# How to prove that the closed convex hull of a compact subset of a Banach space is compact?

Can anyone help me with this problem?

Prove that if $K$ is a compact subset of a Banach space $X$, then the closed convex hull of $K$ (that is, the closure of the set of all elements of the form $\lambda_1 x_1+ \dots + \lambda_n x_n$, where $n \geq 1, x_i \in K, \lambda_i \geq 0, \sum_i \lambda_i = 1$) is compact.

Any help appreciated!

#### Solutions Collecting From Web of "How to prove that the closed convex hull of a compact subset of a Banach space is compact?"

Since $X$ is complete it is enough to show that $\mathrm{hull}(K)$ is completely bounded.

The proof of this fact you can find in theorem 3.24 in Rudin’s Functional analysis. This proof follows the same steps proposed by Harald Hanche-Olsen.