泛函分析学习笔记1 The Analytic Form of the Hahn–Banach Theorem: Extension of Linear Functionals

Theorem 1.1 (Helly, Hahn–Banach analytic form). Let p : E → R be a function satisfying

　　(1) p(λx)=λp(x) ∀x∈E and ∀λ>0, (2) p(x+y)≤p(x)+p(y) ∀x,y∈E.

　　Let G ⊂ E be a linear subspace and let g : G → R be a linear functional such that (3) g(x) ≤ p(x) ∀x ∈ G.

　　Under these assumptions, there exists a linear functional f defined on all of E that extends g, i.e., g(x) = f (x) ∀x ∈ G, and such that

　　　　(4) f(x) ≤ p(x) ∀x ∈ E.

 

We say that P is inductive if every totally ordered subset Q in P has an upper bound.

 

Lemma 1.1 (Zorn). Every nonempty ordered set that is inductive has a maximal element.