Further Functional Analysis

Module aims

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

Course outline

1. 01 Banach spaces and operators 20h

   Learning result Verify completeness or boundedness on a named example.

   Formative check Four-option check

2. 02 Hahn–Banach (taught grain) 25h

   Learning result Apply an extension or separation form with hypotheses.

   Formative check Written derivation

3. 03 Uniform boundedness 25h

   Learning result Deduce a uniform bound from pointwise bounds on a Banach space.

   Formative check Written derivation

4. 04 Open mapping and closed graph 25h

   Learning result Apply closed graph and name completeness of domain and codomain.

   Formative check Written derivation

5. 05 Incomplete counterexamples 20h

   Learning result Give a setting where a Banach-space conclusion fails without completeness.

   Formative check Counterexample

6. 06 Synthesis 35h

   Learning result Write one operator-theoretic argument that uses a named theorem and a sharpness remark.

   Formative check Written derivation

Learning results

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

Subject skills

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

Transferable skills

Track completeness; do not quote a Banach theorem on an incomplete space.