Hahn–Banach theorem
E414346
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Hahn–Banach theorem canonical | 9 |
| analytic Hahn–Banach theorem | 1 |
| complex Hahn–Banach theorem | 1 |
| geometric Hahn–Banach theorem | 1 |
| real Hahn–Banach theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4092295 — 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: Hahn–Banach theorem Context triple: [Banach space, hasConcept, Hahn–Banach theorem]
-
A.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
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B.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
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C.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
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D.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
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E.
Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
- 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: Hahn–Banach theorem Target entity description: The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
-
A.
Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
-
B.
Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
-
C.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
D.
Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
-
E.
Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in functional analysis ⓘ |
| assumes |
sublinear domination condition in its general form
ⓘ
vector space over the real or complex numbers ⓘ |
| concerns |
bounded linear functionals
ⓘ
extension of linear functionals ⓘ linear functionals ⓘ normed vector spaces ⓘ seminorms ⓘ sublinear functionals ⓘ topological vector spaces ⓘ |
| ensures |
density of the canonical embedding into the bidual for normed spaces
ⓘ
existence of nonzero continuous linear functionals on infinite-dimensional normed spaces ⓘ |
| field |
functional analysis
ⓘ
functional analysis on normed spaces ⓘ functional analysis on topological vector spaces ⓘ |
| guarantees |
existence of continuous linear extensions
ⓘ
extension of bounded linear functionals from a subspace to the whole space ⓘ preservation of norm of extended linear functionals ⓘ |
| hasVersion |
Hahn–Banach theorem
self-linksurface differs
ⓘ
surface form:
analytic Hahn–Banach theorem
Hahn–Banach theorem self-linksurface differs ⓘ
surface form:
complex Hahn–Banach theorem
Hahn–Banach theorem self-linksurface differs ⓘ
surface form:
geometric Hahn–Banach theorem
locally convex space version ⓘ normed space version ⓘ Hahn–Banach theorem self-linksurface differs ⓘ
surface form:
real Hahn–Banach theorem
|
| historicalDevelopment | proved independently by Hahn and Banach in the early 20th century ⓘ |
| implies |
existence of many continuous linear functionals on normed spaces
ⓘ
existence of supporting hyperplanes to convex sets ⓘ geometric separation theorems in locally convex spaces ⓘ separation of points by continuous linear functionals ⓘ |
| importance |
cornerstone of the theory of Banach spaces
ⓘ
fundamental tool in modern functional analysis ⓘ |
| namedAfter |
Hans Hahn
ⓘ
Stefan Banach NERFINISHED ⓘ |
| relatedTo |
Riesz representation theorem
ⓘ
Closed Graph Theorem ⓘ
surface form:
closed graph theorem
open mapping theorem ⓘ uniform boundedness principle ⓘ |
| usedIn |
Banach space theory
ⓘ
L^p space theory ⓘ construction of dual spaces ⓘ convex analysis ⓘ distribution theory ⓘ duality theory in functional analysis ⓘ locally convex space theory ⓘ measure theory ⓘ optimization theory ⓘ partial differential equations ⓘ representation of continuous linear functionals ⓘ |
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: Hahn–Banach theorem Description of subject: The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Banach space
subject surface form:
Functional analysis
this entity surface form:
real Hahn–Banach theorem
this entity surface form:
complex Hahn–Banach theorem
this entity surface form:
geometric Hahn–Banach theorem
this entity surface form:
analytic Hahn–Banach theorem