Closed Convex Hull in Normed Vector Space is Convex

Definition

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\R$.

Let $U \subseteq X$.

Let $C$ be the closed convex hull of $U$.

Then:

$C$ is convex.

Proof

From the definition of closed convex hull, we have:

$C$ is the closure of the convex hull $\map {\operatorname {conv} } U$ of $U$.

From Convex Hull is Smallest Convex Set containing Set, we have:

$\map {\operatorname {conv} } U$ is convex.

So, from Closure of Convex Subset in Normed Vector Space is Convex, we have:

$C$ is convex.

$\blacksquare$