{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:737defc6-6789-4cdd-87f7-63952cdef118",
"labels": {
"zh-CN": "三角不等式",
"en": "Triangle inequality"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "c9b6d2edc12655ab2bc9c677483d2b964958ec3740c2508d10f05187e54b47e0",
"occurrences": [
{
"concept_id": "urn:uuid:737defc6-6789-4cdd-87f7-63952cdef118",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-05/02-complex-inner-product-spaces",
"anchor": "first-1cc72a03538ddc6bda1d-zh",
"chapter": "5",
"section": "5.2",
"offset": 10731,
"range": [
17976,
18929
],
"source_revision": "130978731f44c1f1399a4088c5e4ff3a50730c13937bd888ca64abf512d0946a",
"source_category": "translation",
"review_status": "unreviewed",
"role": "theorem",
"role_basis": {
"kind": "registered-source-environment",
"registered_position": {
"route": "zh/chapter-05/02-complex-inner-product-spaces",
"lang": "zh-CN",
"section": "5.2",
"signal": "named-theorem",
"label": "三角不等式",
"english": "Triangle inequality",
"offset": 9271
}
},
"label": "> [!theorem] 定理 5.6 · 三角不等式 (Triangle inequality)",
"context": {
"range": [
17976,
18929
],
"quote": "\n\n> [!theorem] 定理 5.6 · 三角不等式 (Triangle inequality)\n>\n> （三角不等式）对于内积空间中的任意一对向量 $u$ 和 $v$，\n>\n> \n>\n> $$\n> \\| u + v \\| \\leq \\| u \\| + \\| v \\|.\\tag{5.14}\n> $$\n\n> [!proof]- 证明\n>\n> $$\n> \\begin{aligned}\n> (\\| u + v \\|) ^ {2} &= \\langle u + v | u + v \\rangle \\\\[0.4em]\n> &= \\langle u | u \\rangle + \\langle v | v \\rangle + \\langle v | u \\rangle + \\langle u | v \\rangle \\\\[0.4em]\n> &= \\| u \\| ^ {2} + \\| v \\| ^ {2} + 2 \\mathrm{Re} (\\langle u | v \\rangle) \\\\[0.4em]\n> &\\leq \\| u \\| ^ {2} + \\| v \\| ^ {2} + 2 | \\langle u | v \\rangle | \\\\[0.4em]\n> &\\leq \\| u \\| ^ {2} + \\| v \\| ^ {2} + 2 \\| u \\| \\| v \\| \\quad \\text { by Eq. (5.13) } \\\\[0.4em]\n> &= (\\| u \\| + \\| v \\|) ^ {2}.\n> \\end{aligned}\n> $$\n>\n> 对式 (5.14) 的三角不等式取平方根即得。\n",
"quote_sha256": "2462f761e1d0e943ae438bbf9c529a3111dc465b80b3a09be8d6e815f2d8c8d6",
"environment": "theorem",
"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-24a996b7b801af10d6c8db24"
},
{
"concept_id": "urn:uuid:737defc6-6789-4cdd-87f7-63952cdef118",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "en",
"route": "en/chapter-05/02-complex-inner-product-spaces",
"anchor": "first-1cc72a03538ddc6bda1d-en",
"chapter": "5",
"section": "5.2",
"offset": 11882,
"range": [
18620,
19668
],
"source_revision": "c6710ca54ea0de3d78370164543f39eebd289c4b4fdf4952c69aa17383aadd72",
"source_category": "original_transcription",
"review_status": "unreviewed",
"role": "theorem",
"role_basis": {
"kind": "registered-source-environment",
"registered_position": {
"route": "en/chapter-05/02-complex-inner-product-spaces",
"lang": "en",
"section": "5.2",
"signal": "named-theorem",
"label": "Triangle inequality",
"english": "Triangle inequality",
"offset": 11670
}
},
"label": "> [!theorem] Theorem 5.6 · Triangle inequality",
"context": {
"range": [
18620,
19668
],
"quote": "\n\n> [!theorem] Theorem 5.6 · Triangle inequality\n>\n> (Triangle inequality) For any pair of vectors $u$ and $v$ in an inner product space,\n>\n> \n>\n> $$\n> \\| u + v \\| \\leq \\| u \\| + \\| v \\|.\\tag{5.14}\n> $$\n\n> [!proof]- Proof\n>\n> $$\n> \\begin{aligned}\n> (\\| u + v \\|) ^ {2} &= \\langle u + v | u + v \\rangle \\\\[0.4em]\n> &= \\langle u | u \\rangle + \\langle v | v \\rangle + \\langle v | u \\rangle + \\langle u | v \\rangle \\\\[0.4em]\n> &= \\| u \\| ^ {2} + \\| v \\| ^ {2} + 2 \\mathrm{Re} (\\langle u | v \\rangle) \\\\[0.4em]\n> &\\leq \\| u \\| ^ {2} + \\| v \\| ^ {2} + 2 | \\langle u | v \\rangle | \\\\[0.4em]\n> &\\leq \\| u \\| ^ {2} + \\| v \\| ^ {2} + 2 \\| u \\| \\| v \\| \\quad \\text { by Eq. (5.13) } \\\\[0.4em]\n> &= (\\| u \\| + \\| v \\|) ^ {2}.\n> \\end{aligned}\n> $$\n>\n> The triangle inequality Eq. (5.14) follows on taking square roots.\n",
