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Homomorphisms: contractivity, isometry and automatic continuity", "declaration_line": 227, "section_through_line": 230, "named_passage_sha256": "0ad53f3521e3f4b6f2c99f169b0b7116d00f5d75a6f1faf9a039f989f51f177e", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "CF", "result": "Proposition 7.3", "source": "../src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md", "source_sha256": "1caf1ba3fdee1409e497fbaa4884317cde491805a8ceba6f7293ae88f56793b1", "reader": "../c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html", "reader_sha256": "387036fd038fb08f317b45384dc3c10fb42086773d10c81561d339b488f39771", "anchor": "OA-FND-CF-09", "reader_heading": "7. Absolute value, Jordan decomposition and unitaries", "declaration_line": 388, "section_through_line": 415, "named_passage_sha256": "11b4ee4f65bd519b4c87cb6d8fc0b15312e8b31fa97d4beb7c2d4d4794a21ede", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "CF", "result": "Theorem 8.2", "source": "../src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md", "source_sha256": "1caf1ba3fdee1409e497fbaa4884317cde491805a8ceba6f7293ae88f56793b1", "reader": "../c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html", "reader_sha256": "387036fd038fb08f317b45384dc3c10fb42086773d10c81561d339b488f39771", "anchor": "OA-FND-CF-15", "reader_heading": "8. The positive cone and the order", "declaration_line": 428, "section_through_line": 450, "named_passage_sha256": "caf5e03475c0230d27cfcf80bc5dafcbbf065a850096dff25feebc3b6bb25618", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "CF", "result": "Proposition 8.5", "source": "../src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md", "source_sha256": "1caf1ba3fdee1409e497fbaa4884317cde491805a8ceba6f7293ae88f56793b1", "reader": "../c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html", "reader_sha256": "387036fd038fb08f317b45384dc3c10fb42086773d10c81561d339b488f39771", "anchor": "OA-FND-CF-15", "reader_heading": "8. The positive cone and the order", "declaration_line": 457, "section_through_line": 483, "named_passage_sha256": "a503414de56c2a916b8a3312be88027c0af976e0bbbdb7bd9943a4ecd070aeaa", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "GNS", "result": "Proposition 3.2", "source": "../src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md", "source_sha256": "8e8363c7e7c7be8f96e89e44e56dd88f4519f5822d6dff36dbf0a38cc46cb15f", "reader": "../representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html", "reader_sha256": "b2583e2f74d9c120e930b960dc5165baa670e80c0333b41bd7715e3db6bfb089", "anchor": "OA-FND-GN-03", "reader_heading": "3. Positive functionals", "declaration_line": 85, "section_through_line": 104, "named_passage_sha256": "32c631cc951990eaeb67fa60b60c7d0807e214cb412bde96c09b96b29e65d992", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "GNS", "result": "Theorem 4.7", "source": "../src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md", "source_sha256": "8e8363c7e7c7be8f96e89e44e56dd88f4519f5822d6dff36dbf0a38cc46cb15f", "reader": "../representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html", "reader_sha256": "b2583e2f74d9c120e930b960dc5165baa670e80c0333b41bd7715e3db6bfb089", "anchor": "c--algebras", "reader_heading": "C*-algebras", "declaration_line": 344, "section_through_line": 356, "named_passage_sha256": "dca22f8f2cdf4b91c48f17c14f8574d6b7beb18fa4a8f2acc14d6865f50bc555", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "GNS", "result": "Proposition 7.1a", "source": "../src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md", "source_sha256": "8e8363c7e7c7be8f96e89e44e56dd88f4519f5822d6dff36dbf0a38cc46cb15f", "reader": "../representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html", "reader_sha256": "b2583e2f74d9c120e930b960dc5165baa670e80c0333b41bd7715e3db6bfb089", "anchor": "faithful-representations-of-c--algebras", "reader_heading": "Faithful representations of C*-algebras", "declaration_line": 462, "section_through_line": 475, "named_passage_sha256": "a2bed1553320ea45c6c5745ac8141457e62b430e4e6b583c68efc4d608526338", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "GNS", "result": "Theorem 