泛函分析中一些基本概念及定理的证明技巧.

$L^{\infty}[a,b]$:定义在可测集E上的可测函数是本性有界的,是指除去E中的某个零测度集外,在它的补集上是有界的.$L^{\infty}[a,b]$表示在$[a,b]$上本性有界可测函数的全体:

\[\rho(x,y) = \inf_{\substack{mE_0=0\\E_0\subset[a,b]}}\{\sup_{t \in [a,b]-E_0}{|x(t)-y(t)|}\}.\]

$\inf$是对$E_0$取的,$\sup$是对$t$取的.

在证明$\rho(x,y)\le\rho(x,z)+\rho(z,y)$时,(1)运用了$\inf$的定义及性质;(2)用了一点小技巧:在一个更大的空间中取最大值时,最大值不减.即$A \subset B$,$\sup{A} \le \sup{B}$.

在这里$\inf$的定义:$x,y,z \in L^{\infty}[a,b]$,$\forall \epsilon$,$\exists E_0,E_1 \subset[a,b]$,$mE_0=mE_1=0$,使

\[\begin{aligned}\sup_{t \in [a,b]-E_0}{|x(t)-z(t)|} \le \rho(x,z) + \frac{\epsilon}{2} \\\sup_{t \in [a,b]-E_1}{|z(t)-y(t)|} \le \rho(z,y) + \frac{\epsilon}{2}\end{aligned}\]

\[\begin{aligned}\rho(x,y) &\le \sup_{t \in [a,b]-(E_0 \cup E_1)}{|x(t)-y(t)|}\\&\le \sup_{t \in [a,b]-(E_0 \cup E_1)}{|x(t)-z(t)|} + \sup_{t \in [a,b]-(E_0 \cup E_1)}{|z(t)-y(t)|} \\&\le \sup_{t \in [a,b]-E_0}{|x(t)-z(t)|} + \sup_{t \in [a,b]-E_1}{|z(t)-y(t)|} \\&\le \rho(x,z)+\rho(z,y) + \epsilon.\end{aligned}\]

由此得到结论.