[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"skill:01M6NVC2HDFCT2MB8HBM4X9K9Q":3,"skill-same:01M6NVC2HDFCT2MB8HBM4X9K9Q":59},{"skillNo":4,"name":5,"nameZh":6,"description":7,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":16,"categories":17,"keywords":18,"publishedAt":30,"linkOnly":31,"markdownContent":32,"markdownBody":33,"frontMatter":34,"refName":37,"commitSha":38,"skillDirectory":39,"forksCount":14,"repositoryHtmlUrl":40,"repositoryHomepageUrl":6,"ownerHtmlUrl":41,"ownerBlogUrl":42,"ownerDescription":43,"updatedAt":44,"install":45,"resources":58},"01M6NVC2HDFCT2MB8HBM4X9K9Q","banach-spaces",null,"Problem-solving strategies for banach spaces in functional analysis","rubicanjr\u002FFinCognis","rubicanjr","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F234912519?v=4","01M6NT156NND8J474G34SB3923","Rubi","01M6NT15B70RWXE1N3YV0BZY8A",0,"MIT","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Ffunctional-analysis\u002Fbanach-spaces\u002FSKILL.md",[],[19,20,21,22,23,24,25,26,27,28,29],"spaces","banach spaces","analysis","Bash","read","problem-solving","strategies","problem","functional","for","functional analysis","2026-11-10 10:12:30",false,"---\nname: banach-spaces\ndescription: \"Problem-solving strategies for banach spaces in functional analysis\"\nallowed-tools: [Bash, Read]\n---\n\n# Banach Spaces\n\n## When to Use\n\nUse this skill when working on banach-spaces problems in functional analysis.\n\n## Decision Tree\n\n\n1. **Verify Banach space**\n   - Complete normed vector space\n   - Check: every Cauchy sequence converges\n   - `z3_solve.py prove \"completeness\"`\n\n2. **Hahn-Banach Theorem**\n   - Extend bounded linear functionals\n   - Separate convex sets\n   - `z3_solve.py prove \"extension_exists\"`\n\n3. **Open Mapping Theorem**\n   - Surjective bounded operator between Banach spaces is open\n   - Consequence: bounded inverse exists\n   - `z3_solve.py prove \"open_mapping\"`\n\n4. **Closed Graph Theorem**\n   - T: X -> Y has closed graph implies T bounded\n   - Strategy: verify graph closure, conclude boundedness\n   - `z3_solve.py prove \"closed_graph_implies_bounded\"`\n\n5. **Uniform Boundedness Principle**\n   - Pointwise bounded family of operators is uniformly bounded\n   - Application: prove operator families are bounded\n\n\n## Tool Commands\n\n### Z3_Completeness\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"cauchy_sequence implies convergent\"\n```\n\n### Z3_Open_Mapping\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"T_surjective_bounded implies T_open\"\n```\n\n### Z3_Closed_Graph\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"graph_closed implies T_bounded\"\n```\n\n### Sympy_Norm\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py simplify \"norm(alpha*x + beta*y)\"\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [Introductory Functional Analysis with Applications] If (X, d) is a pseudometric space, we call a set B(xo; r) = {x E X I d(x, xo) \u003C r} an open ball in X with center Xo and radius r. Note that this is analogous to 1. What are open balls of radius 1 in Prob.\n- [Measure, Integration  Real Analysis (... (Z-Library)] Section 5C Lebesgue Integration on Rn 11 Suppose E is a subset of Rm Rn and Rm : (x, y) x E for some y . Dene f : R2 R by = (0, 0), (a) Prove that D1(D2 f ) and D2(D1 f ) exist everywhere on R2. Show that D1(D2 f ) (c) Explain why (b) does not violate 5.\n- [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] The Hahn-Banach Theorem has a rather humble nature. The only mathematical con- cepts needed for its statement are linear spaces and linear, subadditive, and positively homogeneous functionals. Besides Zorn’s Lemma, its proof relies on nothing more than the rudimentary properties of the real numbers.\n- [Introductory Functional Analysis with Applications] If in a normed space X, absolute convergence of any series always implies convergence of that series, show that X is complete. Show that in a Banach space, an absolutely convergent series is convergent. Schauder basis) Show that if a normed space has a Schauder basis, it is separable.