Theorem 12.3.1 (Mazur).label Let $E$ be a locally convex space over $K \in \RC$ and $A \subset E$ be compact, then
- (1)
$\conv(A)$ is totally bounded in $E$.
- (2)
If $E$ is complete, then $\conv(A)$ is relatively compact.
Proof. (1): Let $U \in \cn_{E}(0)$ be convex and circled, then since $A$ is compact, there exists $B \subset A$ finite such that $A \subset B + U$. As such, $\conv(A) \subset \conv(B + U) = \conv(B) + U$. Since $B$ is finite, $\conv(B)$ is compact. Thus there exists $C \subset \conv(B)$ finite such that
\[\conv(A) \subset \conv(B + U) = \conv(B) + U \subset C + U\]
which yields a finite covering of $\conv(A)$ using $U$.
(2): By Proposition 6.4.3.$\square$
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