Normed space complete in norm. Hahn–Banach extends functionals; uniform boundedness (Banach–Steinhaus). Core of classical FA.
graph TD
D1["Def: Normed space\n(X, ‖·‖)"]
D2["Def: Banach space\ncomplete normed"]
D3["Def: Dual X*\nbounded linear functionals"]
D4["Def: Operator norm\n‖T‖ = sup ‖Tx‖/‖x‖"]
T1["Thm: Hahn–Banach\nextend ℓ: Y→ℝ\npreserving ‖ℓ‖"]
T2["Thm: Uniform boundedness\npointwise bounded ⇒\nnorm bounded"]
T3["Thm: Open mapping\nT surjective ⇒ T open"]
T4["Thm: Closed graph\nGr T closed ⇒ T bounded"]
T5["Thm: ℓ(X*,X) separation\nx≠0 ⇒ ∃ℓ ℓ(x)≠0"]
D1 --> D2
D1 --> D3
D3 --> D4
D1 --> T1
D2 --> T2
T1 --> T5
D2 --> T3
D2 --> T4
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: 12
- Definitions: 4
- Theorems: 5
Frontier: Nonstandard Banach spaces, random matrices, concentration. math.FA