{ "artifactId": "artifact.library.mathlib.hahn-banach-theorem.v001", "candidateOnly": false, "category": "Functional analysis", "claimBoundary": "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.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "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‖", "family": "Extension theorems", "id": "library.mathlib.hahn-banach-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/hahn-banach-theorem/", "entryData": "/data/library-theorems/hahn-banach-theorem.json", "evidence": "/proofs/artifact.library.mathlib.hahn-banach-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.hahn-banach-theorem.v001.evidence.json", "source": "/sources/upstream/hahn-banach-theorem/" }, "landmark": { "completeFormalType": "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‖", "concepts": [ "continuous linear functionals", "subspaces", "norm-preserving extension", "duality", "functional analysis" ], "plainLanguage": "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.", "poster": { "alt": "An ivory functional-analysis poster shows a subspace inside an ambient space, a scalar-valued linear functional f on the subspace, an extension g on the whole space, agreement on the subspace, and equal norms.", "byteSize": 2565605, "caption": "A continuous scalar-valued linear functional extends to the ambient space with exact agreement and no norm increase.", "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.", "derivatives": [ { "byteSize": 112448, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hahn-banach-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:278b0777e3e2d64a9b1dcd8beaafc978a62a61a8457d69a948ad217c56897f59", "width": 640 }, { "byteSize": 239426, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hahn-banach-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:6cb794ddcf26e6490d31c3ae9bf56b0f3294e1cceb4b795f44c182473dd61b75", "width": 1024 } ], "generatedAt": "2026-07-25T05:58:51.177833318Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hahn-banach-theorem/theorem-poster-v1.png", "sha256": "sha256:50e86d596585a2920298eb5b2786acc5ca54295914f9723057519434edc09215", "title": "Hahn–Banach Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "FUNCTIONAL ANALYSIS · NORM-PRESERVING EXTENSION\nHAHN–BANACH THEOREM\np a subspace of E\nf : p → 𝕜 continuous and linear\nThere exists g : E → 𝕜 continuous and linear\nFor every x in p, g(x) = f(x)\n‖g‖ = ‖f‖\nEXTEND THE DOMAIN · PRESERVE THE NORM\nEXACT SCOPE\nScalar-valued over an RCLike normed field. No constructive algorithm is claimed." }, "visual": { "alt": "A linear height field defined on a narrow subspace continues across a larger ambient space while its slope and supporting norm boundary remain unchanged.", "byteSize": 2834710, "caption": "A continuous linear functional extends from the subspace to the whole space without increasing its norm.", "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.", "derivatives": [ { "byteSize": 37638, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hahn-banach-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:377343a143eca6e36d2454babe834b7462aca920e0dea615df74e05ee0a4625f", "width": 640 }, { "byteSize": 125358, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/hahn-banach-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:2c16dacffd3ec4366e500cdcaf76510fe1b88362176a081aa13212f0fae1725c", "width": 1200 } ], "generatedAt": "2026-07-25T05:28:42.731759901Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/hahn-banach-theorem/theorem-schematic-v1.png", "sha256": "sha256:ac568e5f94b631cd791a6ac2a36a5eb77b5ee10b66f9c1c6e27f50b6707c54f2", "title": "Norm-preserving functional extension schematic", "width": 1536 }, "walkthrough": null, "whyLandmark": "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." }, "nonClaims": [ "Proof Atlas did not originate the Hahn–Banach theorem or Mathlib's declaration.", "The selected declaration is the norm-preserving scalar-functional extension form over real-like or complex-like normed fields.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.hahn-banach-theorem.v001", "statusAxes": { "acceptedAtlasResult": "Not recorded for the preferred artifact", "locallyReproduced": "Exact upstream declaration replayed", "reviewedPage": "Current public presentation reviewed", "upstreamIndexed": "Pinned source bytes verified locally" }, "summary": "Every 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.", "targetId": "target.library.mathlib.hahn-banach-theorem.v001", "title": "Hahn–Banach Theorem", "upstreamOrigin": { "declarationName": "exists_extension_norm_eq", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Normed/Module/HahnBanach.lean#L75", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:610783c4be059d6727deb6ab8e7cc149fe379250c9d74ebcff77b547dd9b53e4", "sourceByteLength": 8485, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Analysis/Normed/Module/HahnBanach.lean", "sourceLine": 75, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }