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rdfs:label
Espacio euclídeo Euklidischer Raum Espace euclidien Euclidische ruimte Przestrzeń euklidesowa Euklidiskt rum Euclidean space Евклідів простір Espaço euclidiano Ruang Euklides 유클리드 공간 Spazio euclideo Евклидово пространство Euklidear espazio فضاء إقليدي ユークリッド空間 欧几里得空间 Eŭklida spaco Espai euclidià Eukleidovský prostor Ευκλείδειος χώρος
rdfs:comment
In matematica, uno spazio euclideo è uno spazio affine in cui valgono gli assiomi e i postulati della geometria euclidea. Si tratta dello spazio di tutte le n-uple di numeri reali, che viene munito di un prodotto interno reale (prodotto scalare) per definire i concetti di distanza, lunghezza e angolo. È un particolare esempio di spazio affine reale che fornisce una generalizzazione degli spazi a due e a tre dimensioni studiati dalla geometria euclidea. Lo spazio euclideo è uno spazio di Hilbert reale a dimensione finita. In de meetkunde, een deelgebied van de wiskunde, is de euclidische ruimte het euclidische vlak en de driedimensionale ruimte binnen de euclidische meetkunde, alsmede de generalisaties van deze begrippen naar hogere dimensies. De term “euclidisch” wordt gebruikt om deze ruimten te onderscheiden van ruimten waarin afstanden en/of hoeken geen betekenis hebben, en van de gekromde ruimten uit de niet-euclidische meetkunde en de ruimtetijd uit Einsteins algemene relativiteitstheorie. Ett euklidiskt rum är ett reellt vektorrum där en skalärprodukt är definierad. Ibland krävs också att rummet är . Jämför inre produktrum. Euklidiska rum är så kallade för att de bland den moderna matematikens många typer av rum närmast motsvarar Euklides rumsbegrepp. I ett euklidiskt rum kan man definiera vinklar, avstånd, linjer, plan och så vidare; och dessa objekt har de egenskaper de tillskrivs i euklidisk geometri. Däremot skulle Euklides själv inte känts vid definitionen av ett modernt euklidiskt rum, eftersom denna involverar begrepp som preciserades först på 1800-talet. Dalam matematika, ruang Euklides adalah geometri euklides, serta generalisasi dari konsep-konsep . Di dalam ruang Euklides dua dimensi, titik dinyatakan oleh pasangan terurut, , bilangan, di mana bilangan pertama yang menurut konvensi menyatakan horizontal dan sering dituliskan sebagai , dan bilangan kedua secara konvensi menyatakan dan sering dituliskan sebagai . Gagasan ini mudah diperumum ke dalam ruang Euclid tiga dimensi, di mana titik dinyatakan oleh pasangan terurut ganda-tiga, , dengan bilangan tambahan ketiga menyatakan kedalaman dan diwakili oleh z. Perumumuman lebih lanjut dinyatakan oleh pasangan terurut ganda-n, di mana n adalah dimensi ruang tempat titik berada. 欧几里得几何是在约公元前300年,由古希腊数学家欧几里得建立的角和空间中距离之间联系的法则。欧几里得首先开发了处理平面上二维物体的“平面几何”,他接着分析三维物体的“立体几何”,所有欧几里得的公理被编排到幾何原本。 这些数学空间可以被扩展来应用于任何有限维度,而这种空间叫做 n维欧几里得空间(甚至简称  维空间)或有限维实内积空间。 这些数学空间还可被扩展到任意维的情形,称为实内积空间(不一定完备),希尔伯特空间在高等代数教科书中也被称为欧几里得空间。为了开发更高维的欧几里得空间,空间的性质必须非常仔细的表达并被扩展到任意维度。尽管结果的数学非常抽象,它却呈现了我们熟悉的欧几里得空间的根本本质,根本性质是它的平面性。另外也存在其他種類的空间,例如球面非欧几里得空间,相对论所描述的四维时空在重力出现的时候也不是欧几里得空间。 Espaço euclidiano é um espaço vetorial real de dimensão finita munido de um produto interno. Por volta de 300 a.C., o matemático grego Euclides estabeleceu as leis do que veio a ser chamado “Geometria euclidiana”, que é o estudo das relações entre ângulos e distâncias no espaço. Euclides desenvolveu primeiramente “a geometria plana” que trata da geometria de objetos bidimensionais em uma superfície plana. Ele então desenvolveu a “geometria sólida”, com que analisou a geometria de objetos tridimensionais. Todos os axiomas de Euclides foram codificados em um espaço matemático abstrato conhecido como espaço euclidiano bi ou tridimensional. Estes espaços matemáticos podem ser estendidos a qualquer dimensão, e tal espaço é chamado espaço euclidiano n-dimensional ou um n-espaço. Este