Further Functional Analysis

模块目标

Prove or apply a named Banach-space theorem with complete hypotheses, and give a counterexample in a merely normed incomplete setting when relevant.

课程大纲

1. 01 Banach spaces and operators 20h

   学习结果 Verify completeness or boundedness on a named example.

   形成性检查 四选一检查

2. 02 Hahn–Banach (taught grain) 25h

   学习结果 Apply an extension or separation form with hypotheses.

   形成性检查 书面推导

3. 03 Uniform boundedness 25h

   学习结果 Deduce a uniform bound from pointwise bounds on a Banach space.

   形成性检查 书面推导

4. 04 Open mapping and closed graph 25h

   学习结果 Apply closed graph and name completeness of domain and codomain.

   形成性检查 书面推导

5. 05 Incomplete counterexamples 20h

   学习结果 Give a setting where a Banach-space conclusion fails without completeness.

   形成性检查 反例

6. 06 Synthesis 35h

   学习结果 Write one operator-theoretic argument that uses a named theorem and a sharpness remark.

   形成性检查 书面推导

学习结果

State a closed-graph / open-mapping / uniform-boundedness application and the completeness hypothesis it uses.

学科技能

Banach spaces, bounded operators, duals and introductory weak-topology language.

可迁移技能

Track completeness; do not quote a Banach theorem on an incomplete space.