7.2", "source": "../src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md", "source_sha256": "8e8363c7e7c7be8f96e89e44e56dd88f4519f5822d6dff36dbf0a38cc46cb15f", "reader": "../representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html", "reader_sha256": "b2583e2f74d9c120e930b960dc5165baa670e80c0333b41bd7715e3db6bfb089", "anchor": "faithful-representations-of-c--algebras", "reader_heading": "Faithful representations of C*-algebras", "declaration_line": 476, "section_through_line": 482, "named_passage_sha256": "b2d2a55ce85a5b2f5fe4fd471a51abef3d51d60353c938c7db19e362d723eb3c", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "ST", "result": "Theorem 3.1", "source": "../src/the-spectral-theorem-for-bounded-self-adjoint-operators.md", "source_sha256": "e02703c40a76cacb47915244bd333da0ad2a900b41475f01074e89586a6f36d0", "reader": "../the-spectral-theorem-for-bounded-self-adjoint-operators.html", "reader_sha256": "62a412b5f7a698e469006caeae927e10ea11ad4c00bd764c9e8dd3d85112594c", "anchor": "OA-FND-ST-03", "reader_heading": "3. The Borel functional calculus", "declaration_line": 101, "section_through_line": 202, "named_passage_sha256": "725ae6fbb455b10cae9640f1a3dc96896bbc3328e1a8e1dcec05cc2c29dc0852", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "ST", "result": "Proposition 5.1", "source": "../src/the-spectral-theorem-for-bounded-self-adjoint-operators.md", "source_sha256": "e02703c40a76cacb47915244bd333da0ad2a900b41475f01074e89586a6f36d0", "reader": "../the-spectral-theorem-for-bounded-self-adjoint-operators.html", "reader_sha256": "62a412b5f7a698e469006caeae927e10ea11ad4c00bd764c9e8dd3d85112594c", "anchor": "OA-FND-ST-05", "reader_heading": "5. Spectral projections and polar decomposition", "declaration_line": 302, "section_through_line": 341, "named_passage_sha256": "32d617c18a08d1215406e9d2c8eaf073184ae3a491f581915b0698d114200583", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "ST", "result": "Theorem 6.1", "source": "../src/the-spectral-theorem-for-bounded-self-adjoint-operators.md", "source_sha256": "e02703c40a76cacb47915244bd333da0ad2a900b41475f01074e89586a6f36d0", "reader": "../the-spectral-theorem-for-bounded-self-adjoint-operators.html", "reader_sha256": "62a412b5f7a698e469006caeae927e10ea11ad4c00bd764c9e8dd3d85112594c", "anchor": "OA-FND-ST-06", "reader_heading": "6. Monotone convergence of operators", "declaration_line": 353, "section_through_line": 381, "named_passage_sha256": "5ee32f734d7bba4a896a59a88ff4594047a12d07bc36acc23140e2378b7e6ea8", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "DC", "result": "Theorem 4.4", "source": "../src/the-double-commutant-theorem.md", "source_sha256": "b44cb7d14cc9f206cb69a4d8f4afaca733dcfff2f1aa8d2d1c7eaf515c5b88b0", "reader": "../the-double-commutant-theorem.html", "reader_sha256": "ce5fb1b87a42c5f716d9733fb2261046284fbc171e1440ae32e23a340ee3dc8e", "anchor": "OA-FND-BI-06", "reader_heading": "4. The double commutant theorem", "declaration_line": 252, "section_through_line": 319, "named_passage_sha256": "e1a5fbf0c2f655c55a2a16cc6c8626ba3a1b1af1c4e32c121d77db51ce31656f", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "DC", "result": "Theorem 5.8", "source": "../src/the-double-commutant-theorem.md", "source_sha256": "b44cb7d14cc9f206cb69a4d8f4afaca733dcfff2f1aa8d2d1c7eaf515c5b88b0", "reader": "../the-double-commutant-theorem.html", "reader_sha256": "ce5fb1b87a42c5f716d9733fb2261046284fbc171e1440ae32e23a340ee3dc8e", "anchor": "OA-FND-BI-04", "reader_heading": "5. Direct sums, reduced and induced algebras", "declaration_line": 355, "section_through_line": 404, "named_passage_sha256": "8ab6349017fa74dcb5449e589ee02ce4a1acc9f7ef0bfce16114feab5d46204c", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 3.1", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-01", "reader_heading": "3. The projection lattice and equivalence of projections", "declaration_line": 51, "section_through_line": 56, "named_passage_sha256": "c555ef2937734ab96646d2aa194a25d683af04aa95cc271d804e6e51edff5d2a", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Lemma 3.3", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-01", "reader_heading": "3. The