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- In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. Bornological spaces were first studied by George Mackey. The name was coined by Bourbaki after borné, the French word for "bounded". (en)
- Bornologische Räume sind in dem mathematischen Teilgebiet Funktionalanalysis spezielle lokalkonvexe Räume, für deren lineare Operatoren die aus der Theorie der normierten Räume bekannte Äquivalenz von Stetigkeit und Beschränktheit gilt. Diese Räume lassen sich durch ihre Nullumgebungsbasen charakterisieren und haben weitere Eigenschaften mit normierten Räumen gemeinsam. (de)
- 수학에서 유계형 집합(有界型集合, 영어: bornological set)은 유계 부분 집합들의 집합족이 명시된 집합이다. (ko)
- 数学、特に函数解析学における有界型空間（ゆうかいけいくうかん、ゆうかいがたくうかん界相空間（かいそうくうかん、英: bornological space; ボルノロジー空間）は、集合や函数の有界性の問題をある意味で考えるのに最低限必要な構造というものを抽出した空間のクラスである（これは位相空間が連続性の問題を考えるのに最低限必要な構造を抽出したものであったことと同様の考え方である）。界相空間を初めて考えたのはマッキーで、命名はブルバキによる（フランス語で有界を意味する borné (と位相 topology) に由来）。 (ja)
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- The product of a collection locally convex bornological spaces is bornological if and only if does admit an Ulam measure. (en)
- Let and be locally convex TVS and let denote endowed with the topology induced by von Neumann bornology of Define similarly. Then a linear map is a bounded linear operator if and only if is continuous.
Moreover, if is bornological, is Hausdorff, and is continuous linear map then so is If in addition is also ultrabornological, then the continuity of implies the continuity of where is the ultrabornological space associated with (en)
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- Theorem (en)
- Mackey-Ulam theorem (en)
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- Bornologische Räume sind in dem mathematischen Teilgebiet Funktionalanalysis spezielle lokalkonvexe Räume, für deren lineare Operatoren die aus der Theorie der normierten Räume bekannte Äquivalenz von Stetigkeit und Beschränktheit gilt. Diese Räume lassen sich durch ihre Nullumgebungsbasen charakterisieren und haben weitere Eigenschaften mit normierten Räumen gemeinsam. (de)
- 수학에서 유계형 집합(有界型集合, 영어: bornological set)은 유계 부분 집합들의 집합족이 명시된 집합이다. (ko)
- 数学、特に函数解析学における有界型空間（ゆうかいけいくうかん、ゆうかいがたくうかん界相空間（かいそうくうかん、英: bornological space; ボルノロジー空間）は、集合や函数の有界性の問題をある意味で考えるのに最低限必要な構造というものを抽出した空間のクラスである（これは位相空間が連続性の問題を考えるのに最低限必要な構造を抽出したものであったことと同様の考え方である）。界相空間を初めて考えたのはマッキーで、命名はブルバキによる（フランス語で有界を意味する borné (と位相 topology) に由来）。 (ja)
- In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by the property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator. (en)
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- Bornologischer Raum (de)
- Bornological space (en)
- 有界型空間 (ja)
- 유계형 집합 (ko)
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