{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:34c6c0e8-a933-4a51-bbf6-7721075b239c", "labels": { "zh-CN": "希尔伯特空间", "en": "Hilbert space" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "98da4498e5a20988a1f19f773e3215fc09a372e60420a1ea65711682e872541c", "occurrences": [ { "concept_id": "urn:uuid:34c6c0e8-a933-4a51-bbf6-7721075b239c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-05/02-complex-inner-product-spaces", "anchor": "concept-5-hilbert-space", "chapter": "5", "section": "5.2", "offset": 2569, "range": [ 9814, 10048 ], "source_revision": "130978731f44c1f1399a4088c5e4ff3a50730c13937bd888ca64abf512d0946a", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "希尔伯特空间(Hilbert space)", "context": { "range": [ 9533, 10091 ], "quote": "带有内积的复向量空间将简称为 内积空间(inner product space)。如果 $V$ 是有限维的，则常称为有限维 希尔伯特空间(Hilbert space)，但对于无限维空间，希尔伯特空间这一术语仅适用于该空间是完备的情形（见第 13 章）。", "quote_sha256": "3d965b4420e4a074e1785d13fb2a3df1274b1a2fef8fc2d0e4430d7ed64df090", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-05/02-complex-inner-product-spaces", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-7595185e9a6c4a275abb242d" }, { "concept_id": "urn:uuid:34c6c0e8-a933-4a51-bbf6-7721075b239c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-13/01-definitions-and-examples", "anchor": "concept-13-hilbert-space", "chapter": "13", "section": "13.1", "offset": 1264, "range": [ 6187, 6422 ], "source_revision": "f0914f256f99a6e02d1927be1f285ad9b238d6e4cb01419d5d49e86fa0d4b82d", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "希尔伯特空间(Hilbert space)", "context": { "range": [ 5993, 6587 ], "quote": "三角不等式是定理5.6的结果，这些性质在无穷维空间也保持，希尔伯特空间(Hilbert space)($\\mathcal{H},\\langle \\cdot\\mid \\cdot\\rangle$)是在这样内积空间的诱导范数下完备的空间。也就是说，($\\mathcal{H},\\left\\lVert \\cdot\\right\\rVert$)是巴拿赫空间。这章水平的希尔伯特空间介绍可以参看[1-6],而更进一步的主题在[7-11]中有讨论", "quote_sha256": "5f9567d5065bc22262d5eb313508d13e93ac180c4b86c4296842d9d5bf3fda5f", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-13/01-definitions-and-examples", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-92d24b52c04b936966834195" }, { "concept_id": "urn:uuid:34c6c0e8-a933-4a51-bbf6-7721075b239c", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-13/01-definitions-and-examples", "anchor": "concept-13-hilbert-space", "chapter": "13", "section": "13.1", "offset": 2086, "range": [ 6501, 6771 ], "source_revision": "968f1c0d16644d86921119722c7a39446387235a239cb76b69485ab3d79eb071", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Hilbert space", "context": { "range": [ 6190, 7071 ], "quote": "The triangle inequality (Norm3) is a consequence of Theorem 5.6. These properties hold equally in finite or infinite dimensional vector spaces. A Hilbert space $(\\mathcal H,\\langle\\cdot\\mid\\cdot\\rangle)$ is an inner product space that is complete in the induced norm; that is, $(\\mathcal H,\\|\\cdot\\|)$ is a Banach space. 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