projection lattice and equivalence of projections", "declaration_line": 65, "section_through_line": 79, "named_passage_sha256": "ab8685841203be124b8e7326382a45030c2f84af35e8cb9cba550cebe2e9ff78", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 3.5", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-01", "reader_heading": "3. The projection lattice and equivalence of projections", "declaration_line": 82, "section_through_line": 96, "named_passage_sha256": "42f45e4dfba54ece66ccf57d39737d5bb44d368b5f534efe4cf92ae41424a5cf", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 4.4", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-04", "reader_heading": "4. Supports, cyclic projections and the parallelogram law", "declaration_line": 111, "section_through_line": 116, "named_passage_sha256": "b9b06a2b9bbc4581de9eeb414011bf4eb0ae134960d4cf732846fb5692f15039", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 5.1", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-03", "reader_heading": "5. The comparison theorem", "declaration_line": 136, "section_through_line": 162, "named_passage_sha256": "e5d61309e4c8247a7f6cb2c1f7e209991c461dc8c3b6bcba0ac4a823dfcdd430", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Lemma 6.2", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-09", "reader_heading": "6. Finite, infinite and abelian projections", "declaration_line": 251, "section_through_line": 265, "named_passage_sha256": "646c530fbb3acd4b4dbb4b7641a1328428955a2a650f36958c35ecb8c0dc880e", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Lemma 6.3", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-09", "reader_heading": "6. Finite, infinite and abelian projections", "declaration_line": 266, "section_through_line": 271, "named_passage_sha256": "ecbd4381baae5f59441984c58321fb09d1473fa77c0b145e3d4adbf06f674d8d", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Definition 7.1", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-10", "reader_heading": "7. The type decomposition", "declaration_line": 274, "section_through_line": 283, "named_passage_sha256": "9656d102edeee67628f9ff86b84bab402acedf5e82a4b773a317b7add0d8f9ea", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Theorem 7.2", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-10", "reader_heading": "7. The type decomposition", "declaration_line": 284, "section_through_line": 295, "named_passage_sha256": "e6180b00f79f83643f316c974d8ce2550f917837137c2ed824b216a12f0158c0", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 8.4", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-11", "reader_heading": "8. Matrix units and tensor splitting", "declaration_line": 347, "section_through_line": 367, "named_passage_sha256": "2be03ac7e46dfd02d42905c62423b3d2a639bd5f5edf9e6ae6a504885b12598d", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 9.1", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-12", "reader_heading": "9. Comparing and counting abelian projections", "declaration_line": 376, "section_through_line": 396, "named_passage_sha256": "f00d2d52b72c0f7e1b8e5e04a9edf14ac16e42501bd595068ca28dc5b88e0dbe", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Theorem 10.3", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-14", "reader_heading": "10. Homogeneous algebras and the structure of type I algebras", "declaration_line": 445, "section_through_line": 464, "named_passage_sha256": "ef640522166e4906ae6460d75cf68e11d55361461e4dff7036c6d1d740badfd6", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 11.2", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-15", "reader_heading": "11. Type I algebras and their commutants", "declaration_line": 500, "section_through_line": 505, "named_passage_sha256": "2c84770754fb12bbdeb3ce3cdc4139098c2d2f64938b3ef92c7345a1f5fc8ddd", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 13.4", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-17", "reader_heading": "13. Orthogonal families of equivalent projections, and halving", "declaration_line": 709, "section_through_line": 720, "named_passage_sha256": "472dfdfd7e622748950a967dcca4c8b2774f491b17ff7786eb6bac9d9d41d6d9", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 