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)
   • sympy_compute.py simplify "x - projection"

2. Projection Theorem

   • For closed convex C, unique nearest point exists
   • P_C is nonexpansive: ||P_C(x) - P_C(y)|| <= ||x - y||
   • z3_solve.py prove "projection_exists_unique"