artigo se r الفضاء الإقليدي (بالإنجليزية: Euclidean space)‏ أو الفضاء المتجهي الإقليدي هو فضاء متجهي معرف على حقل الأعداد الحقيقية مزود بجداء سلمي (لكل عنصر و من ) و بُعده منتهٍ. الفضاء المتجهي مثال على الفضاء الإقليدي. الفضاء الإقليدي هو الفضاء الرئيسي للهندسة الكلاسيكية. في الأصل كان هذا الفضاء معرفا فضاء ثنائيَ وثلاثيَ الأبعاد. فيما بعد، عُمم ليصبح من الدرجة n. In der Mathematik ist der euklidische Raum zunächst der „Raum unserer Anschauung“ („Anschauungsraum“), wie er in Euklids Elementen durch Axiome und Postulate beschrieben wird (vgl. euklidische Geometrie). Bis ins 19. Jahrhundert wurde davon ausgegangen, dass dadurch der uns umgebende physikalische Raum beschrieben wird. Der Zusatz „euklidisch“ wurde nötig, nachdem in der Mathematik allgemeinere Raumkonzepte (z. B. hyperbolischer Raum, riemannsche Mannigfaltigkeiten) entwickelt wurden und es sich im Rahmen der speziellen und allgemeinen Relativitätstheorie zeigte, dass zur Beschreibung des Raums in der Physik andere Raumbegriffe benötigt werden (Minkowski-Raum, Lorentz-Mannigfaltigkeit). Un espai euclidià és un espai vectorial normat de dimensió finita, en què la norma és heretada d'un producte escalar. Евклідів простір — скінченновимірний дійсний векторний простір зі скалярним добутком. Названий на честь давньогрецького математика Евкліда із Александрії. Розширює дво-вимірну евклідову площину до тривимірного простору, і є поняттям Евклідової геометрії. Термін "евклідовий" дозволяє відрізняти ці простори від інших типів просторів, що можуть розглядатися в сучасній геометрії. Евклідів простір також узагальнюють і до більшої кількості вимірів. Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. 수학에서 유클리드 공간(영어: Euclidean space)은 유클리드가 연구했던 평면과 공간을 일반화한 것이다. 이 일반화는 유클리드가 생각했던 거리와 길이와 각도를 좌표계를 도입하여, 임의 차원의 공간으로 확장한 것이다. 이는 표준적인 유한 차원, 실수, 내적 공간이다. 경우에 따라서는 민코프스키 공간에 대비되는 말로서, 피타고라스의 정리에 의한 길이소의 제곱의 계수가 모두 양수인 공간을 이야기한다. ユークリッド空間(ユークリッドくうかん、英: Euclidean space)とは、数学における概念の1つで、エウクレイデス(ユークリッド)が研究したような幾何学(ユークリッド幾何学)の場となる平面や空間、およびその高次元への一般化である。エウクレイデスが研究した平面や空間はそれぞれ、2次元ユークリッド空間、3次元ユークリッド空間に当たり、これらは通常、ユークリッド平面、ユークリッド空間などとも呼ばれる。「ユークリッド的」という修飾辞は、これらの空間が非ユークリッド幾何やアインシュタインの相対性理論に出てくるような曲がった空間ではないことを示唆している。 古典的なギリシャ数学では、ユークリッド平面や(三次元)ユークリッド空間は所定の公準によって定義され、そこからほかの性質が定理として演繹されるものであった。現代数学では、デカルト座標と解析幾何学の考え方にしたがってユークリッド空間を定義するほうが普通である。そうすれば、幾何学の問題に代数学や解析学の道具を持ち込んで調べることができるようになるし、三次元以上のユークリッド空間への一般化も容易になるといった利点が生まれる。 Eukleidovský prostor je matematický výraz pro člověku nejbližší, intuitivní představu prostoru. V tomto pojetí prostoru, formalizovaném Eukleidovými axiomy, začíná školní vzdělávací proces; týká se především geometrie, ale také fyziky a algebry. Pojmu se užívá zejména v kontrastu k jiným prostorům. El espacio euclídeo (también llamado espacio euclidiano) es un tipo de espacio geométrico donde se satisfacen los axiomas de Euclides de la geometría. La recta real, el plano euclídeo y el espacio n-dimensional de la geometría euclidiana son casos especiales de espacios euclidianos de dimensiones 1, 2 y 3 respectivamente. El concepto abstracto de espacio euclídeo generaliza esas construcciones a más dimensiones. Un espacio euclídeo es un espacio vectorial completo dotado de un producto interno (lo cual lo convierte además en un espacio afín, un espacio métrico y una variedad riemanniana al mismo tiempo).... Ο Ευκλείδειος χώρος είναι ο θεμελιώδης χώρος της κλασικής γεωμετρίας. Αρχικά ήταν ο τρισδιάστατος χώρος της Ευκλείδειας γεωμετρίας, αλλά στα σύγχρονα μαθηματικά υπάρχουν Ευκλείδειοι χώροι οποιουδήποτε αριθμού (ακέραιου μη αρνητικού) διαστάσεων, συμπεριλαμβανομένου του τρισδιάστατου χώρου και του Ευκλείδειου επιπέδου (δύο διαστάσεων). Εισήχθη ως έννοια από τον αρχαίο Έλληνα μαθηματικό Ευκλείδη της Αλεξάνδρειας και ο χαρακτηρισμός Ευκλείδειος χρησιμοποιείται για να τον διακρίνει από άλλους χώρους που ανακαλύφθηκαν αργότερα