Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Functional analysis

Hahn–Banach Theorem

A bounded linear measurement defined only on a subspace can be continued to every vector in the ambient space while agreeing on the original subspace and keeping exactly the same norm.

Exact theorem

Exact Mathlib statement

theorem exists_extension_norm_eq {𝕜 : Type*} [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (p : Subspace 𝕜 E) (f : StrongDual 𝕜 p) : ∃ g : StrongDual 𝕜 E, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖

The theorem at a glance

Hahn–Banach Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

A continuous scalar-valued linear functional extends to the ambient space with exact agreement and no norm increase. Explanatory diagram.
Detailed visual description

The poster makes the nested domains, extension relationship, and norm equality explicit. A continuous family of level sets grows from the subspace into the ambient space without changing its controlling envelope, while the boundary footer excludes arbitrary-codomain and constructive claims.

Statement structure

Statement and scope

Statement map for Hahn–Banach TheoremEvery continuous scalar-valued linear functional on a subspace of a real or complex seminormed space extends to the whole space without changing its operator norm. Claim boundary: This page indexes Mathlib's analytic Hahn–Banach extension theorem for continuous scalar-valued linear functionals over a field satisfying IsRCLikeNormedField. The ambient space may be seminormed, and the conclusion gives an extension agreeing on the subspace with exactly the same operator norm. It is not the general sublinear domination form, an extension theorem for arbitrary codomains, or a constructive extension algorithm. The pinned upstream declaration is exists_extension_norm_eq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingEvery continuousscalar-valued linearfunctional on a subspaceof a real or complexseminormed space extendsto the whole spacewithout changing itsoperator norm.Claim boundaryThis page indexesMathlib's analyticHahn–Banach extensiontheorem for continuousscalar-valued linearfunctionals over a fieldsatisfyingIsRCLikeNormedField. Theambient space may beseminormed, and theconclusion gives anextension agreeing onthe subspace withexactly the sameoperator norm. It is notthe general sublineardomination form, anextension theorem forarbitrary codomains, ora constructive extensionalgorithm.Pinned declarationmathlib ·exists_extension_norm_eqStatement map for Hahn–Banach TheoremEvery continuous scalar-valued linear functional on a subspace of a real or complex seminormed space extends to the whole space without changing its operator norm. Claim boundary: This page indexes Mathlib's analytic Hahn–Banach extension theorem for continuous scalar-valued linear functionals over a field satisfying IsRCLikeNormedField. The ambient space may be seminormed, and the conclusion gives an extension agreeing on the subspace with exactly the same operator norm. It is not the general sublinear domination form, an extension theorem for arbitrary codomains, or a constructive extension algorithm. The pinned upstream declaration is exists_extension_norm_eq. The complete formal statement is available in the Exact Mathlib statement section.Mathematical readingEvery continuousscalar-valued linearfunctional on a subspaceof a real or complexseminormed space extendsto the whole spacewithout changing itsoperator norm.Claim boundaryThis page indexesMathlib's analyticHahn–Banach extensiontheorem for continuousscalar-valued linearfunctionals over a fieldsatisfyingIsRCLikeNormedField. Theambient space may beseminormed, and theconclusion gives anextension agreeing onthe subspace withexactly the sameoperator norm. It is notthe general sublineardomination form, anextension theorem forarbitrary codomains, ora constructive extensionalgorithm.Pinned declarationmathlib ·exists_extension_norm_eq

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A continuous linear functional extends from the subspace to the whole space without increasing its norm. Explanatory diagram.
Detailed visual description

A narrow emerald plane sits inside a larger forest-green ambient space. Ivory level lines defined on the subspace extend seamlessly across the ambient field, while one antique-gold norm envelope keeps the same opening and slope on both domains.

Why it matters

A mathematical landmark

Hahn–Banach is a foundational extension and separation principle in functional analysis. It creates continuous linear functionals with controlled norm and powers duality, separation arguments, and many existence theorems.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's analytic Hahn–Banach extension theorem for continuous scalar-valued linear functionals over a field satisfying IsRCLikeNormedField. The ambient space may be seminormed, and the conclusion gives an extension agreeing on the subspace with exactly the same operator norm. It is not the general sublinear domination form, an extension theorem for arbitrary codomains, or a constructive extension algorithm.

Source and local evidence

Where the theorem comes from

Existing declaration
exists_extension_norm_eq in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.hahn-banach-theorem.v001
Source
Open the pinned upstream reference