{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2", "labels": { "zh-CN": "闭的", "en": "closed" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "39edad544de124afa4b1de6edea40131f9c5eef5d0d5b6c5fc9c60928970a603", "occurrences": [ { "concept_id": "urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-13/06-unbounded-operators", "anchor": "concept-13-closed", "chapter": "13", "section": "13.6", "offset": 6902, "range": [ 13504, 13721 ], "source_revision": "d45e53fc27c5bf1d2c2121c20ab0e61aeb57067de09a90fd6173850a6fa3d4f2", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "闭的(closed)", "context": { "range": [ 13362, 14157 ], "quote": "对于任意一个满足$u_{n} \\rightarrow u$ 和 $A u_{n} \\rightarrow v$ 的序列 $u_{n} \\in$ $D_{A}$，如果有 $u \\in D_{A}$ 且 $A u=v$，则我们称稠定算符 $\\left(A, D_{A}\\right)$ 是闭的(closed)。另一种表述是，一个算符是闭的，当且仅当它的图像(graph) $G_{A}=\\{(x, A x) \\mid x \\in$ $\\left.D_{A} \\right\\}$ 是积集 $\\mathcal{H} \\times \\mathcal{H}$ 的闭子集。封闭性的概念类似于连续性，不同之处在于，对于封闭性，我们必须保证极限 $A u_{n} \\rightarrow v$ 必须成立；而对于连续性，这一性质可以推导出来。每个连续算符显然都是闭的，但反之未必成立。", "quote_sha256": "a40bdb2d211ac0b081bdbbd684797d557b0bb943a450b3e2e42d940e43e6abd9", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-13/06-unbounded-operators", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-a9973e71f27c64cf3c8d86f3" }, { "concept_id": "urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-13/06-unbounded-operators", "anchor": "first-df2cde65096927db28d1-en", "chapter": "13", "section": "13.6", "offset": 9172, "range": [ 15267, 15492 ], "source_revision": "c9f6aaaf4583a87d844433adf5551a93a8faeb812cbe88bab00cc9cdaa02d709", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "closed", "context": { "range": [ 15190, 16313 ], "quote": "We say a densely defined operator $( A , D _ { A } )$ in $\\mathcal { H }$ is closed if for every sequence $u _ { n } \\in$ $D _ { A }$ such that $u _ { n } \\to u$ and $Au_n\\to v$ it follows that $u \\in D _ { A }$ and $A u = v$ . Another way of expressing this is to say that an operator is closed if and only if its graph $G _ { A } = \\{ ( x , A x ) | x \\in$ $D _ { A } \\}$ is a closed subset of the product set $\\mathcal { H } \\times \\mathcal { H }$ . The notion of closedness is similar to continuity, but differs in that we must assert the limit $Au_n\\to v$ , while for continuity it is deduced. Clearly every continuous operator is closed, but the converse does not hold in general.", "quote_sha256": "a1bb6fab59bbb6f330337390fb669bf5e59e5a2c69913aca87b9196b5385b103", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-13/06-unbounded-operators", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-1fcfad472054d5caa5706871" }, { "concept_id": "urn:uuid:8da45318-5944-4aa7-afd5-245f6894ccb2", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-17/04-homology-and-cohomology", "anchor": "first-e2e602e31df1891319af-en", "chapter": "17", "section": "17.4", "offset": 16665, "range": [ 20489, 20679 ], "source_revision": "bcdaffd9c93fa5d4dbb16ac427fef782512ff58c5d5472ea282c7769a2173a3d", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "closed", "context": { "range": [ 20442, 21907 ], "quote": "A differential $r$-form $\\alpha$ is said to be closed if $\\mathrm{d}\\alpha=0$, and it is said to be exact if there exists an $(r-1)$-form $\\beta$ such that $\\alpha = \\mathrm { d } \\beta$ . Clearly every exact $r$-form is closed since $\\mathbf { d } ^ { 2 } { \\boldsymbol { \\beta } } = 0$ . In the language of cochains these definitions can be expressed as follows: an $r$-cochain $\\alpha$ is called an $r$-cocycle if it is a closed differential form, while it is an $r$-coboundary if it is exact. We denoted the vector subspace of $r$-cocycles by $Z ^ { r } ( M )$ , and the subspace of $r$-coboundaries by $B ^ { r } ( M )$", "quote_sha256": "437a575acf05d0338cf7236057db2e31447114343d90578117d43911cd53e738", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-17/04-homology-and-cohomology", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-165aa4b6c3f7aa2100f53e36" } ], "registered_positions": [ { "route": "en/chapter-13/06-unbounded-operators", "lang": "en", "section": "13.6", "signal": "term-mark", "label": "closed", "english": "closed", "offset": 7573 }, { "route": "zh/chapter-13/06-unbounded-operators", "lang": "zh-CN", "section": "13.6", "signal": "term-mark", "label": "闭的", "english": "closed", "offset": 5583 }, { "route": "en/chapter-17/04-homology-and-cohomology", "lang": "en", "section": "17.4", "signal": "term-mark", "label": "closed", "english": "closed", "offset": 13991 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-a9973e71f27c64cf3c8d86f3", "occ-1fcfad472054d5caa5706871", "occ-165aa4b6c3f7aa2100f53e36" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 3, "located_occurrences": 3, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "11a5387bf8b74d2dc6dcff7f13cdb1304d57798ccaee325c699ff251f77d3230", "optional_enrichments": [] }