría de la medida y de la integral de Lebesgue. Reciben también el nombre de espacios de Lebesgue por el matemático Henri Lebesgue. (es) In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. (en) En mathématiques, un espace Lp est un espace vectoriel de classes des fonctions dont la puissance d'exposant p est intégrable au sens de Lebesgue, où p est un nombre réel strictement positif. Le passage à la limite de l'exposant aboutit à la construction des espaces L∞ de fonctions bornées. Les espaces Lp sont appelés espaces de Lebesgue. Identifiant les fonctions qui ne diffèrent que sur un ensemble négligeable, chaque espace Lp est un espace de Banach lorsque l'exposant est supérieur ou égal à 1. Lorsque 0 < p < 1, l'intégrale définit une quasi-norme qui en fait un espace complet. Il existe en outre une dualité entre les espaces d'exposants p et q conjugués, c'est-à-dire tels que 1⁄p + 1⁄q = 1. Les espaces Lp généralisent les espaces L2 des fonctions de carré intégrable, mais aussi les espaces ℓp de suites de puissance p-ième sommable. Diverses constructions étendent encore cette définition à l'aide de distributions ou en se contentant d'une intégrabilité locale. Tous ces espaces constituent un outil fondamental de l'analyse fonctionnelle en permettant la résolution d'équations par approximation avec des solutions non nécessairement dérivables ni même continues. (fr) 数学の分野における Lp 空間（エルピーくうかん、英: Lp space）とは、有限次元ベクトル空間に対する p-ノルムの自然な一般化を用いることで定義される関数空間である。アンリ・ルベーグの名にちなんでルベーグ空間としばしば呼ばれる が、 によると初めて導入されたのは とされている。Lp 空間は関数解析学におけるバナッハ空間や、線型位相空間の重要なクラスを形成する。物理学や統計学、金融、工学など様々な分野で応用されている。 (ja) In matematica, e più precisamente in analisi funzionale, lo spazio è lo spazio delle funzioni a p-esima potenza sommabile. Si tratta di uno spazio funzionale i cui elementi sono particolari classi di funzioni misurabili. Lo spazio delle successioni a p-esima potenza sommabile è inoltre detto spazio . In particolare, lo spazio l2 delle successioni a quadrato sommabile rappresenta un caso di notevole importanza. Gli spazi , con , sono spazi di Banach. In particolare, è anche uno spazio di Hilbert. (it) 함수해석학에서 르베그 공간(Lebesgue空間, 영어: Lebesgue space) 또는 Lp 공간(영어: Lp-space)은 절댓값의 제곱이 르베그 적분 가능한 가측 함수들의 동치류들로 구성된 노름 공간이다. (ko) In de functionaalanalyse, een deelgebied van de wiskunde, zijn Lp-ruimten functieruimten die zijn gedefinieerd door gebruik te maken van natuurlijke veralgemeningen van -normen voor eindig--dimensionale vectorruimten. Zij worden soms ook Lebesgue-ruimtes genoemd naar Henri Lebesgue, hoewel zij volgens Bourbaki in 1910 voor het eerst door Riesz werden geïntroduceerd. Zij vormen een belangrijke klasse van voorbeelden van banachruimten in de functionaalanalyse en van topologische vectorruimten. Lebesgue-ruimten vinden toepassingen in de natuurkunde, statistiek, financiën, techniek en andere disciplines. (nl) Przestrzenie – dla ustalonej liczby dodatniej – klasy przestrzeni liniowo-topologicznych, odpowiednio: takich ciągów liczbowych, że szereg -tych potęg modułów ich wyrazów jest zbieżny oraz funkcji mierzalnych, całkowalnych w -tej potędze na ustalonym zbiorze (utożsamia się funkcje równe prawie wszędzie). W przypadku to w przestrzeniach tych można w naturalny sposób zdefiniować normę i są one wtedy