Schauder fixed-point theorem
E121350
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Schauder fixed-point theorem canonical | 4 |
| Schauder fixed point | 1 |
| Schauder fixed-point theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1056914 — 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: Schauder fixed-point theorem Context triple: [Brouwer fixed-point theorem, relatedTo, Schauder fixed-point theorem]
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A.
Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
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B.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
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C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
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E.
Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
- 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: Schauder fixed-point theorem Target entity description: The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
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A.
Glicksberg fixed-point theorem
The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
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B.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
-
C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
E.
Banach spaces
Banach spaces are complete normed vector spaces that provide a fundamental framework for functional analysis and the study of infinite-dimensional linear phenomena.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
fixed-point theorem
ⓘ
mathematical theorem ⓘ |
| appliesTo |
Banach spaces
ⓘ
infinite-dimensional spaces ⓘ |
| assumption | Banach space is real or complex normed linear space ⓘ |
| category |
Lefschetz fixed-point theorem
ⓘ
surface form:
topological fixed-point theorem
|
| conclusion |
existence of a fixed point
ⓘ
there exists x such that T(x) = x ⓘ |
| contrastWith |
Banach fixed-point theorem
ⓘ
surface form:
Banach fixed-point theorem which assumes contraction
|
| doesNotRequire | contraction condition ⓘ |
| domainCondition |
bounded subset
ⓘ
closed subset ⓘ convex subset ⓘ nonempty subset of a Banach space ⓘ subset is nonempty, closed, bounded, and convex ⓘ |
| field | functional analysis ⓘ |
| formalStatement | Every continuous compact map from a nonempty closed bounded convex subset of a Banach space into itself has a fixed point. ⓘ |
| generalizes | Brouwer fixed-point theorem ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| implies | existence of solutions to certain boundary value problems ⓘ |
| mapCondition |
compact mapping
ⓘ
continuous mapping ⓘ mapping from the subset into itself ⓘ |
| namedAfter | Juliusz Schauder ⓘ |
| relatedTo |
Banach fixed-point theorem
ⓘ
Kakutani fixed-point theorem ⓘ Leray–Schauder degree ⓘ Tychonoff theorem for products of compact spaces ⓘ
surface form:
Tychonoff fixed-point theorem
|
| requires |
boundedness of the domain
ⓘ
closedness of the domain ⓘ compactness of the operator ⓘ continuity of the operator ⓘ convexity of the domain ⓘ |
| strengthens | Brouwer fixed-point theorem to infinite dimensions ⓘ |
| toolFor | proving existence without uniqueness ⓘ |
| typeOfCompactness | image of bounded sets is relatively compact ⓘ |
| usedIn |
existence theory for differential equations
ⓘ
integral equations ⓘ nonlinear functional analysis ⓘ partial differential equations ⓘ topological methods in analysis ⓘ |
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: Schauder fixed-point theorem Description of subject: The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Schauder fixed point
this entity surface form:
Schauder fixed-point theory