14.2", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-18", "reader_heading": "14. Finite projections: the modular law and unitary equivalence", "declaration_line": 767, "section_through_line": 783, "named_passage_sha256": "71b944030f5b55fdd36d2cd7d079fa9829dcad8ab987ddfe3b15965209c68d16", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Definition 15.1", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-19", "reader_heading": "15. Countable sums and properly infinite algebras", "declaration_line": 788, "section_through_line": 791, "named_passage_sha256": "36d719218a4e2f8ca35ae5294bbbc5ea6e6e50a7f33d7fabe475668fe00a3f1a", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PT", "result": "Proposition 15.2", "source": "../src/projections-and-types-of-von-neumann-algebras.md", "source_sha256": "6cf4b11633049669b90c1f247bb954f4e9e8b418ce1f04d094d5929c6242fbe7", "reader": "../projections-and-types-of-von-neumann-algebras.html", "reader_sha256": "029b4ae847c8348836e1bce53de5b0974651a79236ee90c820675ce1bcf665ad", "anchor": "OA-FND-TY-19", "reader_heading": "15. Countable sums and properly infinite algebras", "declaration_line": 792, "section_through_line": 808, "named_passage_sha256": "0e1bddc2e6f0d23fe5127bb7fa032b49db120af64ff8d69b42d2f865a2e5349d", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "PW", "result": "Theorem 10.2", "source": "../src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md", "source_sha256": "16875e65b1ea176c7e1ac95329892b348e6ac399c099fa104661880de9007412", "reader": "../polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html", "reader_sha256": "7a0f03bd8ad2f0433226f9b42f0dbc2deccbcc5699741ae20ea503bdd043f256", "anchor": "OA-FND-PD-16", "reader_heading": "10. Weak compactness in the predual", "declaration_line": 813, "section_through_line": 869, "named_passage_sha256": "045253c647069f524d3d2aeba930972b3817094d8272b234d8596d95e3b33d7e", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "UE", "result": "Lemma 2.1", "source": "../src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md", "source_sha256": "e07ea555c07d14fa449cf2a415a417d4157e4527f3bc6436609b2e305c035f1b", "reader": "../the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html", "reader_sha256": "5f80bdbc4027d8a52ba453d99317e614245541f730d94da8fc4898e26ca154de", "anchor": "OA-FND-WA-02", "reader_heading": "2. Extending a representation to the bidual", "declaration_line": 194, "section_through_line": 227, "named_passage_sha256": "e9c5ef3fe96c0a2eeb55ba909f8698a9ca4661c1d9cc7959fc02e30eb137bbfa", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "UE", "result": "Theorem 3.3", "source": "../src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md", "source_sha256": "e07ea555c07d14fa449cf2a415a417d4157e4527f3bc6436609b2e305c035f1b", "reader": "../the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html", "reader_sha256": "5f80bdbc4027d8a52ba453d99317e614245541f730d94da8fc4898e26ca154de", "anchor": "OA-FND-WA-03", "reader_heading": "3. The universal enveloping von Neumann algebra", "declaration_line": 254, "section_through_line": 274, "named_passage_sha256": "d07e775a2daf20d6517336a7b7cb8aac81b94dcc8ff3d78432638e4ab5a07408", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "UE", "result": "Lemma 6.3", "source": "../src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md", "source_sha256": "e07ea555c07d14fa449cf2a415a417d4157e4527f3bc6436609b2e305c035f1b", "reader": "../the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html", "reader_sha256": "5f80bdbc4027d8a52ba453d99317e614245541f730d94da8fc4898e26ca154de", "anchor": "OA-FND-WA-13", "reader_heading": "6. Minimal projections and finite-dimensional von Neumann algebras", "declaration_line": 484, "section_through_line": 487, "named_passage_sha256": "8708538d16854025bf30b600747b3c40b95200e6c3416c0e56e36c6d47945521", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "UE", "result": "Lemma 