Hahn–Banach theorem

All labels observed (7)

Label	Occurrences
Hahn–Banach theorem canonical	12
Hahn–Banach separation theorem	1
analytic Hahn–Banach theorem	1
complex Hahn–Banach theorem	1
geometric Hahn–Banach theorem	1
real Hahn–Banach theorem	1
separating hyperplane 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.
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
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	analytic Hahn–Banach theorem ⓘ linked to: Hahn–Banach theorem complex Hahn–Banach theorem ⓘ linked to: Hahn–Banach theorem geometric Hahn–Banach theorem ⓘ linked to: Hahn–Banach theorem locally convex space version ⓘ normed space version ⓘ real Hahn–Banach theorem ⓘ linked to: 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 ⓘ
relatedTo	Riesz representation theorem ⓘ closed graph theorem ⓘ linked to: 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

Referenced by (18)

Full triples — surface form annotated when it differs from this entity's canonical label.

Banach space → hasConcept → Hahn–Banach theorem ⓘ

subject linked to: Banach spaces

Stefan Banach → notableWork → Hahn–Banach theorem ⓘ

Stefan Banach → eponymOf → Hahn–Banach theorem ⓘ

Hans Hahn → notableWork → Hahn–Banach theorem ⓘ

Banach inverse mapping theorem → relatedTo → Hahn–Banach theorem ⓘ

Functional analysis → fieldOfStudy → Hahn–Banach theorem ⓘ

subject linked to: "Functional Analysis"

Banach–Steinhaus theorem → relatedTo → Hahn–Banach theorem ⓘ

Banach–Alaoglu theorem → relatedTo → Hahn–Banach theorem ⓘ

Hahn–Banach theorem → hasVersion → real Hahn–Banach theorem ⓘ