στη φυσική και τα σύγχρονα μαθηματικά. En matematiko, eŭklida spaco estas ĝeneraligo de la 2- kaj 3-dimensiaj spacoj kiujn studis Eŭklido. La ĝeneraligo aplikas eŭklida koncepto de distanco, kaj la rilatantajn konceptoj de longo kaj angulo, al koordinatsistemo kiu konsistas el nombraj dimensioj. Ĝi estas la "normo" ekzemplo por findimensia reela . Eŭklida spaco estas aparta metrika spaco kiu kapabligas la esploron de topologiaj aferoj kiel kompakteco. Ena produta spaco estas ĝeneraligo de Eŭklida spaco. Ambaŭ enaj produtaj spacoj kaj metrikaj spacoj estas esploritaj de . En mathématiques, un espace euclidien est un objet algébrique permettant de généraliser de façon naturelle la géométrie traditionnelle développée par Euclide, dans ses Éléments. Une géométrie de cette nature modélise, en physique classique, le plan ainsi que l'espace qui nous entoure. Un espace euclidien permet également de traiter les dimensions supérieures ; il est défini par la donnée d'un espace vectoriel sur le corps des réels, de dimension finie, muni d'un produit scalaire, qui permet de « mesurer » distances et angles. Espazio euklidearra matematikan espazio geometriko bat da, zeinetan Euklidesen axiomak bete ahal diren. zuzena, planoa eta espazio tridimentsionala euklidear espazioaren kasu bereziak dira, 1, 2 eta 3 dimentsiokoak hurrenez hurren. Euklidear espazioan kontzeptu abstraktu hori dimentsio gehigarrietara eraman daiteke. Euklidear hitza erabiltzen da beste espazio mota batzuetatik bereizteko, adibidez eta Einsteinen erlatibitatearen teorian. Euklidear espazio batek n dimentsio izan ditzakeenez euklidear espazio n-dimentsional deitu ohi zaio . Przestrzeń euklidesowa – przestrzeń opisywana przez geometrię euklidesową. Model ten stanowi dobre przybliżenie przestrzeni fizycznej, jeśli za jej pomocą opisuje się odległości makroskopowe. Nie nadaje się do opisu przestrzeni fizycznej w odległościach bardzo małych, atomowych, gdy rolę zaczynają odgrywać efekty kwantowe lub w pobliżu masywnych obiektów astronomicznych, jak Słońce, czarne dziury – gdy rolę zaczynają grać efekty zakrzywienia przestrzeni i geometria staje się nieeuklidesowa. Евкли́дово простра́нство (также эвкли́дово пространство) в изначальном смысле — это пространство, свойства которого описываются аксиомами евклидовой геометрии. В этом случае предполагается, что пространство имеет размерность, равную 3, то есть является трёхмерным. -мерное евклидово пространство обычно обозначается ;также часто используется обозначение , когда из контекста ясно, что пространство снабжено естественной евклидовой структурой.
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Euclidean space
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In der Mathematik ist der euklidische Raum zunächst der „Raum unserer Anschauung“ („Anschauungsraum“), wie er in Euklids Elementen durch Axiome und Postulate beschrieben wird (vgl. euklidische Geometrie). Bis ins 19. Jahrhundert wurde davon ausgegangen, dass dadurch der uns umgebende physikalische Raum beschrieben wird. Der Zusatz „euklidisch“ wurde nötig, nachdem in der Mathematik allgemeinere Raumkonzepte (z. B. hyperbolischer Raum, riemannsche Mannigfaltigkeiten) entwickelt wurden und es sich im Rahmen der speziellen und allgemeinen Relativitätstheorie zeigte, dass zur Beschreibung des Raums in der Physik andere Raumbegriffe benötigt werden (Minkowski-Raum, Lorentz-Mannigfaltigkeit). Im Laufe der Zeit wurde Euklids Geometrie auf verschiedene Arten präzisiert und verallgemeinert: * axiomatisch durch Hilbert (siehe Hilberts Axiomensystem der euklidischen Geometrie), * als euklidischer Vektorraum (einem über definierten Vektorraum mit Skalarprodukt), * als euklidischer Punktraum (einem affinen Raum, der über einem euklidischen Vektorraum modelliert ist), * als Koordinatenraum mit dem Standardskalarprodukt. Wenn vom