11.1", "source": "../src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md", "source_sha256": "e07ea555c07d14fa449cf2a415a417d4157e4527f3bc6436609b2e305c035f1b", "reader": "../the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html", "reader_sha256": "5f80bdbc4027d8a52ba453d99317e614245541f730d94da8fc4898e26ca154de", "anchor": "OA-FND-WA-18", "reader_heading": "11. Singular functionals and the uniqueness of the predual", "declaration_line": 782, "section_through_line": 793, "named_passage_sha256": "c3146e7fc98a3661211e34b2eacbd2acb3756986d6da1ad96399774c5ad71b26", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "UE", "result": "Corollary 11.4", "source": "../src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md", "source_sha256": "e07ea555c07d14fa449cf2a415a417d4157e4527f3bc6436609b2e305c035f1b", "reader": "../the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html", "reader_sha256": "5f80bdbc4027d8a52ba453d99317e614245541f730d94da8fc4898e26ca154de", "anchor": "OA-FND-WA-18", "reader_heading": "11. Singular functionals and the uniqueness of the predual", "declaration_line": 807, "section_through_line": 812, "named_passage_sha256": "a75049f17ed1eac8110aaababf21e790c2710c7ce4b12a47c8728d7a1792e269", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Lemma 2.2", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-02", "reader_heading": "2. Tychonoff's theorem", "declaration_line": 60, "section_through_line": 65, "named_passage_sha256": "3669e248a5c225e611bd1907d1709f92e1b79293e96efb87ebf3806f6b23d2e0", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Theorem 2.3", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-02", "reader_heading": "2. Tychonoff's theorem", "declaration_line": 66, "section_through_line": 71, "named_passage_sha256": "1f5a3e5d169ae0d8bd04f92f24a620dc9c8e9a6c9d9c5a1f26b7d38733473411", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Theorem 3.1", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-03", "reader_heading": "3. The Banach–Alaoglu theorem", "declaration_line": 74, "section_through_line": 83, "named_passage_sha256": "7ffd2480f462af712ba9a67188c15672f451293ec13315321c0323b0a48c7b63", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Theorem 5.2", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-05", "reader_heading": "5. Annihilators, bipolars and Goldstine's theorem", "declaration_line": 121, "section_through_line": 128, "named_passage_sha256": "69ee00af45a6355b01e12d1681fbc642e4170dcfb3527644b355f2605eeac70f", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Proposition 5.3", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-05", "reader_heading": "5. Annihilators, bipolars and Goldstine's theorem", "declaration_line": 129, "section_through_line": 155, "named_passage_sha256": "dbed45be32f03d3c7973de14ec3e140c38f86e91058dcf7e4a0d680fc4e21920", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Theorem 6.1", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-06", "reader_heading": "6. The Krein–Milman theorem", "declaration_line": 167, "section_through_line": 181, "named_passage_sha256": "86c5db2309496cd2700cd8032a69a2dca5286fe8991cc556ed11e27c704a85c5", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "WT", "result": "Theorem 6.2", "source": "../src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md", "source_sha256": "0b07a69ced011f1b54a1a6e657f752466eff4202ed1cc43a85a209e645d4a8da", "reader": "../weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html", "reader_sha256": "77d892ff83b479d040f75c6d356f14e7e53bc93a99c60964bb254d87e772a11f", "anchor": "OA-FND-WT-06", "reader_heading": "6. The Krein–Milman theorem", "declaration_line": 182, "section_through_line": 194, "named_passage_sha256": "ad103fb247e20913e07d78823bf997a076c254d8b34105a096d4ef66bb83ef1f", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "OT", "result": "Theorem 6.2", "source": "../src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md", "source_sha256": "65e04118c1d72459208f78f956003075cf7d453c6dfb79b3a4e35b94512ba974", "reader": "../compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html", "reader_sha256": "1e1975be393568f7e6baee04d1dd627f5de418293ad9ad5d33ec95a83bb457f2", "anchor": "OA-FND-LT-07", "reader_heading": "6. The predual of B(H)", "declaration_line": 697, "section_through_line": 