"quote_sha256": "d40edf64963961178b6e7983f003daee2a4f29e318a5cd97a962819ddd363cc1",
"environment": "theorem",
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "zh/chapter-05/02-complex-inner-product-spaces",
"lang": "zh-CN",
"source_category": "translation",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-211ed4f21cd3d18ec5827b43"
},
{
"concept_id": "urn:uuid:737defc6-6789-4cdd-87f7-63952cdef118",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-10/03-metric-spaces",
"anchor": "first-19bd3976430bde51d2c9-zh",
"chapter": "10",
"section": "10.3",
"offset": 1119,
"range": [
4687,
4897
],
"source_revision": "78dafe40e9e2ba9929ed744a89f06897db2fe96f8434fc563f89b7f142503655",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "三角不等式(triangle inequality)",
"context": {
"range": [
4674,
5022
],
"quote": "条件 (Met4) 称为 三角不等式(triangle inequality)——三角形 $xyz$ 任意一边的长度小于另外两边之和。对于度量空间 $( M , d )$ 中的每个 $x$ 以及正实数 $a > 0$，我们定义开球 $B _ { a } ( x ) = \\{ y \\mid d ( x , y ) < a \\}$",
"quote_sha256": "2fbfd0a62f0a23a311215c16a449d8051cb68dc69f8c358c1a6af42c53143bcc",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "en/chapter-10/03-metric-spaces",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-352a98c26be9ccf85ddb022d"
}
],
"registered_positions": [
{
"route": "en/chapter-05/02-complex-inner-product-spaces",
"lang": "en",
"section": "5.2",
"signal": "named-theorem",
"label": "Triangle inequality",
"english": "Triangle inequality",
"offset": 11670
},
{
"route": "zh/chapter-05/02-complex-inner-product-spaces",
"lang": "zh-CN",
"section": "5.2",
"signal": "named-theorem",
"label": "三角不等式",
"english": "Triangle inequality",
"offset": 9271
},
{
"route": "zh/chapter-10/03-metric-spaces",
"lang": "zh-CN",
"section": "10.3",
"signal": "term-mark",
"label": "三角不等式",
"english": "triangle inequality",
"offset": 737
}
],
"statements": [],
"resolved_references": [
{
"raw": "href=\"#theorem-5-6\"",
"href": "#theorem-5-6",
"range": [
18019,
18038
],
"relation": "references",
"target_route": "zh/chapter-05/02-complex-inner-product-spaces",
"target_anchor": "theorem-5-6",
"resolved": true,
"reason": null,
"source_revision": "130978731f44c1f1399a4088c5e4ff3a50730c13937bd888ca64abf512d0946a",
"source_route": "zh/chapter-05/02-complex-inner-product-spaces",
"occurrence_id": "occ-24a996b7b801af10d6c8db24"
},
{
"raw": "href=\"#equation-5-14\"",
"href": "#equation-5-14",
"range": [
18781,
18802
],
"relation": "references",
"target_route": "zh/chapter-05/02-complex-inner-product-spaces",
"target_anchor": "equation-5-14",
"resolved": true,
"reason": null,
"source_revision": "130978731f44c1f1399a4088c5e4ff3a50730c13937bd888ca64abf512d0946a",
"source_route": "zh/chapter-05/02-complex-inner-product-spaces",
"occurrence_id": "occ-24a996b7b801af10d6c8db24"
},
{
"raw": "href=\"#theorem-5-6\"",
"href": "#theorem-5-6",
"range": [
18663,
18682
],
"relation": "references",
"target_route": "en/chapter-05/02-complex-inner-product-spaces",
"target_anchor": "theorem-5-6",
"resolved": true,
"reason": null,
"source_revision": "c6710ca54ea0de3d78370164543f39eebd289c4b4fdf4952c69aa17383aadd72",
"source_route": "en/chapter-05/02-complex-inner-product-spaces",
"occurrence_id": "occ-211ed4f21cd3d18ec5827b43"
},
{
"raw": "href=\"#equation-5-14\"",
"href": "#equation-5-14",
"range": [
19500,
19521
],
"relation": "references",
"target_route": "en/chapter-05/02-complex-inner-product-spaces",
"target_anchor": "equation-5-14",
"resolved": true,
"reason": null,
"source_revision": "c6710ca54ea0de3d78370164543f39eebd289c4b4fdf4952c69aa17383aadd72",
"source_route": "en/chapter-05/02-complex-inner-product-spaces",
"occurrence_id": "occ-211ed4f21cd3d18ec5827b43"
}
],
"reading_order": [
"occ-24a996b7b801af10d6c8db24",
"occ-211ed4f21cd3d18ec5827b43",
"occ-352a98c26be9ccf85ddb022d"
],
"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": "5626683247a9283ed3e5ef13ca0a3188ee0751e2f858893770cdf96c5917688e",
"optional_enrichments": []
}