Complete inner product space. Riesz: H* ≅ H. Orthogonal projection, best approximation. Spectral theorem for self-adjoint operators.
graph TD
D1["Def: Inner product\n⟨·,·⟩ sesquilinear"]
D2["Def: Hilbert space\ncomplete inner product"]
D3["Def: Orthogonal M⊥"]
D4["Def: Self-adjoint\nT* = T"]
T1["Thm: Cauchy–Schwarz\n|⟨x,y⟩| ≤ ‖x‖‖y‖"]
T2["Thm: Riesz representation\nℓ ∈ H* ⇒ ℓ = ⟨·,y⟩"]
T3["Thm: Orthogonal projection\nH = M ⊕ M⊥"]
T4["Thm: Spectral thm\nT self-adjoint compact\n⇒ T = Σ λⱼ Pⱼ"]
T5["Thm: Bessel, Parseval"]
D1 --> D2
D1 --> T1
D2 --> T2
D2 --> T3
D3 --> T3
D4 --> T4
T1 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 14
- Edges: 11
- Definitions: 4
- Theorems: 5
Frontier: Free probability, operator algebras, quantum information. math.FA