Hilbert Spaces When to Use Use this skill when working on hilbert-spaces problems in functional analysis. Decision Tree 1. Orthogonal decomposition - For closed subspace M: H = M + M^perp (direct sum) - Every x = P M(x) + P {M^perp}(x) - 2. Projection Theorem - For closed convex C, unique nearest point exists - P C is nonexpansive: ||P C(x) - P C(y)|| <= ||x - y|| - 3. Riesz Representation - Every f in H has form f(x) = <x, y f for unique y f - ||f|| = ||y f|| - 4. Parseval's Identity - For orthonormal basis {e n}: ||x||^2 = sum|<x, e n |^2 - 5. Bessel's Inequality - sum|<x, e n |^2 <= ||x||^…