Suppose $X$ is a Hausdorff topological vector space, and let $K \subseteq X$ be a non-empty subset that is compact and convex.
Let $\mathcal{F}$ be a commuting family of continuous affine transformations of $K$.
For $T \in \mathcal{F}$ define $T_n=\frac{1}{n}(T^0+T^1+\dots T^{n-1})$ for $n \in \mathbb{N}$
Show that $T_{n}$ is also a continuous transformation.
Attempt:
Since $T$ is continuous, each $T^{i}$ for $i \in (0,...,n-1)$ is continuous as it is the composition of continuous functions.
Furthermore we have that addition and scalar multiplication are continuous operations by definition of a topological vector space.
Does this mean that adding two continuous functions together gives another continuous function? If it does then we are done, but I am not sure!