Compact maps problem in Lax

Question

In Functional Analysis of Peter Lax there are the following exercise

Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$.

Assumptions are that ${\bf C}: {\bf X} \rightarrow {\bf X},{\bf M_n}: {\bf X} \rightarrow {\bf X}$ where $\bf X$ is a Banach space

I was thinking that one could use that if let $x_i$ be such that $|({\bf M_i-M})x_i|\ge||{\bf M_i-M}|| - \epsilon $. Then from compactness of $\bf C$ we have that there is a finite sequence $j=1\ldots,n$ of ${\bf C}({\bf M_j-M})x_j$ s.t $\min_j ||{\bf C}({\bf M_j-M})x_j - {\bf C}({\bf M_i-M})x_i||<\epsilon$ for all i. But I dont get anywhere.

Answer 1

I believe that the problem is wrong. Here is an counter example:

Assume that ${\bf X}=l^2$, ${\bf C}x=(x,e_1)e_1$ and ${\bf M}_nx=(x,e_n)e_1$.

${\bf M_n} x\rightarrow 0$ for all $x$, so ${\bf M_n} \rightarrow {\bf 0}$.

But $||{\bf C M}_n - {\bf C 0}|| = ||{\bf C M}_n|| \geq ||{\bf C M}_ne_n||=||(e_n,e_n)(e_1,e_1)e_1||=1.$

Answer 2

As Jonas Wallin points out in his answer, it's not necessarily true that $CM_n-MC\to 0$ uniformly even in the context of Hilbert spaces. It's however true in finite dimensional vector spaces, as strong convergence is the same thing as uniform convergence in this particular case.

However, maybe Lax meant to show that $M_nC-MC\to 0$ uniformly. In this case, we can follow the following steps.

1. We assume that $M=0$, otherwise consider $M_n-M$.
2. Using the principle of uniform boundedness, we obtain that $R:=\sup_{n\in\Bbb N}M_n$ is finite.
3. Let $\delta$ such that $\delta\leqslant\limsup_{n\to +\infty}\lVert M_nC\rVert$. Let $\{n_k\}$ a strictly increasing sequence of integers such that $\delta\leqslant \lVert M_{n_k}C\rVert$, and $x_k$ of norm $1$ such that $\lVert M_{n_k}C\rVert\leqslant k^{-1}+\lVert M_{n_k}Cx_k\rVert$.
4. The sequence $\{Cx_k\}$ lies in a compact set, hence we extract a converging subsequence $\{x_{k'}\}$ to some $y$.
5. We have for all $k'$ that $$\delta\leqslant k'^{-1}+\lVert M_{n_{k'}}(Cx_{k'}-y)+M_{n_{k'}}y\rVert\leqslant k'^{-1}+R\lVert Cx_{k'}-y\rVert+\lVert M_{n_{k'}}y\rVert.$$
6. Taking the limit $k'\to +\infty$ in the later inequality, we get that $\delta=0$.