Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: Hahn–Banach theorem | Statement: [Banach limit, existenceDependsOn, Hahn–Banach theorem]

Target entity: Hahn–Banach theorem
Context triple: [Banach limit, existenceDependsOn, Hahn–Banach theorem]

• A. Hahn–Banach theorem chosen
  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.
• B. 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.
• C. Banach–Alaoglu theorem
  The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
• D. Krein–Milman theorem
  The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.
• E. Banach–Mazur theorem
  The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target predicate: existenceDependsOn
Context triple: [Banach limit, existenceDependsOn, Hahn–Banach theorem]

• A. dependency chosen
  Indicates that one entity relies on, is conditioned by, or cannot function or exist properly without another entity.
• B. representationDependsOn
  Indicates that one representation is reliant on or derived from another representation in order to be valid, complete, or interpretable.
• C. statusDependsOn
  Indicates that the status or state of one entity is determined or influenced by the status or state of another entity.
• D. positionDependsOn
  Indicates that the spatial or ordered position of one entity is determined or constrained by the position of another entity.
• E. existsAs
  Indicates that one entity has the same identity or manifestation as another, effectively asserting they are the same existing thing or state.
• F. None of above.