euklidischen Raum die Rede ist, dann kann jede dieser Definitionen gemeint sein oder auch eine höherdimensionale Verallgemeinerung. Den zweidimensionalen euklidischen Raum nennt man auch euklidische Ebene. In diesem zweidimensionalen Fall wird der Begriff in der synthetischen Geometrie etwas allgemeiner gefasst: Euklidische Ebenen können dort als affine Ebenen über einer allgemeineren Klasse von Körpern, den euklidischen Körpern definiert werden. Diese Körper sind (je nach Auffassung) Teilkörper oder isomorph zu Teilkörpern von Vom affinen Raum unterscheidet sich der euklidische dadurch, dass man Längen und Winkel messen kann. Man zeichnet deshalb die Abbildungen aus, die Längen und Winkel erhalten. Diese nennt man traditionell Kongruenzabbildungen, andere Bezeichnungen sind Bewegungen und Isometrien. Der einem pseudoeuklidischen Raum (en. Pseudo-Euclidean space) zugrunde liegende Vektorraum besitzt ein Pseudoskalarprodukt, d. h. eine im Allgemeinen nicht positiv definite symmetrische Bilinearform. In den nichteuklidischen Räumen, so dem hyperbolischen und dem elliptischen Raum, gilt das Parallelenaxiom nicht. Eukleidovský prostor je matematický výraz pro člověku nejbližší, intuitivní představu prostoru. V tomto pojetí prostoru, formalizovaném Eukleidovými axiomy, začíná školní vzdělávací proces; týká se především geometrie, ale také fyziky a algebry. Pojmu se užívá zejména v kontrastu k jiným prostorům. En matematiko, eŭklida spaco estas ĝeneraligo de la 2- kaj 3-dimensiaj spacoj kiujn studis Eŭklido. La ĝeneraligo aplikas eŭklida koncepto de distanco, kaj la rilatantajn konceptoj de longo kaj angulo, al koordinatsistemo kiu konsistas el nombraj dimensioj. Ĝi estas la "normo" ekzemplo por findimensia reela . Eŭklida spaco estas aparta metrika spaco kiu kapabligas la esploron de topologiaj aferoj kiel kompakteco. Ena produta spaco estas ĝeneraligo de Eŭklida spaco. Ambaŭ enaj produtaj spacoj kaj metrikaj spacoj estas esploritaj de . Eŭklida spaco ludas rolon en la difino de sternaĵo kiu kunigas konceptojn de ambaŭ eŭklida geometrio kaj neeŭklida geometrio. Unu matematika motivado por difinanta distanca funkcio estas ebleco por difini malfermitan pilkon ĉirkaŭ punktoj en la spaco. Ĉi tiu fundamenta koncepto similigas diferencialan kalkulon inter eŭklida spaco kaj aliaj sternaĵoj. Diferenciala geometrio enkondukas tian diferencialan kalkulo, kaj ankaŭ teknikon de movebla, loka eŭklida spaco, por esplori propraĵojn de neeŭklidaj sternaĵoj. El espacio euclídeo (también llamado espacio euclidiano) es un tipo de espacio geométrico donde se satisfacen los axiomas de Euclides de la geometría. La recta real, el plano euclídeo y el espacio n-dimensional de la geometría euclidiana son casos especiales de espacios euclidianos de dimensiones 1, 2 y 3 respectivamente. El concepto abstracto de espacio euclídeo generaliza esas construcciones a más dimensiones. Un espacio euclídeo es un espacio vectorial completo dotado de un producto interno (lo cual lo convierte además en un espacio afín, un espacio métrico y una variedad riemanniana al mismo tiempo).... El término euclídeo se utiliza para distinguir estos espacios de los espacios "curvos", de las geometrías no euclidianas y del espacio de la teoría de la relatividad de Einstein. Para resaltar el hecho de que un espacio euclídeo puede poseer n dimensiones, se suele hablar de "espacio euclídeo n-dimensional" (denotado , o incluso ). Espaço euclidiano é um espaço vetorial real de dimensão finita munido de um produto interno. Por volta de 300 a.C., o matemático grego Euclides estabeleceu as leis do que veio a ser chamado “Geometria euclidiana”, que é o estudo das relações entre ângulos e distâncias no espaço. Euclides desenvolveu primeiramente “a geometria plana” que trata da geometria de objetos bidimensionais em uma superfície plana. Ele