710, "named_passage_sha256": "3f644bbefcaf60793f735ce03158d012af3bb70a23ee8b6007c5250169003e42", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "OT", "result": "Lemma 8.5", "source": "../src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md", "source_sha256": "65e04118c1d72459208f78f956003075cf7d453c6dfb79b3a4e35b94512ba974", "reader": "../compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html", "reader_sha256": "1e1975be393568f7e6baee04d1dd627f5de418293ad9ad5d33ec95a83bb457f2", "anchor": "OA-FND-LT-09", "reader_heading": "8. Seven topologies on B(H)", "declaration_line": 883, "section_through_line": 893, "named_passage_sha256": "fcb90bd1b7caef2ddf4f3c48c7b2e7363580a657d320743b1246704aa321532f", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "OT", "result": "Proposition 8.6", "source": "../src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md", "source_sha256": "65e04118c1d72459208f78f956003075cf7d453c6dfb79b3a4e35b94512ba974", "reader": "../compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html", "reader_sha256": "1e1975be393568f7e6baee04d1dd627f5de418293ad9ad5d33ec95a83bb457f2", "anchor": "OA-FND-LT-09", "reader_heading": "8. Seven topologies on B(H)", "declaration_line": 894, "section_through_line": 916, "named_passage_sha256": "33cb79f9efbf1c18db3519223f2c281aa81066bf2b63887ba86719a9033d2690", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "OT", "result": "Theorem 9.4", "source": "../src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md", "source_sha256": "65e04118c1d72459208f78f956003075cf7d453c6dfb79b3a4e35b94512ba974", "reader": "../compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html", "reader_sha256": "1e1975be393568f7e6baee04d1dd627f5de418293ad9ad5d33ec95a83bb457f2", "anchor": "OA-FND-LT-10", "reader_heading": "9. Continuous functionals and closed convex sets", "declaration_line": 984, "section_through_line": 1010, "named_passage_sha256": "b8419c6a115319f2ba28879a3695963722d7a17c51090cd9b6b53003b5005151", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 1.1", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-01", "reader_heading": "1. Zorn's lemma", "declaration_line": 34, "section_through_line": 46, "named_passage_sha256": "ee3fc3789e0e25b5ad9e734cd36246d59f9981d4e7e029a85282f786d8716053", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 2.1", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-02", "reader_heading": "2. The Hahn–Banach theorem", "declaration_line": 51, "section_through_line": 73, "named_passage_sha256": "07d62a3e815ec3f5b5bcc84b90573cd5b68c4503edd3ef61acb1ee826f9e4ce9", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 2.2", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-02", "reader_heading": "2. The Hahn–Banach theorem", "declaration_line": 74, "section_through_line": 97, "named_passage_sha256": "4fac8b03d5e9fa6462a0e62ae3a4be97fb44675bad968e9051c112232899c34e", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Corollary 2.3", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-02", "reader_heading": "2. The Hahn–Banach theorem", "declaration_line": 98, "section_through_line": 118, "named_passage_sha256": "e1120b9592d2b6e1d2aad6e95d33e747a8bc5f7515f850e273d575d5edf8c14d", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 3.1", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-03", "reader_heading": "3. The Baire category theorem", "declaration_line": 121, "section_through_line": 130, "named_passage_sha256": "c2a04a3d14aaedea983350c7751d8c95ea2ab063832123ddaea0b5ae78a727d2", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 3.2", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-03-LCH", "reader_heading": "Locally compact Hausdorff spaces and weak compactness", "declaration_line": 135, "section_through_line": 161, "named_passage_sha256": "70d99d77cc8353ae959eb6a1a181e8411b543fb2c4d247ed8eabe2c4c054c9f5", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Corollary 3.3", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-03-LCH", "reader_heading": "Locally compact Hausdorff spaces and weak compactness", "declaration_line": 162, "section_through_line": 