{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:d9cfc0e8-4d07-4144-9b86-a6fa7b076397", "labels": { "zh-CN": "乘法算符", "en": "multiplication operator" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "4dfa8dd609b0a821c6c774ab644f611a03ae0169d84f9c2e3d57bb0d24d48db5", "occurrences": [ { "concept_id": "urn:uuid:d9cfc0e8-4d07-4144-9b86-a6fa7b076397", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-13/04-bounded-linear-operators", "anchor": "concept-13-multiplication-operator", "chapter": "13", "section": "13.4", "offset": 2842, "range": [ 8720, 8973 ], "source_revision": "194ca7b68e093551807e302c52f29521e73d76a77f31f6fd5b10ca6c3ee642c4", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "乘法算符(multiplication operator)", "context": { "range": [ 8592, 9463 ], "quote": "> [!example] 例 13.9 \n>\n> $\\alpha$是测度空间$X$上的平方可积函数构成的希尔伯特空间$L^2(X)$上的有界可测函数，乘法算符(multiplication operator)$A_\\alpha:L^2(X)\\rightarrow L^2(X)$，定义成$A_\\alpha (f)=\\alpha f$是一个有界线性算符，因为$\\alpha f$对于每个$f\\in L^2(X)$，而且是平方可积的因为\n>\n> $$\n> |\\alpha f|^{2} \\leq M^{2}|f|^{2} \\quad \\text { 其中 } \\quad M=\\sup _{x \\in X}|\\alpha(x)| .\n> $$\n>\n> 乘法算符在$L^2(X)$空间上是良定义的，因为$f$和$f'$几乎处处相等，$f\\sim f'$于是$\\alpha f\\sim \\alpha f'$，于是我们用$A_\\alpha f$来代替$A_\\alpha [f]$并无歧义，线性性显然，而有界性源于\n>\n> $$\n> \\left\\|A_{\\alpha} f\\right\\|^{2}=\\int_{X}|\\alpha f|^{2} \\mathrm{~d} \\mu \\leq M^{2} \\int_{X}|f|^{2} \\mathrm{~d} \\mu=M^{2}\\|f\\|^{2}\n> $$", "quote_sha256": "dcf37c770c3b9fe9e76a4d043f2b94b4c08f02e5a8d3d76556e4bd9fd9624a25", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-13/04-bounded-linear-operators", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d3d144c821aeac38fa5cf9dd" }, { "concept_id": "urn:uuid:d9cfc0e8-4d07-4144-9b86-a6fa7b076397", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-13/04-bounded-linear-operators", "anchor": "first-bf78e829304ec69ff626-en", "chapter": "13", "section": "13.4", "offset": 3671, "range": [ 9040, 9282 ], "source_revision": "eb0725f6ab7fc44fa4ee614b878f2ec644f55a6b84b31a4afac5555a1c5df299", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "multiplication operator", "context": { "range": [ 8810, 10107 ], "quote": "> [!example] Example 13.9 \n>\n> Let $\\alpha$ be any bounded measurable function on the Hilbert space $L ^ { 2 } ( X )$ of square integrable functions on a measure space $X$. The multiplication operator $A_{\\alpha}:L^2(X)\\to L^2(X)$ defined by $A _ { \\alpha } ( f ) = \\alpha f$ is a bounded linear operator, for $\\alpha f$ is measurable for every $f \\in L ^ { 2 } ( X )$ , and it is square integrable since\n>\n> $$\n> | \\alpha f | ^ {2} \\leq M ^ {2} | f | ^ {2} \\quad \\text { where } \\quad M = \\sup _ {x \\in X} | \\alpha (x) |.\n> $$\n>\n> The multiplication operator is well-defined on $L ^ { 2 } ( X )$ , for if $f$ and $f ^ { \\prime }$ are equal almost everywhere, $f \\sim f ^ { \\prime }$ , then $\\alpha f \\sim \\alpha f^{\\prime}$ ; thus there is no ambiguity in writing $A _ { \\alpha } f$ for $A _ { \\alpha } [ f ]$ Linearity is trivial, while boundedness follows from\n>\n> $$\n> \\| A _ {\\alpha} f \\| ^ {2} = \\int_ {X} | \\alpha f | ^ {2} \\mathrm{d} \\mu \\leq M ^ {2} \\int_ {X} | f | ^ {2} \\mathrm{d} \\mu = M ^ {2} \\| f \\| ^ {2}.\n> $$", "quote_sha256": "44eb90663f1e69779db6ebacb32869cc87a84e7d8a1270ba15d33e3022aeb077", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-13/04-bounded-linear-operators", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-75c15821c117b34ce5794028" } ], "registered_positions": [ { "route": "en/chapter-13/04-bounded-linear-operators", "lang": "en", "section": "13.4", "signal": "term-mark", "label": "multiplication operator", "english": "multiplication operator", "offset": 3218 }, { "route": "zh/chapter-13/04-bounded-linear-operators", "lang": "zh-CN", "section": "13.4", "signal": "term-mark", "label": "乘法算符", "english": "multiplication operator", "offset": 2459 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#example-13-9\"", "href": "#example-13-9", "range": [ 8633, 8653 ], "relation": "references", "target_route": "zh/chapter-13/04-bounded-linear-operators", "target_anchor": "example-13-9", "resolved": true, "reason": null, "source_revision": "194ca7b68e093551807e302c52f29521e73d76a77f31f6fd5b10ca6c3ee642c4", "source_route": "zh/chapter-13/04-bounded-linear-operators", "occurrence_id": "occ-d3d144c821aeac38fa5cf9dd" }, { "raw": "href=\"#example-13-9\"", "href": "#example-13-9", "range": [ 8851, 8871 ], "relation": "references", "target_route": "en/chapter-13/04-bounded-linear-operators", "target_anchor": "example-13-9", "resolved": true, "reason": null, "source_revision": "eb0725f6ab7fc44fa4ee614b878f2ec644f55a6b84b31a4afac5555a1c5df299", "source_route": "en/chapter-13/04-bounded-linear-operators", "occurrence_id": "occ-75c15821c117b34ce5794028" } ], "reading_order": [ "occ-d3d144c821aeac38fa5cf9dd", "occ-75c15821c117b34ce5794028" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "ecbf11d30a323768043b0ede9b2198a3befc5b0010d11b3cb46469a1ccea50d0", "optional_enrichments": [] }