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Currently I'm self studying functional analysis, namely compact operators. In the text, the author gives the following definitions:

Definition 1: A set $K\subseteq X$ is called a compact set if and only if for every sequence $x_n\in K$ there exists an $x\in K$ and a subsequence $x_{n_k}$ of $x_n$ so that $x_{n_k}\to x$.

Definition 2: A set $K\subseteq X$ is called a relatively compact set if and only if every sequence $x_n\in K$ has a Cauchy subsequence $x_{n_k}$.

The author then gives the following remark:

Remark: If $X$ is complete and $K$ is relatively compact, then the closure $\overline{K}$ is compact.

I decided to try to show this myself, but I'm having trouble. Any help would be appreciated.


My attempt, though unfinished, goes as follows:

Proof. Let $x_n\in\overline{K}$. To show that $\overline{K}$ is a compact set we need to show that there exists an $x\in\overline{K}$ and a subsequence $x_{n_k}$ of $x_n$ so that $x_{n_k}\to x$. By defining of the closure operation, $x_n\in \overline{K}$ implies there exists a sequence $y_m\in K$ so that $y_m\to x_n$. Since $K$ is relatively compact, the sequence $y_m$ has a Cauchy subsequence $y_{m_K}$. Since $X$ is complete $y_{m_k}\to x\in\overline{K}$, as $\overline{K}$ is closed.

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The argument is a little more subtle - the sequence $\{y_m\}$ you construct in the question converges to $x_n$ ($n$ fixed), so you can't use that to get at a subsequence of $\{x_n\}$.

For each $n \in \mathbb{N}$, since $x_n \in \overline{K}$, there is a point $y_n \in K$ such that $d(y_n,x_n) < 1/n$. Now $\{y_n\}$ has a Cauchy subsequence $\{y_{n_k}\}$. Now, use the triangle inequality to show that $\{x_{n_k}\}$ is also Cauchy.

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  • $\begingroup$ Triangle inequality (norm notation): $$ ||x_{n_k}-x_{n_j}||=||x_{n_k}-y_n+y_n-x_{n_j}||\leq||y_n-x_{n_k}||+||y_n-x_{n_j}||<2/n\to0. $$ Correct? To me, my argument seems off as I don't see where I made use of $\{y_{n_k}\}$ being a Cauchy subsequence... $\endgroup$
    – D.Math
    Jun 22 '21 at 1:23
  • $\begingroup$ What you need is $\|x_{n_k} - x_{n_j}\| \leq \|x_{n_k} - y_{n_k}\| + \|y_{n_k} - y_{n_j}\| + \|y_{n_j} - x_{n_j}\|$. $\endgroup$ Jun 22 '21 at 1:32
  • $\begingroup$ Ah okay, I see it now. You're up is very appreciated! $\endgroup$
    – D.Math
    Jun 22 '21 at 1:37
  • $\begingroup$ Just to add, so by the completeness of $X$, there exists an $x\in X$ for which $x_{n_k}\to x$. By the closedness of $\overline{K}$, and $\{x_{n_k}\}\in\overline{K}$, it follows that $x\in\overline{K}$, and hence $\overline{K}$ is compact. Correct? $\endgroup$
    – D.Math
    Jun 22 '21 at 1:40

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