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I'm trying to prove this result. Could you verify if my attempt is fine?

In a Banach space, the closed convex hull of a compact set is compact.

I post my proof separately as below answer. If other people post an answer, of course I will happily accept theirs. Otherwise, this allows me to subsequently remove this question from unanswered list.

Analyst
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    Please stop doing this. Why would one prefer your proof to the one in any book on FA? – Kavi Rama Murthy Jun 03 '22 at 09:29
  • @KaviRamaMurthy I just come across this result and would like to give it a try. I guess this is more beneficial (to me) than directly looking up the proof on some books. Sometimes, there are subtle errors that I could not recognize. That's why I post my proof and ask for a verification/comments/suggestions. – Analyst Jun 03 '22 at 09:33
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    If you are asking for a verification of your proof that that proof should be part of the question. – Martin R Jun 03 '22 at 09:34
  • For proof verification you are supposed to post the answer inside the question. If you think you are making a significant contribution and others would be benefitted you can post your own answer. – Kavi Rama Murthy Jun 03 '22 at 09:36
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    @KaviRamaMurthy Of course, including the proof into the question is no problem to me. Usually, people only post comments under the question. This makes my question rest in the unanswered list even though I do get an answer (in a form of a comment). This kind of separation allows me to subsequently remove this question from unanswered list. – Analyst Jun 03 '22 at 09:38

1 Answers1

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Let $X$ is a Banach space and $K$ its compact subset. We want to prove that $\overline{\operatorname{conv} K}$ is compact. It suffices to show that $\overline{\operatorname{conv} K}$ is totally bounded.

Lemma 1: Let $(E, d)$ be a metric space. If $K \subset E$ is totally bounded, then so is its closure $\overline K$.

Proof: Fix $r>0$. There is $x_1, \ldots, x_n \in K$ such that $\{B(x_i, r/2)\}_{i=1}^n$ covers $K$. Clearly, $$ \overline K \subset \overline{\bigcup_{i=1}^n B(x_i, r/2)} \subset \overline{\bigcup_{i=1}^n \overline B(x_i, r/2)}= \bigcup_{i=1}^n \overline B(x_i, r/2) \subset \bigcup_{i=1}^n B(x_i, r). $$ The claim then follows.

By our Lemma 1, it suffices to show $\operatorname{conv} K$ is totally bounded. By Lemma 2 below, it suffices to prove that $K$ is totally bounded. This is trivially true because $K$ is compact.

Lemma 2: Let $(E, d)$ be a metric space. If $K \subset E$ is totally bounded, then so is its convex hull $\operatorname{conv} K$.

Analyst
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