{
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  "exactFormalStatement": "theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖",
  "family": "Inner-product geometry",
  "id": "library.mathlib.cauchy-schwarz-inequality.v001",
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    "completeFormalType": "theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖",
    "concepts": [
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      "projection",
      "inequality"
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    "plainLanguage": "The amount by which two real or complex inner-product-space vectors align can never exceed the product of their lengths.",
    "poster": {
      "alt": "An ivory editorial poster states the normed inner-product inequality and explains its four reading steps with a projection diagram and an exact-scope footer.",
      "byteSize": 1958102,
      "caption": "Vector alignment is bounded by the product of vector lengths.",
      "description": "The poster presents Mathlib's real-or-complex inner-product-space inequality, a projection-based interpretation, and the boundary that this declaration alone does not characterize equality.",
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      "generatedAt": "2026-07-22T04:45:00Z",
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      "sha256": "sha256:634aae7a412f6c7c5dfbd2155d27aa430b89eb4973ec51dedb7a88bc296cde9a",
      "title": "Cauchy–Schwarz inequality at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "INNER-PRODUCT SPACES · NORM CONTROL\nCAUCHY–SCHWARZ INEQUALITY\n‖inner(x, y)‖ ≤ ‖x‖ · ‖y‖\nFor vectors x and y over the real or complex numbers\nALIGNMENT CANNOT EXCEED THE PRODUCT OF LENGTHS\nHOW TO READ IT\n1 · Choose vectors x and y\n2 · Form their inner product\n3 · Take its norm\n4 · Bound it by the product of vector norms\nEXACT SCOPE\nNorm inequality only. This declaration does not characterize equality."
    },
    "visual": {
      "alt": "Green and cobalt vectors share an origin while a shorter gold projection lies on the cobalt direction and a perpendicular drop records the controlled alignment.",
      "byteSize": 1043162,
      "caption": "The inner-product magnitude cannot exceed the product of the two vector norms.",
      "description": "A sparse inner-product-space construction isolates two vectors, the projected component, and a perpendicular remainder without adding an equality condition not present in the selected declaration.",
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      "generatedAt": "2026-07-22T01:04:16Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/cauchy-schwarz-inequality/theorem-schematic-v1.png",
      "sha256": "sha256:bfb9a59462d9a2122c1a1d60128a65e5f2eccb90c781b0fe2a91df414ee63e55",
      "title": "Cauchy–Schwarz inequality schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Cauchy–Schwarz is the basic inequality of inner-product geometry. It controls projections, yields the triangle inequality, and supports foundational estimates across linear algebra, analysis, probability, and geometry."
  },
  "nonClaims": [
    "Proof Atlas did not originate the Cauchy–Schwarz inequality or Mathlib's declaration.",
    "The selected declaration does not characterize when equality holds.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.cauchy-schwarz-inequality.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "2 of 3 presentation reviews recorded",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "The norm of an inner product is at most the product of the two vector norms.",
  "targetId": "target.library.mathlib.cauchy-schwarz-inequality.v001",
  "title": "Cauchy–Schwarz Inequality",
  "upstreamOrigin": {
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    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/InnerProductSpace/Basic.lean#L455",
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    "sourceFile": "Mathlib/Analysis/InnerProductSpace/Basic.lean",
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  "publicPresentationReview": {
    "status": "reviewed"
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}