então desenvolveu a “geometria sólida”, com que analisou a geometria de objetos tridimensionais. Todos os axiomas de Euclides foram codificados em um espaço matemático abstrato conhecido como espaço euclidiano bi ou tridimensional. Estes espaços matemáticos podem ser estendidos a qualquer dimensão, e tal espaço é chamado espaço euclidiano n-dimensional ou um n-espaço. Este artigo se refere a tais espaços matemáticos. Para desenvolver esses espaços euclidianos de dimensões mais elevadas, as propriedades dos espaços euclidianos conhecidos devem ser expressas e então estendidas a uma dimensão arbitrária. Embora a matemática resultante seja um tanto abstrata, ela captura a natureza essencial dos espaços euclidianos com que todos nós estamos familiarizados. Uma propriedade essencial de um espaço euclidiano é sua planitude. Existem outros espaços que não são euclidianos. Por exemplo, o espaço-tempo quadridimensional descrito pela teoria da relatividade quando a gravidade está presente não é euclidiano. Евклідів простір — скінченновимірний дійсний векторний простір зі скалярним добутком. Названий на честь давньогрецького математика Евкліда із Александрії. Розширює дво-вимірну евклідову площину до тривимірного простору, і є поняттям Евклідової геометрії. Термін "евклідовий" дозволяє відрізняти ці простори від інших типів просторів, що можуть розглядатися в сучасній геометрії. Евклідів простір також узагальнюють і до більшої кількості вимірів. В класичній давньогрецькій геометрії існує визначення евклідової площини і тривимірного евклідового простору, що ґрунтується на певних постулатах, в той час як інші властивості цих просторів виведені як теореми. Також використовувалися геометричні побудови для визначення раціональних чисел, що є відношеннями [джерело?]. Коли алгебра і математичний аналіз набули достатнього розвиту, цей зв'язок зберігся і тепер більш загальним стало визначення Евклідового простору на основі векторних просторів, що дозволяють використовувати декартові координати і методи алгебри та диференціального та інтегрального числення. Це означає, що точки визначають за допомогою трійок дійсних чисел, які називаються координатними векторами, а геометричні фігури описують рівняннями і нерівностями, що визначають співвідношення цих координат. Цей підхід також дозволяє легко узагальнити w. геометрію до евклідових просторів до просторів більшої розмірності. Евклідів простір визначено за допомогою аксіом, які не вказують як саме мають бути представлені точки цього простору. Евклідів простір може бути побудований за допомогою декартової системи координат, як один із можливих способів його представлення. В такому випадку, Евклідів простір моделюють застосовуючи дійсний простір координат, що має таку ж розмірність. Для одного виміру це була б шкала дійсних чисел; для двох вимірів, він представляється декартовою системою координат на площині; і для більшої кількості вимірів, це є із трьома або більше координатами, що представлені дійсними числами. Математики позначають -вимірний Евклідів простір як , якщо вони хочуть підкреслити його природу та властивості, але також використовують позначення , оскільки ці дві структури мають подібні властивості і їх як правило не розрізняють. Евклідові простори мають скінченну кількість вимірів. Dalam matematika, ruang Euklides adalah geometri euklides, serta generalisasi dari konsep-konsep . Di dalam ruang Euklides dua dimensi, titik dinyatakan oleh pasangan terurut, , bilangan, di mana bilangan pertama yang menurut konvensi menyatakan horizontal dan sering dituliskan sebagai , dan bilangan kedua secara konvensi menyatakan dan sering dituliskan sebagai . Gagasan ini mudah diperumum ke dalam ruang Euclid tiga dimensi, di mana titik dinyatakan oleh pasangan terurut ganda-tiga, , dengan bilangan tambahan ketiga menyatakan kedalaman dan diwakili oleh