{
  "artifactId": "artifact.library.mathlib.banach-steinhaus-theorem.v001",
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  "category": "Functional analysis",
  "claimBoundary": "This target indexes Mathlib's Banach–Steinhaus theorem for an arbitrary family of continuous semilinear maps from a complete seminormed 𝕜-space E to a seminormed 𝕜₂-space F, over an isometric ring homomorphism between nontrivially normed fields. If the family is pointwise norm-bounded, then one real constant uniformly bounds every operator norm. The pointwise constant may depend on x. The declaration does not assert convergence, compactness, injectivity, surjectivity, a finite index type, or completeness of the codomain.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem banach_steinhaus {E F 𝕜 𝕜₂ : Type*} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {ι : Type*} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C'",
  "family": "Banach-space structure",
  "id": "library.mathlib.banach-steinhaus-theorem.v001",
  "links": {
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    "entry": "/library-theorems/banach-steinhaus-theorem/",
    "entryData": "/data/library-theorems/banach-steinhaus-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.banach-steinhaus-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.banach-steinhaus-theorem.v001.evidence.json",
    "source": "/sources/upstream/banach-steinhaus-theorem/"
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  "landmark": {
    "completeFormalType": "theorem banach_steinhaus {E F 𝕜 𝕜₂ : Type*} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {ι : Type*} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C'",
    "concepts": [
      "uniform boundedness principle",
      "continuous semilinear maps",
      "operator norms",
      "pointwise boundedness",
      "equicontinuity",
      "barrelled spaces"
    ],
    "plainLanguage": "For each vector x, the family may need its own bound. If the source space is complete, those separate pointwise bounds force a single constant that bounds the operator norm of every map in the family.",
    "poster": {
      "alt": "An ivory functional-analysis poster states pointwise boundedness for a family of continuous semilinear maps, the resulting common operator-norm bound, and the checked route through equicontinuity and the barrelled-space theorem.",
      "byteSize": 2488761,
      "caption": "Pointwise boundedness on a complete source forces one uniform bound on every operator norm.",
      "description": "The poster keeps the source and codomain seminormed boundary visible, places the vector-dependent pointwise estimate above the uniform operator estimate, and summarizes the exact source bridge from norm bounds to equicontinuity, through the barrelled-space Banach–Steinhaus theorem, and back.",
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      "generatedAt": "2026-07-25T20:30:41Z",
      "height": 1536,
      "mediaType": "image/png",
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      "publicPath": "/assets/library-theorems/banach-steinhaus-theorem/theorem-poster-v1.png",
      "sha256": "sha256:783208cfcaba12650e512d7a0ccf95197556226502b45738fd569dd449a29e17",
      "title": "Banach–Steinhaus theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "FUNCTIONAL ANALYSIS · UNIFORM BOUNDEDNESS\nBANACH–STEINHAUS THEOREM\n\nE complete · F seminormed\ngᵢ : E →SL[σ] F\n\nPOINTWISE BOUNDED\n∀ x, ∃ C, ∀ i, ‖gᵢ(x)‖ ≤ C\n\nUNIFORMLY BOUNDED OPERATOR NORMS\n∃ C′, ∀ i, ‖gᵢ‖ ≤ C′\n\nTHE CHECKED ROUTE\n1 · Operator-norm bounds ↔ equicontinuity\n2 · Completeness supplies the barrelled-space setting\n3 · Pointwise bounds give bounded seminorm ranges\n4 · Banach–Steinhaus gives equicontinuity\n5 · Translate back to one operator-norm bound\n\nEXACT SCOPE\nContinuous semilinear maps over an isometric scalar homomorphism.\nC may depend on x.\nNo convergence, compactness, or finite-family claim."
    },
    "visual": {
      "alt": "A text-free engraved field shows many continuous ribbons meeting vector probes with different local bound rings, passing through an equicontinuity aperture, and ending as operator columns below one shared norm ceiling.",
      "byteSize": 3077176,
      "caption": "Vector-dependent pointwise bounds become one common operator-norm ceiling when the source is complete.",
      "description": "Fading ribbons represent an arbitrary family rather than a finite list. Three probes carry visibly different local bound rings, a restrained central aperture evokes the barrelled-space equicontinuity step, and every output column remains beneath the same antique-gold operator-norm ceiling.",
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          "height": 427,
          "mediaType": "image/webp",
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      "generatedAt": "2026-07-25T20:30:41Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/banach-steinhaus-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:d1aa1c09913db16b3ef3fc6ebbed1278855f6697402ce402fcb1f5822ebfe5df",
      "title": "Many local bounds, one operator-norm ceiling",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Banach–Steinhaus is one of the foundational principles of functional analysis. It converts pointwise control of an entire operator family into uniform norm control, with completeness supplying the decisive global structure."
  },
  "nonClaims": [
    "The pointwise hypothesis permits its bounding constant to depend on x; it does not assume one pointwise bound works for the whole domain.",
    "The index type is arbitrary, and the declaration makes no finite-family, convergence, or compactness claim.",
    "The selected scalar context allows continuous semilinear maps between spaces over different nontrivially normed fields linked by an isometric ring homomorphism.",
    "Only the domain E is assumed complete; the codomain F is seminormed but need not carry a CompleteSpace instance.",
    "Proof Atlas is indexing an existing Mathlib theorem, not claiming new mathematics, and the generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.banach-steinhaus-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": "Pointwise boundedness of an arbitrary family of continuous semilinear maps from a complete seminormed space forces a uniform operator-norm bound.",
  "targetId": "target.library.mathlib.banach-steinhaus-theorem.v001",
  "title": "Banach–Steinhaus Theorem",
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    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Normed/Operator/BanachSteinhaus.lean#L36",
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    "sourceFile": "Mathlib/Analysis/Normed/Operator/BanachSteinhaus.lean",
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}