banach-spaces

# Banach Spaces ## When to Use Use this skill when working on banach-spaces problems in functional analysis. ## Decision Tree 1. **Verify Banach space** - Complete normed vector space - Check: every Cauchy sequence converges - `z3_solve.py prove "completeness"` 2. **Hahn-Banach Theorem** - Extend bounded linear functionals - Separate convex sets - `z3_solve.py prove "extension_exists"` 3. **Open Mapping Theorem** - Surjective bounded operator between Banach spaces is open - Consequence: bounded inverse exists - `z3_solve.py prove "open_mapping"` 4. **Closed Graph Theorem** - T: X -> Y has closed graph implies T bounded - Strategy: verify graph closure, conclude boundedness - `z3_solve.py prove "closed_graph_implies_bounded"` 5. **Uniform Boundedness Principle** - Pointwise bounded family of operators is uniformly bounded - Application: prove operator families are bounded ## Tool Commands ### Z3_Completeness ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "cauchy_sequence implies convergent" ``` ### Z3_Open_Mapping ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "T_surjective_bounded implies T_open" ``` ### Z3_Closed_Graph ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "graph_closed implies T_bounded" ``` ### Sympy_Norm ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "norm(alpha*x + beta*y)" ``` ## Key Techniques *From indexed textbook...