\n- [Introductory Functional Analysis with Applications] What are the adjoints of a zero operator 0 and an identity operator I? Annihllator) Let X and Y be normed spaces, T: X - Y a bounded linear operator and -M = (¥t( T), the closure of the range of T. Fundamental Theorems for Normed and Banach Spaces To complete this discussion, we should also list some of the main differences between the adjoint operator T X of T: X ~ Y and the Hilbert-adjoint operator T* of T: Hi ~ H 2 , where X, Yare normed spaces and Hi> H2 are Hilbert spaces.\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n","# Banach Spaces\n\n## When to Use\n\nUse this skill when working on banach-spaces problems in functional analysis.\n\n## Decision Tree\n\n\n1. **Verify Banach space**\n   - Complete normed vector space\n   - Check: every Cauchy sequence converges\n   - `z3_solve.py prove \"completeness\"`\n\n2. **Hahn-Banach Theorem**\n   - Extend bounded linear functionals\n   - Separate convex sets\n   - `z3_solve.py prove \"extension_exists\"`\n\n3. **Open Mapping Theorem**\n   - Surjective bounded operator between Banach spaces is open\n   - Consequence: bounded inverse exists\n   - `z3_solve.py prove \"open_mapping\"`\n\n4. **Closed Graph Theorem**\n   - T: X -> Y has closed graph implies T bounded\n   - Strategy: verify graph closure, conclude boundedness\n   - `z3_solve.py prove \"closed_graph_implies_bounded\"`\n\n5. **Uniform Boundedness Principle**\n   - Pointwise bounded family of operators is uniformly bounded\n   - Application: prove operator families are bounded\n\n\n## Tool Commands\n\n### Z3_Completeness\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"cauchy_sequence implies convergent\"\n```\n\n### Z3_Open_Mapping\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"T_surjective_bounded implies T_open\"\n```\n\n### Z3_Closed_Graph\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"graph_closed implies T_bounded\"\n```\n\n### Sympy_Norm\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py simplify \"norm(alpha*x + beta*y)\"\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [Introductory Functional Analysis with Applications] If (X, d) is a pseudometric space, we call a set B(xo; r) = {x E X I d(x, xo) \u003C r} an open ball in X with center Xo and radius r. Note that this is analogous to 1. What are open balls of radius 1 in Prob.\n- [Measure, Integration  Real Analysis (... (Z-Library)] Section 5C Lebesgue Integration on Rn 11 Suppose E is a subset of Rm Rn and Rm : (x, y) x E for some y . Dene f : R2 R by = (0, 0), (a) Prove that D1(D2 f ) and D2(D1 f ) exist everywhere on R2. Show that D1(D2 f ) (c) Explain why (b) does not violate 5.\n- [Real Analysis (Halsey L. Royden, Patr... (Z-Library)] The Hahn-Banach Theorem has a rather humble nature. The only mathematical con- cepts needed for its statement are linear spaces and linear, subadditive, and positively homogeneous functionals. Besides Zorn’s Lemma, its proof relies on nothing more than the rudimentary properties of the real numbers.\n- [Introductory Functional Analysis with Applications] If in a normed space X, absolute convergence of any series always implies convergence of that series, show that X is complete. Show that in a Banach space, an absolutely convergent series is convergent. Schauder basis) Show that if a normed space has a Schauder basis, it is separable.\n- [Introductory Functional Analysis with Applications] What are the adjoints of a zero operator 0 and an identity operator I? Annihllator) Let X and Y be normed spaces, T: X - Y a bounded linear operator and -M = (¥t( T), the closure of the range of T. Fundamental Theorems for Normed and Banach Spaces To complete this discussion, we should also list some of the main differences between the adjoint operator T X of T: X ~ Y and the Hilbert-adjoint operator T* of T: Hi ~ H 2 , where X, Yare normed spaces and Hi> H2 are Hilbert spaces.\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n",{"name":5,"description":7,"allowed-tools":35},[22,36],"Read","main","70e1c350500ee750987420bbae48837afa906b7e","skills\u002Fmath\u002Ffunctional-analysis\u002Fbanach-spaces","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis","https:\u002F\u002Fgithub.com\u002Frubicanjr","linktr.ee\u002Ffinanskodu","Industrial Engineer & CIO @FinansKodu. Architecting AI-driven financial agents and autonomous workflows. 