z. Perumumuman lebih lanjut dinyatakan oleh pasangan terurut ganda-n, di mana n adalah dimensi ruang tempat titik berada. En mathématiques, un espace euclidien est un objet algébrique permettant de généraliser de façon naturelle la géométrie traditionnelle développée par Euclide, dans ses Éléments. Une géométrie de cette nature modélise, en physique classique, le plan ainsi que l'espace qui nous entoure. Un espace euclidien permet également de traiter les dimensions supérieures ; il est défini par la donnée d'un espace vectoriel sur le corps des réels, de dimension finie, muni d'un produit scalaire, qui permet de « mesurer » distances et angles. La donnée d'un produit scalaire permet par exemple de définir la notion de bases particulières dites orthonormales, d'établir une relation canonique entre l'espace et son dual, ou de préciser des familles d'endomorphismes faciles à réduire. Il permet aussi de définir une norme et par conséquent une distance donc une topologie, ce qui met à disposition les méthodes d'analyse. Les espaces euclidiens possèdent une longue histoire ainsi que de nombreuses applications. Les relations entre cet outil et le reste des mathématiques sont multiples et variées, depuis la logique et l'algèbre jusqu'aux géométries non euclidiennes. Cet aspect est traité dans l'article « Géométrie euclidienne ». Евкли́дово простра́нство (также эвкли́дово пространство) в изначальном смысле — это пространство, свойства которого описываются аксиомами евклидовой геометрии. В этом случае предполагается, что пространство имеет размерность, равную 3, то есть является трёхмерным. В современном понимании, в более общем смысле, может обозначать один из сходных и тесно связанных объектов: конечномерное вещественное векторное пространство с введённым на нём положительно определённым скалярным произведением; либо метрическое пространство, соответствующее такому векторному пространству. Некоторые авторы ставят знак равенства между евклидовым и предгильбертовым пространством. В этой статье за исходное будет взято первое определение. -мерное евклидово пространство обычно обозначается ;также часто используется обозначение , когда из контекста ясно, что пространство снабжено естественной евклидовой структурой. In de meetkunde, een deelgebied van de wiskunde, is de euclidische ruimte het euclidische vlak en de driedimensionale ruimte binnen de euclidische meetkunde, alsmede de generalisaties van deze begrippen naar hogere dimensies. De term “euclidisch” wordt gebruikt om deze ruimten te onderscheiden van ruimten waarin afstanden en/of hoeken geen betekenis hebben, en van de gekromde ruimten uit de niet-euclidische meetkunde en de ruimtetijd uit Einsteins algemene relativiteitstheorie. In de klassieke Griekse meetkunde werden het euclidische vlak en de euclidische driedimensionale ruimte met behulp van bepaalde postulaten gedefinieerd. De andere eigenschappen van deze ruimtes werden vervolgens als stellingen gededuceerd. In de moderne wiskunde is het gebruikelijker om de euclidische ruimte met behulp van cartesiaanse coördinaten en de ideeën van de analytische meetkunde te definiëren. Deze aanpak maakt het mogelijk bij het beantwoorden van meetkundige vragen gebruik te maken van de instrumenten uit de algebra en de analyse. Tevens heeft deze werkwijze als voordeel dat het niet moeilijk is om euclidische ruimten te generaliseren naar meer dan drie dimensies. In matematica, uno spazio euclideo è uno spazio affine in cui valgono gli assiomi e i postulati della geometria euclidea. Si tratta dello spazio di tutte le n-uple di numeri reali, che viene munito di un prodotto interno reale (prodotto scalare) per definire i concetti di distanza, lunghezza e angolo. È un particolare esempio di spazio affine reale che fornisce una generalizzazione degli spazi a due e a tre dimensioni studiati dalla