Convex Set is Contractible

Theorem
Let $V$ be a topological vector space over $\R$ or $\C$.

Let $A\subset V$ be a convex subset.

Then $A$ is contractible.

Proof
Let $x_0\in A$.

Define $H:A\times[0,1]\to A$ by
 * $H(x,t)=tx_0+(1-t)x$.

This yields a homotopy between the identity map $\operatorname{id}_A$ and the constant map $x_0$.