🏗️ Building systems, not just prompts.","2026-06-09 13:50:15",{"source":8,"skillName":5,"commands":46,"prompt":56,"zipAvailable":57},[47,50,53],{"tool":48,"command":49},"npx","npx skills add rubicanjr\u002FFinCognis --skill banach-spaces",{"tool":51,"command":52},"bunx","bunx skills add rubicanjr\u002FFinCognis --skill banach-spaces",{"tool":54,"command":55},"pnpm","pnpm dlx skills add rubicanjr\u002FFinCognis --skill banach-spaces","安装技能，技能地址：https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Ffunctional-analysis\u002Fbanach-spaces\u002FSKILL.md",true,[],[60,81,90,109,138,157],{"skillNo":61,"name":62,"nameZh":6,"description":63,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":64,"categories":65,"keywords":71,"publishedAt":30},"01M6NWS9DAXJTADT4T5MB49X8W","workflow-router","Goal-based workflow orchestration - routes tasks to specialist agents based on user goals","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fworkflow-router\u002FSKILL.md",[66],{"categoryNo":67,"categoryName":68,"taxonomyType":69,"taxonomyTypeDesc":70},"01JBSEEDCAT000000000000023","Skill与智能体开发","SCENARIO","使用场景",[72,73,74,75,76,77,78,79,80],"workflow router","workflow","router","orchestration","goals","goal","user","workflow orchestration","tasks",{"skillNo":82,"name":83,"nameZh":6,"description":84,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":85,"categories":86,"keywords":87,"publishedAt":30},"01M6NWS3RETSC0SHP16HP66E14","wiring","Wiring Verification","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fwiring\u002FSKILL.md",[],[83,88,89],"verification","false",{"skillNo":91,"name":92,"nameZh":6,"description":93,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":94,"categories":95,"keywords":99,"publishedAt":30},"01M6NWS0XW9CVQZBDA1EGW0TWC","websocket-patterns","Connection management, room patterns, reconnection strategies, message buffering, and binary protocol design.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fwebsocket-patterns\u002Fskill.md",[96],{"categoryNo":97,"categoryName":98,"taxonomyType":69,"taxonomyTypeDesc":70},"01JBSEEDCAT000000000000015","视觉设计",[100,101,102,103,104,105,106,25,107,108],"websocket","patterns","websocket patterns","design","protocol","management","message","binary","connection",{"skillNo":110,"name":111,"nameZh":6,"description":112,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":113,"categories":114,"keywords":119,"publishedAt":30},"01M6NWRDQDX9ZNETAWRRMQWV52","visual-verdict","Screenshot comparison QA for frontend development. Takes a screenshot of the current implementation, scores it across multiple visual dimensions, and returns a structured PASS\u002FREVISE\u002FFAIL verdict with concrete fixes. Use when implementing UI from a design reference or verifying visual correctness.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fvisual-verdict\u002FSKILL.md",[115,118],{"categoryNo":116,"categoryName":117,"taxonomyType":69,"taxonomyTypeDesc":70},"01JBSEEDCAT000000000000009","前端开发",{"categoryNo":67,"categoryName":68,"taxonomyType":69,"taxonomyTypeDesc":70},[120,121,122,123,124,125,126,127,128,28,129,130,131,103,132,133,134,135,136,137],"visual","verdict","visual verdict","ui","across","revise","comparison","correctness","implementing","fixes","dimensions","frontend","development","qa","screenshot","returns","use","frontend development",{"skillNo":139,"name":140,"nameZh":6,"description":141,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":142,"categories":143,"keywords":144,"publishedAt":30},"01M6NWR8HSNRS7MNDK2J02QYMN","verification-loop","Comprehensive verification system covering build, types, lint, tests, security, and diff review before a PR.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fverification-loop\u002FSKILL.md",[],[145,146,88,147,148,149,150,151,152,153,154,155,156],"verification loop","loop","lint","review","security","pr","system","diff","build","tests","diff review","types",{"skillNo":158,"name":159,"nameZh":6,"description":160,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":161,"categories":162,"keywords":163,"publishedAt":30},"01M6NWR4HHSFR0VR5EXFRWA747","vector-db-patterns","Embedding strategies, ANN algorithms, hybrid search, RAG chunking strategies, and reranking for semantic search and retrieval.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fvector-db-patterns\u002Fskill.md",[],[164,165,166,167,101,168,169,170,171,25,172,173,174,175,176,28,177,178],"vector db patterns","vector db","db","vector","semantic search","search","embedding strategies","algorithms","chunking","semantic","retrieval","embedding","Hybrid Search","hybrid","rag"]