geometria euclidea. Lo spazio euclideo è uno spazio di Hilbert reale a dimensione finita. Ο Ευκλείδειος χώρος είναι ο θεμελιώδης χώρος της κλασικής γεωμετρίας. Αρχικά ήταν ο τρισδιάστατος χώρος της Ευκλείδειας γεωμετρίας, αλλά στα σύγχρονα μαθηματικά υπάρχουν Ευκλείδειοι χώροι οποιουδήποτε αριθμού (ακέραιου μη αρνητικού) διαστάσεων, συμπεριλαμβανομένου του τρισδιάστατου χώρου και του Ευκλείδειου επιπέδου (δύο διαστάσεων). Εισήχθη ως έννοια από τον αρχαίο Έλληνα μαθηματικό Ευκλείδη της Αλεξάνδρειας και ο χαρακτηρισμός Ευκλείδειος χρησιμοποιείται για να τον διακρίνει από άλλους χώρους που ανακαλύφθηκαν αργότερα στη φυσική και τα σύγχρονα μαθηματικά. Οι αρχαίοι Έλληνες γεωμέτρες εισήγαγαν την έννοια του Ευκλείδειου χώρου για να περιγράψουν τον χώρο του αντιληπτού φυσικού κόσμου, όπως παρατηρείται στην καθημερινή ζωή, σύμφωνα με την διαίσθηση και την λογική. Μετά την εισαγωγή των μη Ευκλείδεων γεωμετριών στα τέλη του 19ου αιώνα, παρουσιάστηκε η ανάγκη επαναπροσδιορισμού των Ευκλείδειων χώρων μέσω της αξιωματικής μεθόδου, δηλ. χρησιμοποιώντας θεμελιώδη αξιώματα τα οποία ορίζουν τις ιδιότητες του χώρου. Ένας άλλος ορισμός των Ευκλείδειων χώρων, μέσω των διανυσματικών χώρων και της γραμμικής άλγεβρας, έχει αποδειχθεί ότι είναι ισοδύναμος με τον αξιωματικό ορισμό. Αυτός είναι ο ορισμός που χρησιμοποιείται πιο συχνά στα σύγχρονα μαθηματικά. Γενικότερα, οι Ευκλείδειοι χώροι αποτελούνται από σημεία, τα οποία καθορίζονται μόνο από τις ιδιότητες που πρέπει να έχουν για να σχηματίσουν έναν Ευκλείδειο χώρο. Υπάρχει ουσιαστικά μόνο ένας Ευκλείδειος χώρος της κάθε διάστασης, εννοώντας ότι όλοι οι Ευκλείδειοι χώροι μιας δεδομένης διάστασης είναι ισομορφικοί (άρα και ισοδύναμοι) μεταξύ τους. Επομένως, σε πολλές περιπτώσεις, είναι δυνατόν να εργαστούμε σε ένα συγκεκριμένο Ευκλείδειο χώρο, που είναι γενικά ο πραγματικός n-διάστατος χώρος χρησιμοποιώντας το εσωτερικό γινόμενο. Ένας ισομορφισμός από έναν Ευκλείδειο χώρο στο , συσχετίζει κάθε σημείο με μια ν-άδα (πλειάδα) πραγματικών αριθμών, οι οποίοι ορίζουν αυτό το σημείο στον Ευκλείδειο χώρο και ονομάζονται Καρτεσιανές συντεταγμένες του σημείου. Przestrzeń euklidesowa – przestrzeń opisywana przez geometrię euklidesową. Model ten stanowi dobre przybliżenie przestrzeni fizycznej, jeśli za jej pomocą opisuje się odległości makroskopowe. Nie nadaje się do opisu przestrzeni fizycznej w odległościach bardzo małych, atomowych, gdy rolę zaczynają odgrywać efekty kwantowe lub w pobliżu masywnych obiektów astronomicznych, jak Słońce, czarne dziury – gdy rolę zaczynają grać efekty zakrzywienia przestrzeni i geometria staje się nieeuklidesowa. Jednowymiarową przestrzeń euklidesową nazywa się prostą euklidesową, a dwuwymiarową – płaszczyzną euklidesową. Przestrzenie euklidesowe nazywa się również afinicznymi przestrzeniami euklidesowymi, w odróżnieniu od liniowych przestrzeni euklidesowych, nazywanych też przestrzeniami unitarnymi. Kluczową własnością przestrzeni euklidesowych jest ich „płaskość”. W geometrii wyróżnia się inne przestrzenie, które nie są euklidesowe. Np. sfera jest przestrzenią nieeuklidesową, gdyż kąty trójkąta na sferze sumują się do wartości większej niż 180 stopni, inaczej niż na płaszczyźnie euklidesowej. Geometria rozważa przestrzenie wielowymiarowe. Dla danej liczby naturalnej n istnieje dokładnie jedna przestrzeń euklidesowa o wymiarze n, zaś przestrzeni nieeuklidesowych wymiaru n jest nieskończenie wiele. Te ostatnie można konstruować np. poprzez deformację przestrzeni euklidesowej. ユークリッド空間(ユークリッドくうかん、英: Euclidean space)とは、数学における概念の1つで、エウクレイデス(ユークリッド)が研究したような幾何学(ユークリッド幾何学)の場となる平面や空間、およびその高次元への一般化である。エウクレイデスが研究した平面や空間はそれぞれ、2次元ユークリッド空間、3次元ユークリッド空間に当たり、これらは通常、ユークリッド平面、ユークリッド空間などとも呼ばれる。「ユークリッド的」という修飾辞は、これらの空間が非ユークリッド幾何やアインシュタインの相対性理論に出てくるような曲がった空間ではないことを示唆している。 古典的なギリシャ数学では、ユークリッド平面や(三次元)ユークリッド空間は所定の公準によって定義され、そこからほかの性質が定理として演繹されるものであった。