190, "named_passage_sha256": "20a006e88cb547b470a21aec32bab0be6da30d8de429973456f482b8ee87f090", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 4.2", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-04", "reader_heading": "4. Uniform boundedness", "declaration_line": 197, "section_through_line": 200, "named_passage_sha256": "15fac4a4ad5c7f03ed4c459d5bc1a4122a63d0601189e603e5fb8e7586ce5c1a", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 5.1", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-05", "reader_heading": "5. The open mapping and closed graph theorems", "declaration_line": 214, "section_through_line": 233, "named_passage_sha256": "a88d4f26bd2145af9739d45affec76595cec02fde94f4a5fdf66fc859a7e3741", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 5.3", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-05", "reader_heading": "5. The open mapping and closed graph theorems", "declaration_line": 238, "section_through_line": 243, "named_passage_sha256": "badfbf9912762aee7bb8902878990950f2233458fa6377aaacd44c41691d293b", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HB", "result": "Theorem 7.1", "source": "../src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md", "source_sha256": "2e9b534576a42733482c63da7511ab4a6d169508bcce887032e7d93da60d5b91", "reader": "../hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html", "reader_sha256": "ee88b6e8674f145ca7ca5b91ea7b70c87a34aac053f10984aefd5f838fc2903f", "anchor": "OA-FND-HB-07", "reader_heading": "7. Finite-dimensional spaces", "declaration_line": 320, "section_through_line": 336, "named_passage_sha256": "d2e90cb0c086bb57568d1669a9d580e30c991b42c23dacb6142368f4f5b90149", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "BN", "result": "Proposition 3.4", "source": "../src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md", "source_sha256": "c1cddcf9028d7f51f891ae87f4ee53f270eeaab5c49665bdf5542b43a86bc3c4", "reader": "../banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.html", "reader_sha256": "cfd4737b21bd3a6f4cfc9f0cc318ca248e987d84da0d67cd729972310e709cde", "anchor": "OA-FND-BN-05", "reader_heading": "A C*-norm on the unitization", "declaration_line": 188, "section_through_line": 211, "named_passage_sha256": "78ae772a02a28296c2f17b32a717a32b476d27b1fd4c47a499c68a0ad7507f46", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." }, { "key": "HS", "result": "Lemma (Completion)", "source": "../src/hilbert-spaces-and-compact-operators.md", "source_sha256": "833a2d60bb99e63875ce10bf363953c0bb265f92996ecc8648fc8c6a105cb1fe", "reader": "../hilbert-spaces-and-compact-operators.html", "reader_sha256": "27be849d0131cecb55b41326a21d5aea490064536645e9e92ef4b13756d8a827", "anchor": "completing-normed-and-inner-product-spaces", "reader_heading": "Completing normed and inner product spaces", "declaration_line": 50, "section_through_line": 83, "named_passage_sha256": "689dcc2c00c18249e118b0121962cc40af319aa71dd990432ed80642caa044ce", "scope": "Named result in the supplied complete provider; a declaration-range locator is not a new proof certification." } ], "authorship": [ { "role": "retained proof", "product": "ChatGPT Pro", "model": null, "reasoning_effort": null, "evidence": "Reported origin of the received authored proof; exact generating model/effort not established." }, { "role": "elementary closedness and compactness bridges, checked integration and source bindings", "model": "GPT-6 Astra", "provider": "OpenAI", "reasoning_effort": "ultra" } ], "original_source": { "url": "https://huggingface.co/datasets/KokunoYumeto/open-math-proof-checks/blob/440749e133e23c5b48b743b3ed143ccd3e200fad/episodes/sources/a86986b483f4e974d7162c4cbd6c3ee40117ea9bcd10e605fc44bc2cb37f072d.md", "sha256": "a86986b483f4e974d7162c4cbd6c3ee40117ea9bcd10e605fc44bc2cb37f072d", "preserved": true }, "checking": { "coordinator_selfcheck": true, "independent_review": false, "human_review": false, "formal_verification": false, "scope": "Retained argument, elementary bridge and named programme input routes; not programme-wide certification." }, "mathematical_scope": "All algebraic C*-algebra derivations are bounded. 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