{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:d091fdbd-b578-4a81-85b8-b28b54ceaf3d", "labels": { "zh-CN": "定义域", "en": "domain" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "bc562ce67057aaf936d32b490c6db2667cb8006d7b69fe7510eef04841c1cd7b", "occurrences": [ { "concept_id": "urn:uuid:d091fdbd-b578-4a81-85b8-b28b54ceaf3d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-01/04-mappings", "anchor": "first-1a4e8f2ed8c7a5443668-zh", "chapter": "1", "section": "1.4", "offset": 803, "range": [ 4492, 4722 ], "source_revision": "584fda344e534c6df3d98f8c4bf23c1a7c4695a49d55be41c192759946ba16fc", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "定义域(domain)", "context": { "range": [ 4430, 4962 ], "quote": "通常写 $y=\\varphi(x)$ 或 $\\varphi:x\\mapsto y$。这一定义把函数与其图像识别。$X$ 称为定义域(domain)，$Y$ 称为陪域(codomain)，而", "quote_sha256": "59b2c1ebbffa31779dda3a1436f050f1d08b3475fd563a4d5a7ba98ce821b8d7", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-01/04-mappings", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-700cc8163b83dd53977535fc" }, { "concept_id": "urn:uuid:d091fdbd-b578-4a81-85b8-b28b54ceaf3d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-13/06-unbounded-operators", "anchor": "concept-13-domain", "chapter": "13", "section": "13.6", "offset": 791, "range": [ 7393, 7611 ], "source_revision": "d45e53fc27c5bf1d2c2121c20ab0e61aeb57067de09a90fd6173850a6fa3d4f2", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "定义域(domain)", "context": { "range": [ 6933, 8060 ], "quote": "对一个希尔伯特空间 $\\mathcal{H}$ 上的线性算符$A$ 而言，若满足如果对于任何 $M>0$ ，存在一个向量 $u$ 使得 $\\|A u \\| \\geq M\\|u\\|$，则算符称为无界的(unbounded)。在整个 $\\mathcal{H}$ 上的无界算符中的有趣示例很少 ， 例如自伴算符就根本没有。因此，通常认为无界算子 $A$ 不一定定义在整个 $\\mathcal{H}$ 上，而只定义在某个向量子空间 $D_{A} \\subseteq \\mathcal{H}$ 称为 $A$ 的定义域(domain)。它的值域(range)定义为映射到的向量集合，$R_{A}=A\\left(D_{A}\\right)$。一般来说，我们会用一对 $\\left(A, D_{A}\\right)$，其中 $D_{A}$ 是 $\\mathcal{H}$的向量子空间 且$A: D_{A} \\rightarrow R_{A} \\subseteq \\mathcal{H}$ 是一个线性映射，来指代一个$\\mathcal{H}$ 中的算符。当定义域 $D_{A}$没有歧义时，我们通常会简单地用算符 $A$来指代。", "quote_sha256": "7cc8035df3e7973323763e133d3a9814ea5d1c45e2fddc8bc34dd62e2d630ea7", "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-75cfe3cafe90554dfd49ae36" }, { "concept_id": "urn:uuid:d091fdbd-b578-4a81-85b8-b28b54ceaf3d", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-13/06-unbounded-operators", "anchor": "first-1a4e8f2ed8c7a5443668-en", "chapter": "13", "section": "13.6", "offset": 1041, "range": [ 7136, 7361 ], "source_revision": "c9f6aaaf4583a87d844433adf5551a93a8faeb812cbe88bab00cc9cdaa02d709", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "domain", "context": { "range": [ 6426, 8001 ], "quote": "A linear operator $A$ on a Hilbert space is unbounded if for any $M > 0$ there exists a vector $u$ such that $\\| A u \\| \\geq M \\| u \\|$ . Very few interesting examples of unbounded operators are defined on all of $\\mathcal H$ – for self-adjoint operators, there are none at all. It is therefore usual to consider an unbounded operator $A$ as not being necessarily defined over all of $\\mathcal H$ but only on some vector subspace $D _ { { \\cal A } } \\subseteq { \\mathcal { H } }$ called the domain of $A$. Its range is defined as the set of vectors that are mapped onto, $R _ { A } = A ( D _ { A } )$ . In general we will refer to a pair $( A , D _ { A } )$ , where $D _ { A }$ is a vector subspace of $\\mathcal H$ and $A : D _ { A } \\to R _ { A } \\subseteq \\mathcal { H }$ is a linear map, as being an operator in $\\mathcal H$. 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