現代数学では、デカルト座標と解析幾何学の考え方にしたがってユークリッド空間を定義するほうが普通である。そうすれば、幾何学の問題に代数学や解析学の道具を持ち込んで調べることができるようになるし、三次元以上のユークリッド空間への一般化も容易になるといった利点が生まれる。 現代的な観点では、ユークリッド空間は各次元に本質的に一つだけ存在すると考えられる。たとえば一次元なら実数直線、二次元なら、より高次の場合は実数の組を座標にもつ実座標空間である。つまり、ユークリッド空間の「点」は実数からなる組であり、二点間の距離は二点間の距離の公式に従うものとして定まる。n-次元ユークリッド空間は、(標準的なモデルを与えるものという意味で)しばしば Rn とかかれるが、(余分な構造を想起させない)ユークリッド空間固有の性質を備えたものということを強調する意味で En と書かれることもある。ふつう、ユークリッド空間といえば有限次元であるものをいう。 Ett euklidiskt rum är ett reellt vektorrum där en skalärprodukt är definierad. Ibland krävs också att rummet är . Jämför inre produktrum. Euklidiska rum är så kallade för att de bland den moderna matematikens många typer av rum närmast motsvarar Euklides rumsbegrepp. I ett euklidiskt rum kan man definiera vinklar, avstånd, linjer, plan och så vidare; och dessa objekt har de egenskaper de tillskrivs i euklidisk geometri. Däremot skulle Euklides själv inte känts vid definitionen av ett modernt euklidiskt rum, eftersom denna involverar begrepp som preciserades först på 1800-talet. Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the Euclidean plane (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements, with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate). After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space equipped with the dot product. An isomorphism from a Euclidean space to associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point. Un espai euclidià és un espai vectorial normat de dimensió finita, en què la norma és heretada d'un producte escalar. Espazio euklidearra matematikan espazio geometriko bat da, zeinetan Euklidesen axiomak bete ahal diren. zuzena, planoa eta espazio tridimentsionala euklidear espazioaren kasu bereziak dira, 1, 2 eta 3 dimentsiokoak hurrenez hurren. Euklidear espazioan kontzeptu abstraktu hori dimentsio gehigarrietara eraman daiteke. Euklidear hitza erabiltzen da beste espazio mota batzuetatik bereizteko, adibidez eta Einsteinen erlatibitatearen teorian. Euklidear espazio batek n dimentsio izan ditzakeenez euklidear espazio n-dimentsional deitu ohi zaio . 欧几里得几何是在约公元前300年,由古希腊数学家欧几里得建立的角和空间中距离之间联系的法则。欧几里得首先开发了处理平面上二维物体的“平面几何”,他接着分析三维物体的“立体几何”,所有欧几里得的公理被编排到幾何原本。 这些数学空间可以被扩展来应用于任何有限维度,而这种空间叫做 n维欧几里得空间(甚至简称  维空间)或有限维实内积空间。 这些数学空间还可被扩展到任意维的情形,称为实内积空间(不一定完备),希尔伯特空间在高等代数教科书中也被称为欧几里得空间。为了开发更高维的欧几里得空间,空间的性质必须非常仔细的表达并被扩展到任意维度。尽管结果的数学非常抽象,它却呈现了我们熟悉的欧几里得空间的根本本质,根本性质是它的平面性。另外也存在其他種類的空间,例如球面非欧几里得空间,相对论所描述的四维时空在重力出现的时候也不是欧几里得空间。 수학에서 유클리드 공간(영어: Euclidean space)은 유클리드가 연구했던 평면과 공간을 일반화한 것이다. 이 일반화는 유클리드가 생각했던 거리와 길이와 각도를 좌표계를 도입하여, 임의 차원의 공간으로 확장한 것이다. 이는 표준적인 유한 차원, 실수, 내적 공간이다. 경우에 따라서는 민코프스키 공간에 대비되는 말로서, 피타고라스의 정리에 의한 길이소의 제곱의 계수가 모두 양수인 공간을 이야기한다. الفضاء الإقليدي (بالإنجليزية: Euclidean space)‏ أو الفضاء المتجهي الإقليدي هو فضاء متجهي معرف على حقل الأعداد الحقيقية مزود بجداء سلمي (لكل عنصر و من ) و بُعده منتهٍ. الفضاء المتجهي مثال على الفضاء الإقليدي. الفضاء الإقليدي هو الفضاء الرئيسي للهندسة الكلاسيكية. في الأصل كان هذا الفضاء معرفا فضاء ثنائيَ وثلاثيَ الأبعاد. فيما بعد، عُمم ليصبح من الدرجة n. قدم علماء الهندسة الإغريق مفهوم الفضاء الإقليدي من أجل نمذجة الكون. الإبداع الكبير الذي جاء به هؤلاء العلماء هو البرهان على جميع خصائص هذا الفضاء في شكل مبرهنات توصلوا إليها انطلاقا من مجموعة من الموضوعات أو المسلمات أو البديهيات. قسمت هذه البديهيات إلى صنفين اثنين، أحدهما هو ما هو بديهي ولا يحتاج إلى برهان (منها على سبيل المثال، البديهية التي تنص على أنه لا يمر أكثر من خط مستقيم واحد من نقطتين معلومتين) والثاني هو ما يُعتقد استحالة البرهان عليه كما هو الحال بالنسبة إلى مسلمة التوازي. بعد تقديم الهندسة غير الإقليدية خلال القرن التاسع عشر، أعيدت صياغة الموضوعات القديمة من أجل إعطاء تعريف جديد للفضاءات الإقليدية من خلال . بُين أن تعريفا معتمدا على الفضاءات المتجهية والجبر الخطي يكافئ التعريف المعتمد على نظام البديهيات. هذا التعريف هو الأكثر استعمالا في الرياضيات المعاصرة.
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