Sequence space (original) (raw)
Ein Folgenraum ist ein in der Mathematik betrachteter Vektorraum, dessen Elemente Zahlenfolgen sind.Viele in der Funktionalanalysis auftretende Vektorräume sind Folgenräume oder können durch solche repräsentiert werden.Zu den Beispielen zählen u. a. die wichtigen Räume wie aller beschränkten Folgen oder aller gegen 0 konvergenten Folgen.Die Folgenräume bieten vielfältige Möglichkeiten zur Konstruktion von Beispielen und können daher auch als eine Spielwiese für Funktionalanalytiker betrachtet werden.
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| dbo:abstract | Ein Folgenraum ist ein in der Mathematik betrachteter Vektorraum, dessen Elemente Zahlenfolgen sind.Viele in der Funktionalanalysis auftretende Vektorräume sind Folgenräume oder können durch solche repräsentiert werden.Zu den Beispielen zählen u. a. die wichtigen Räume wie aller beschränkten Folgen oder aller gegen 0 konvergenten Folgen.Die Folgenräume bieten vielfältige Möglichkeiten zur Konstruktion von Beispielen und können daher auch als eine Spielwiese für Funktionalanalytiker betrachtet werden. (de) En mathématiques, l'espace ℓp est un exemple d'espace vectoriel, constitué de suites à valeurs réelles ou complexes et qui possède, pour 1 ≤ p ≤ ∞, une structure d'espace de Banach. (fr) In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important sequence spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space. (en) In matematica, in particolare in analisi funzionale, lo spazio delle successioni è uno spazio funzionale formato da tutte le successioni reali o complesse. Si tratta dell'insieme delle funzioni definite sull'insieme dei numeri naturali a valori in o . Definendo una somma, detta puntuale: e un prodotto per scalari: lo spazio delle successioni viene dotato della struttura di spazio vettoriale. Solitamente, vengono studiati appropriati sottospazi dello spazio di tutte le successioni. Un caso importante è dato dagli spazi lp, solitamente denotati con , cioè gli spazi delle successioni tali che: Essi infatti risultano essere spazi di Banach per . Due sottocasi importanti del precedente sono lo spazio delle successioni limitate e lo spazio delle successioni , che è uno spazio di Hilbert. Un sottospazio vettoriale di è lo spazio c delle successioni convergenti, formato da tutti gli tali che esiste. Si tratta di uno spazio chiuso rispetto alla norma , ed è pertanto uno spazio di Banach. Lo spazio c0 delle successioni convergenti a zero è un sottospazio chiuso di c, e dunque anch'esso uno spazio di Banach. (it) 関数解析学および関連する数学の分野における数列空間（すうれつくうかん、英: sequence space）とは、実数あるいは複素数の無限列を元とするベクトル空間のことを言う。またそれと同値であるが、自然数から実あるいは複素数体 K への関数を元とする関数空間のことでもある。そのような関数すべてからなる集合は、K に元を持つ無限列すべてからなる集合であると自然に認識され、関数の点ごとの和および点ごとのスカラー倍の作用の下で、ベクトル空間と見なされる。すべての数列空間は、この空間の線型部分空間である。通常、数列空間はノルムを備えるものであり、そうでなくとも少なくとも位相ベクトル空間の構造を備えている。 解析学におけるもっとも重要な数列空間のクラスは、p-乗総和可能数列からなる関数空間 ℓp である。それらの空間は p-ノルムを備え、自然数の集合上の数え上げ測度に対するLp空間の特別な場合と見なされる。収束列や零列のような他の重要な数列のクラスも数列空間を構成し、それらの場合はそれぞれ c および c0 と表記され、上限ノルムが備えられる。任意の数列空間は各点収束の位相を備えるものでもあり、その位相の下でのそれらの空間は、と呼ばれるフレシェ空間の特殊な場合となる。 (ja) Em matemática, os espaços , são espaços vetoriais normados cujos vetores são sequências de números pertencentes a um corpo onde ou . Espaços são exemplos de espaços vetoriais de dimensão infinita. (pt) |
