# 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!

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The best proof may depend a bit on what you know. One good idea might be to show that the convex hull (i.e., without the closure) is totally bounded. For that, you should be able to limit your attention to convex combinations using only a fixed number of members of $K$. – Harald Hanche-Olsen Oct 16 '12 at 9:30

## 1 Answer

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.

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