Banach–Mazur distance
E421065
The Banach–Mazur distance is a numerical measure in functional analysis that quantifies how "far apart" two finite-dimensional normed vector spaces are up to linear isomorphism.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Banach–Mazur compactum | 2 |
| Banach–Mazur distance canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T4219682 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Banach–Mazur distance Context triple: [Stefan Banach, eponymOf, Banach–Mazur distance]
-
A.
Kolmogorov distance
Kolmogorov distance is a statistical metric that measures the maximum difference between two cumulative distribution functions, commonly used to quantify convergence in distribution and in goodness-of-fit tests.
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B.
Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
-
C.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Banach–Mazur distance Target entity description: The Banach–Mazur distance is a numerical measure in functional analysis that quantifies how "far apart" two finite-dimensional normed vector spaces are up to linear isomorphism.
-
A.
Kolmogorov distance
Kolmogorov distance is a statistical metric that measures the maximum difference between two cumulative distribution functions, commonly used to quantify convergence in distribution and in goodness-of-fit tests.
-
B.
Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
-
C.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
D.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
-
E.
Minkowski functional
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
functional analysis concept
ⓘ
mathematical concept ⓘ numerical invariant ⓘ |
| appliesTo |
complex normed spaces
ⓘ
finite-dimensional normed vector spaces ⓘ real normed spaces ⓘ |
| cannotBeDirectlyDefinedAs | a metric on infinite-dimensional Banach spaces ⓘ |
| characterizes | equivalence classes of norms up to linear isomorphism ⓘ |
| codomain | [1,∞) ⓘ |
| definitionInWords | infimum over linear isomorphisms of the product of the operator norm and the norm of the inverse ⓘ |
| definitionUses |
inverse operator norm
ⓘ
operator norm of linear isomorphisms ⓘ |
| domain | finite-dimensional normed spaces up to linear isomorphism ⓘ |
| equals | 1 if and only if the spaces are isometric ⓘ |
| field |
Banach space theory
ⓘ
functional analysis ⓘ |
| hasAlternativeFormulation | via equivalence of norms on a finite-dimensional vector space ⓘ |
| isDefinedFor | pairs of isomorphism classes of finite-dimensional normed spaces ⓘ |
| isInvariantUnder | linear isomorphisms ⓘ |
| isLogarithmicallyEquivalentTo | a metric via taking logarithm ⓘ |
| isMetricOn | isomorphism classes of finite-dimensional normed spaces ⓘ |
| isNonNegative | true ⓘ |
| isPartOf | geometric functional analysis ⓘ |
| isQuasiMetricOn | finite-dimensional normed spaces themselves ⓘ |
| isScaleInvariant | true ⓘ |
| isSymmetric | true ⓘ |
| isToolFor |
comparing geometric structure of normed spaces
ⓘ
quantifying distortion of linear isomorphisms ⓘ |
| measures | how far two normed spaces are from being isometric ⓘ |
| minimumValue | 1 ⓘ |
| namedAfter |
Stanisław Mazur
ⓘ
Stefan Banach ⓘ |
| oftenStudiedBetween | an n-dimensional normed space and ℓ₂ⁿ ⓘ |
| relatedConcept |
Banach–Mazur compactum
ⓘ
linked to:
Banach–Mazur distance
isometric Banach spaces ⓘ isomorphic Banach spaces ⓘ normed vector space ⓘ operator norm ⓘ |
| requires | finite dimension for being a true metric ⓘ |
| satisfiesTriangleInequality | true ⓘ |
| typicalQuestion | how close a given normed space is to Euclidean space ⓘ |
| upperBoundGrowth | polynomial in the dimension for many classes of spaces ⓘ |
| usedIn |
asymptotic geometric analysis
ⓘ
classification of finite-dimensional normed spaces ⓘ local theory of Banach spaces ⓘ study of Banach–Mazur compactum ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Banach–Mazur distance Description of subject: The Banach–Mazur distance is a numerical measure in functional analysis that quantifies how "far apart" two finite-dimensional normed vector spaces are up to linear isomorphism.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Banach–Mazur distance
linked to: Banach–Mazur distance