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| dbp:mathStatement | Let be a Fréchet space over Then the following are equivalent: admits no continuous norm . contains a vector subspace TVS-isomorphic to . contains a complemented vector subspace TVS-isomorphic to . (en) |
| dbp:name | Theorem (en) |
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| dct:subject | dbc:Sequences_and_series dbc:Sequence_spaces dbc:Functional_analysis |
| gold:hypernym | dbr:Space |
| rdf:type | owl:Thing yago:WikicatSequenceSpaces yago:WikicatSequencesAndSeries yago:Abstraction100002137 yago:Arrangement107938773 yago:Attribute100024264 yago:Group100031264 yago:Ordering108456993 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 yago:Sequence108459252 yago:Series108457976 yago:Space100028651 yago:WikicatPropertiesOfTopologicalSpaces |
| rdfs:comment | Ein Folgenraum ist ein in der Mathematik betrachteter Vektorraum, dessen Elemente Zahlenfolgen sind.Viele in der Funktionalanalysis auftretende Vektorräume sind Folgenräume oder können durch solche repräsentiert werden.Zu den Beispielen zählen u. a. die wichtigen Räume wie aller beschränkten Folgen oder aller gegen 0 konvergenten Folgen.Die Folgenräume bieten vielfältige Möglichkeiten zur Konstruktion von Beispielen und können daher auch als eine Spielwiese für Funktionalanalytiker betrachtet werden. (de) En mathématiques, l'espace ℓp est un exemple d'espace vectoriel, constitué de suites à valeurs réelles ou complexes et qui possède, pour 1 ≤ p ≤ ∞, une structure d'espace de Banach. (fr) 関数解析学および関連する数学の分野における数列空間（すうれつくうかん、英: sequence space）とは、実数あるいは複素数の無限列を元とするベクトル空間のことを言う。またそれと同値であるが、自然数から実あるいは複素数体 K への関数を元とする関数空間のことでもある。そのような関数すべてからなる集合は、K に元を持つ無限列すべてからなる集合であると自然に認識され、関数の点ごとの和および点ごとのスカラー倍の作用の下で、ベクトル空間と見なされる。すべての数列空間は、この空間の線型部分空間である。通常、数列空間はノルムを備えるものであり、そうでなくとも少なくとも位相ベクトル空間の構造を備えている。 解析学におけるもっとも重要な数列空間のクラスは、p-乗総和可能数列からなる関数空間 ℓp である。それらの空間は p-ノルムを備え、自然数の集合上の数え上げ測度に対するLp空間の特別な場合と見なされる。収束列や零列のような他の重要な数列のクラスも数列空間を構成し、それらの場合はそれぞれ c および c0 と表記され、上限ノルムが備えられる。任意の数列空間は各点収束の位相を備えるものでもあり、その位相の下でのそれらの空間は、と呼ばれるフレシェ空間の特殊な場合となる。 (ja) Em matemática, os espaços , são espaços vetoriais normados cujos vetores são sequências de números pertencentes a um corpo onde ou . Espaços são exemplos de espaços vetoriais de dimensão infinita. (pt) In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. (en) In matematica, in particolare in analisi funzionale, lo spazio delle successioni è uno spazio funzionale formato da tutte le successioni reali o complesse. Si tratta dell'insieme delle funzioni definite sull'insieme dei numeri naturali a valori in o . Definendo una somma, detta puntuale: e un prodotto per scalari: lo spazio delle successioni viene dotato della struttura di spazio vettoriale. Solitamente, vengono studiati appropriati sottospazi dello spazio di tutte le successioni. Un caso importante è dato dagli spazi lp, solitamente denotati con , cioè gli spazi delle successioni tali che: (it) |
| rdfs:label | Folgenraum (de) Espace de suites ℓp (fr) Spazio delle successioni (it) 数列空間 (ja) Sequence space (en) Espaço lp (pt) |
| rdfs:seeAlso | dbr:Lp_space dbr:C_space dbr:Sup>_space |
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