Bourgain–Tzafriri restricted invertibility principle

E547403

mathematical theorem result in functional analysis

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (3)

Label	Occurrences
Bourgain–Tzafriri restricted invertibility principle canonical	1
Bourgain–Tzafriri theorem on restricted invertibility	1
restricted invertibility theorem	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 ⓘ linked to: Bourgain–Tzafriri restricted invertibility principle
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 ⓘ
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 ⓘ Lior Tzafriri ⓘ
originallyPublishedIn	a research article in the late 1980s ⓘ
provedBy	Jean Bourgain ⓘ Lior Tzafriri ⓘ
provides	lower bounds on singular values of selected submatrices ⓘ quantitative bounds on size of invertible submatrices ⓘ
relatedTo	Kadison–Singer problem ⓘ Weaver’s conjecture ⓘ paving conjecture ⓘ
strengthens	earlier results on invertibility of submatrices ⓘ
typeOf	restricted invertibility theorem ⓘ linked to: Bourgain–Tzafriri restricted invertibility principle
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 ⓘ linked to: Banach spaces compressed sensing ⓘ discrepancy theory ⓘ frame theory ⓘ local theory of Banach spaces ⓘ numerical linear algebra ⓘ sparse approximation ⓘ
yearProved	1987 ⓘ

How these facts were elicited

Referenced by (3)

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

Jean Bourgain → knownFor → Bourgain–Tzafriri restricted invertibility principle ⓘ

Bourgain–Tzafriri restricted invertibility principle → typeOf → restricted invertibility theorem ⓘ

linked to: Bourgain–Tzafriri restricted invertibility principle

Bourgain–Tzafriri restricted invertibility principle → alsoKnownAs → Bourgain–Tzafriri theorem on restricted invertibility ⓘ