Corona theorem
E943111
The Corona theorem is a fundamental result in complex analysis that characterizes when bounded analytic functions on the unit disk can be solved in a certain type of division problem, showing that the maximal ideal space of the disk algebra has no "corona."
All labels observed (1)
| Label | Occurrences |
|---|---|
| Corona theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11728192 — 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: Corona theorem Context triple: [Lennart Carleson, knownFor, Corona theorem]
-
A.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
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C.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
D.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
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E.
Bieberbach conjecture
The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
- 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: Corona theorem Target entity description: The Corona theorem is a fundamental result in complex analysis that characterizes when bounded analytic functions on the unit disk can be solved in a certain type of division problem, showing that the maximal ideal space of the disk algebra has no "corona."
-
A.
Montel theorem
Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
-
B.
Picard theorem
Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
-
C.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
-
D.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
E.
Bieberbach conjecture
The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in complex analysis ⓘ |
| appliesTo |
algebra H^∞(D)
ⓘ
disk algebra A(D) ⓘ |
| author | Lennart Carleson NERFINISHED ⓘ |
| concerns |
corona problem
ⓘ
maximal ideal space of the disk algebra ⓘ |
| conclusion | existence of bounded analytic g_1,...,g_n with f_1 g_1+...+f_n g_n=1 ⓘ |
| coreCondition | bounded analytic functions f_1,...,f_n with no common zero on the unit disk ⓘ |
| difficulty | proof is technically complex ⓘ |
| domain | open unit disk in the complex plane ⓘ |
| field |
complex analysis
ⓘ
functional analysis ⓘ operator theory ⓘ |
| generalizationOf | classical division problems in function algebras ⓘ |
| hasAlternativeProof |
methods using Carleson measure estimates
ⓘ
methods using \\bar{\partial}-techniques ⓘ methods using functional-analytic techniques ⓘ |
| hasGeneralization |
corona theorems for several complex variables
ⓘ
corona theorems on Riemann surfaces ⓘ corona theorems on strictly pseudoconvex domains ⓘ operator corona theorem ⓘ |
| implies |
maximal ideal space of H^∞(D) is connected to the unit disk
ⓘ
no additional maximal ideals lying over the boundary of the unit disk ⓘ |
| influenced |
Hardy space theory
ⓘ
development of function algebra theory ⓘ operator-valued function theory ⓘ |
| isCornerstoneOf | corona theory in function algebras ⓘ |
| mainObject |
bounded analytic functions
ⓘ
disk algebra ⓘ unit disk ⓘ |
| namedAfter | corona of the maximal ideal space ⓘ |
| publishedIn | Acta Mathematica NERFINISHED ⓘ |
| relatedOpenProblem |
Carleson’s corona problem in higher dimensions
NERFINISHED
ⓘ
structure of maximal ideal space of H^∞(D) ⓘ |
| relatedTo |
Beurling’s theorem
NERFINISHED
ⓘ
Carleson measures ⓘ Nevanlinna–Pick interpolation NERFINISHED ⓘ interpolation in H^∞ ⓘ |
| statementAbout |
absence of corona in maximal ideal space of disk algebra
ⓘ
solvability of certain division problems in H^∞(D) ⓘ |
| surveyedIn |
monographs on H^∞ spaces
ⓘ
“The corona theorem” by Thomas W. Gamelin NERFINISHED ⓘ |
| usesConcept |
Banach algebras
NERFINISHED
ⓘ
Gelfand transform NERFINISHED ⓘ bounded holomorphic functions ⓘ maximal ideals ⓘ |
| yearProved | 1962 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Corona theorem Description of subject: The Corona theorem is a fundamental result in complex analysis that characterizes when bounded analytic functions on the unit disk can be solved in a certain type of division problem, showing that the maximal ideal space of the disk algebra has no "corona."
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.