przestrzeniami Banacha. Przestrzenie oraz są ponadto przestrzeniami Hilberta z odpowiednio zdefiniowanym iloczynem skalarnym. Przestrzenie są szczególnymi przypadkami przestrzeni Przestrzenie znajdują zastosowanie w statystyce, ekonomii matematycznej i inżynierii. (pl) (также встречается обозначение ; читается «эль-пэ»; также — лебеговы пространства) — это пространства измеримых функций, таких, что их -я степень интегрируема, где . — важнейший класс банаховых пространств. (читается «эль-два») — классический пример гильбертова пространства. (ru) Ett -rum är ett funktionsrum inom matematik. -rummet består av funktioner som är p-integrerbara. Man behöver -rummet till exempel inom måtteori och funktionalanalys. (sv) Em matemática, sobretudo na teoria da medida e na análise funcional, os espaços são um dos mais importantes espaços funcionais. (pt) Просторами в математиці називаються простори вимірних функцій, які при піднесенні до степеня (де ) є інтегровними за Лебегом. — найважливіший клас банахових просторів. Окрім того, — класичний приклад гільбертового простору. (uk) 在数学中，Lp空间是由p次可积函数组成的空间；对应的ℓp空间是由p次可和序列组成的空间。它們有時叫做勒貝格空間。 在泛函分析和拓扑向量空间中，他们构成了巴拿赫空间一类重要的例子。Lp空间在工程学领域的有限元分析中有应用。 (zh) |
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| rdfs:comment | En matemàtiques, els espais Lp són certs espais funcionals definits a partir de generalitzacions naturals de les p-normes dels espais vectorials de dimensió finita. S'anomenen a vegades espais de Lebesgue, en honor d'Henri Lebesgue, encara que potser van ser introduïts abans per Frigyes Riesz el 1910. Formen una classe important d'exemples d'espais de Banach dins l'anàlisi funcional.Els espais Lp tenen aplicacions en física, estadística, finances, enginyeria i altres disciplines. (ca) Lp prostor je v matematické analýze funkcí integrovatelných s p-tou mocninou. (cs) Die -Räume, auch Lebesgue-Räume, sind in der Mathematik spezielle Räume, die aus allen p-fach integrierbaren Funktionen bestehen. Das in der Bezeichnung geht auf den französischen Mathematiker Henri Léon Lebesgue zurück, da diese Räume über das Lebesgue-Integral definiert werden. Im Fall Banachraum-wertiger Funktionen (wie im Folgenden allgemein für Vektorräume dargestellt) bezeichnet man sie auch als Bochner-Lebesgue-Räume. Das in der Bezeichnung ist ein reeller Parameter: Für jede Zahl ist ein -Raum definiert. Die Konvergenz in diesen Räumen wird als Konvergenz im p-ten Mittel bezeichnet. (de) Los espacios son los espacios vectoriales normados más importantes en el contexto de la teoría de la medida y de la integral de Lebesgue. Reciben también el nombre de espacios de Lebesgue por el matemático Henri Lebesgue. (es) In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. (en) 数学の分野における Lp 空間（エルピーくうかん、英: Lp space）とは、有限次元ベクトル空間に対する p-ノルムの自然な一般化を用いることで定義される関数空間である。アンリ・ルベーグの名にちなんでルベーグ空間としばしば呼ばれる が、 によると初めて導入されたのは とされている。Lp 空間は関数解析学におけるバナッハ空間や、線型位相空間の重要なクラスを形成する。物理学や統計学、金融、工学など様々な分野で応用されている。 (ja) In matematica, e più precisamente in analisi funzionale, lo spazio è lo spazio delle funzioni a p-esima potenza sommabile. Si tratta di uno spazio funzionale i cui elementi sono particolari classi di funzioni misurabili. Lo spazio delle successioni a p-esima potenza sommabile è inoltre detto spazio . In particolare, lo spazio l2 delle successioni a quadrato sommabile rappresenta un caso di notevole importanza. Gli spazi , con , sono spazi di Banach. In particolare, è anche uno spazio di Hilbert. (it) 함수해석학에서 르베그 공간(Lebesgue空間, 영어: Lebesgue space) 또는 Lp 공간(영어: Lp-space)은 절댓값의 제곱이 르베그 적분 가능한 가측 함수들의 동치류들로 구성된 노름 공간이다. (ko) In de functionaalanalyse, een deelgebied van de wiskunde, zijn Lp-ruimten functieruimten die zijn gedefinieerd door gebruik te maken van natuurlijke veralgemeningen van -normen voor eindig--dimensionale vectorruimten. Zij worden soms ook Lebesgue-ruimtes genoemd naar Henri Lebesgue, hoewel zij volgens Bourbaki in 1910 voor het eerst door Riesz werden geïntroduceerd. Zij vormen een belangrijke klasse van voorbeelden van banachruimten in de functionaalanalyse en van topologische vectorruimten. Lebesgue-ruimten vinden toepassingen in de natuurkunde, statistiek, financiën, techniek en andere disciplines. (nl) (также встречается обозначение ; читается «эль-пэ»; также — лебеговы пространства) — это пространства измеримых функций, таких, что их -я степень интегрируема, где . — важнейший класс банаховых пространств. (читается «эль-два») — классический пример гильбертова пространства. (ru) Ett -rum är ett funktionsrum inom matematik. -rummet består av funktioner som är p-integrerbara. Man behöver -rummet till exempel inom måtteori och funktionalanalys. (sv) Em matemática, sobretudo na teoria da medida e na análise funcional, os espaços são um dos mais importantes espaços funcionais. (pt) Просторами в математиці називаються простори вимірних функцій, які при піднесенні до степеня (де ) є інтегровними за Лебегом. — найважливіший клас банахових просторів. Окрім того, — класичний приклад гільбертового простору. (uk) 在数学中，Lp空间是由p次可积函数组成的空间；对应的ℓp空间是由p次可和序列组成的空间。它們有時叫做勒貝格空間。 在泛函分析和拓扑向量空间中，他们构成了巴拿赫空间一类重要的例子。Lp空间在工程学领域的有限元分析中有应用。 (zh) En mathématiques, un espace Lp est un espace vectoriel de classes des fonctions dont la puissance d'exposant p est intégrable au sens de Lebesgue, où p est un nombre réel strictement positif. Le passage à la limite de l'exposant aboutit à la construction des espaces L∞ de fonctions bornées. Les espaces Lp sont appelés espaces de Lebesgue. Les espaces Lp généralisent les espaces L2 des fonctions de carré intégrable, mais aussi les espaces ℓp de suites de puissance p-ième sommable. (fr) Przestrzenie – dla ustalonej liczby dodatniej – klasy przestrzeni liniowo-topologicznych, odpowiednio: takich ciągów liczbowych, że szereg -tych potęg modułów ich wyrazów jest zbieżny oraz funkcji mierzalnych, całkowalnych w -tej potędze na ustalonym zbiorze (utożsamia się funkcje równe prawie wszędzie). W przypadku to w przestrzeniach tych można w naturalny sposób zdefiniować normę i są one wtedy przestrzeniami Banacha. Przestrzenie oraz są ponadto przestrzeniami Hilberta z odpowiednio zdefiniowanym iloczynem skalarnym. Przestrzenie są szczególnymi przypadkami przestrzeni (pl) |
| rdfs:label | Espai Lp (ca) Lp prostor (cs) Lp-Raum (de) Espacios Lp (es) Spazio Lp (it) Espace Lp (fr) Lp space (en) 르베그 공간 (ko) Lp空間 (ja) Przestrzeń Lp (pl) Lp-ruimte (nl) Espaço Lp (pt) Lp (пространство) (ru) Lp-rum (sv) Lp空间 (zh) Простір Lp (uk) |
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