Bourgain–Tzafriri restricted invertibility principle
E547403
The Bourgain–Tzafriri restricted invertibility principle is a fundamental result in functional analysis and operator theory that guarantees the existence of large, well-invertible submatrices within certain classes of linear operators.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bourgain–Tzafriri restricted invertibility principle canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5790537 — 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: Bourgain–Tzafriri restricted invertibility principle Context triple: [Jean Bourgain, knownFor, Bourgain–Tzafriri restricted invertibility principle]
-
A.
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.
-
B.
Three regularity results in harmonic analysis
"Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
-
C.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
D.
Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
-
E.
Fefferman–Phong inequality
The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
- 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: Bourgain–Tzafriri restricted invertibility principle Target entity description: The Bourgain–Tzafriri restricted invertibility principle is a fundamental result in functional analysis and operator theory that guarantees the existence of large, well-invertible submatrices within certain classes of linear operators.
-
A.
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.
-
B.
Three regularity results in harmonic analysis
"Three regularity results in harmonic analysis" is the doctoral thesis of mathematician Terence Tao, focusing on advanced problems in harmonic analysis and the study of regularity properties of functions and operators.
-
C.
Khinchin–Kahane type inequalities
Khinchin–Kahane type inequalities are fundamental results in probability and functional analysis that bound moments or norms of random series (often with Rademacher or Gaussian coefficients) in terms of each other, providing powerful tools for studying the geometry of Banach spaces and random processes.
-
D.
Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
-
E.
Fefferman–Phong inequality
The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in functional analysis ⓘ |
| alsoKnownAs | Bourgain–Tzafriri theorem on restricted invertibility NERFINISHED ⓘ |
| appliesTo |
bounded linear operators on Hilbert spaces
ⓘ
matrices with normalized columns under certain conditions ⓘ |
| assumes | operator with columns of comparable norm ⓘ |
| concerns |
restricted invertibility of linear operators
ⓘ
selection of well-conditioned column subsets ⓘ |
| conclusion | there exists a large subset of columns forming a well-conditioned operator ⓘ |
| context |
finite-dimensional operator theory
ⓘ
geometry of Banach spaces ⓘ |
| field |
functional analysis
ⓘ
operator theory ⓘ |
| guarantees |
existence of large subsets of columns with uniformly bounded condition number
ⓘ
existence of large well-invertible submatrices ⓘ |
| hasGeneralization | Spielman–Srivastava restricted invertibility results NERFINISHED ⓘ |
| implies | existence of a large coordinate subset on which an operator is bounded below ⓘ |
| importance |
considered a fundamental tool in modern functional analysis
ⓘ
key ingredient in several major developments related to Kadison–Singer ⓘ |
| inspired |
algorithms for column subset selection
ⓘ
work on spectral sparsification ⓘ |
| involves |
norm estimates
ⓘ
probabilistic methods in analysis ⓘ singular values of linear operators ⓘ |
| namedAfter |
Jean Bourgain
NERFINISHED
ⓘ
Lior Tzafriri NERFINISHED ⓘ |
| originallyPublishedIn | a research article in the late 1980s ⓘ |
| provedBy |
Jean Bourgain
NERFINISHED
ⓘ
Lior Tzafriri NERFINISHED ⓘ |
| provides |
lower bounds on singular values of selected submatrices
ⓘ
quantitative bounds on size of invertible submatrices ⓘ |
| relatedTo |
Kadison–Singer problem
NERFINISHED
ⓘ
Weaver’s conjecture NERFINISHED ⓘ paving conjecture NERFINISHED ⓘ |
| strengthens | earlier results on invertibility of submatrices ⓘ |
| typeOf | restricted invertibility theorem NERFINISHED ⓘ |
| typicalFormulation | for an operator with normalized columns there is a large subset of columns on which the operator is bounded below by a universal constant ⓘ |
| usedIn |
Banach space theory
NERFINISHED
ⓘ
compressed sensing ⓘ discrepancy theory ⓘ frame theory ⓘ local theory of Banach spaces ⓘ numerical linear algebra ⓘ sparse approximation ⓘ |
| yearProved | 1987 ⓘ |
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: Bourgain–Tzafriri restricted invertibility principle Description of subject: The Bourgain–Tzafriri restricted invertibility principle is a fundamental result in functional analysis and operator theory that guarantees the existence of large, well-invertible submatrices within certain classes of linear operators.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.