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 - 2. Hahn-Banach Theorem - Extend bounded linear functionals - Separate convex sets - 3. Open Mapping Theorem - Surjective bounded operator between Banach spaces is open - Consequence: bounded inverse exists - 4. Closed Graph Theorem - T: X - Y has closed graph implies T bounded - Strategy: verify graph closure, conclude boundedness - 5. Uniform Boundedness Principle - Pointwise boun…