关于完备随机内积模中的Lax-Milgram定理的注记

A Note on the Lax-Milgram Theorem in Complete Random Inner Product Modules

设(S,X)为数域K上以σ-有限测度空间(Ω,A,μ)为基的完备的RIP-模,而且α:S×S→L(μ,K)满足如下条件:(A)存在ξ∈L+(μ),使得a(p,q)ξ·X^p·X^q,p,q∈S;(B)a是coercive(即,存在η∈L+(μ),使得a(p,p)η·X^p2,p∈S且μ({ωη(ω)=0})=0);(C)对每个q∈S,a(·,q):S→L(μ,K)是模同态,且对每个p∈S,a(p,ξq1+ηq2)=ξ-a(p,q1)+η-a(p,q2),q1,q2∈S及ξ,η∈L(μ,K).则存在唯一的连续模同态A:S→S使A-1存在且μ-a.s.有界,还满足:(1)a(p,q)=XA(p),q,p,q∈S;(2)X^A-1(p)1ηX^p,p∈S.

Let （S,X） be a complete random inner product module over the scalar field K with base a σ-finite measure space （Ω,A,μ）, and α：S×S→L（μ,K） satisfy the following：（A） there exists ξ∈L^＋（μ such that ｜α（p,q）≤ξ·Xp^-·Xq^- for ang p,q∈S; （B） a is coercive （namely, there exists η∈L^＋（μ） such that ｜α（p,p）｜≥η·Xp^-z p∈S and μ（（ω）｜η（ω）=0））=0）;（C）α（·,q）：S→L（μ,K）is module homomophism for any q∈S,and for every p∈S the following also holds：α（p,ξq1＋ηq2）=ξa（p,q1）＋η^-α（p,q2）,for and q1,q2∈ S, for ang ξ,η∈L（μ,K）.Then there exists s unique continuous module homomorphism A：S→S such that A^-1 exists and is μ-μ.s.bounded,further the following also hold; （1）α（p,q）=XA（p）,q,for ang p,q,∈S;（2）XA^--1（p）≤1/ηxp^-,for ang p∈S.