2017-2018-2偏微分方程复习题解析5

最新推荐文章于 2020-07-10 16:52:58 发布

孙瑞宇

最新推荐文章于 2020-07-10 16:52:58 发布

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原文链接：http://www.cnblogs.com/zhangzujin/p/9017729.html

本文探讨了Banach空间中线性映射的有界性问题，证明了线性映射有界与它将有界集映射为有界集这两个条件的等价性。

Problem: Let $X,Y$ be Banach spaces, $T:X\to Y$ be a linear map. $T$ is said to be bounded, if $\exists\ M>0$, such that $\forall\ x\in X,\ \sen{Tx}\leq M\sen{x}$. Show that $T$ is bounded iff (if and only if) for any bounded subset $B\subset X$, $T(B)$ is a bounded subset of $Y$.

Proof: $\ra$: Let $B$ be a bounded subset of $X$. Then there exists a $R>0$, such that $\sen{x}\leq R,\ \forall\ x\in B$. Hence, $$\bex \sen{Tx}\leq M\sen{x}\leq MR,\ \forall\ x\in B. \eex$$ This verifies that $T(B)=\sed{Tx;x\in B}$ is a bounded subset of $Y$.

$\la$: By the assumption, for $B=\sed{x\in X;\sen{x}\leq 1}$, $T(B)=\sed{Tx; x\in B}$ is a bounded subset of $Y$; that is, $$\bex \exists\ R>0,\st \sen{Tx}\leq R,\ \forall\ x\in B. \eex$$ Consequently, $$\bex x\in X\ra\f{x}{\sen{x}}\in B\ra \sen{T\f{x}{\sen{x}}} \leq R\ra \sen{Tx}\leq R\sen{x}. \eex$$

转载于:https://www.cnblogs.com/zhangzujin/p/9017729.html