Q:

The product of an arbitrary family of locally convex spaces is locally convex.

Let $\{E_\alpha\ : \ \alpha\in I\}$ be a family of a locally convex sets, where $I$ is an index family. I want to prove that
$$E:= \prod_{\alpha\in I}E_\alpha$$
is locally convex.

I know that, by definition, for each $\alpha\in I$, $E_\alpha$ is locally convex, that is, $E_\alpha$ is topological vector space such that there is a basis of neighborhoods in $E_\alpha$ consisting of convex sets. I also know that I must prove that there is a basis of neighborhoods in $E$ formed by convex sets, but I do not know how to prove it from hypotheses.

A:

Let $C$ be a collection of convex spaces.
Let $a = \prod_{s \in C}a_s$
and $b = \prod_{s \in C}b_s$
be two points in $\prod C$.
Show that the line from $a$ to $b$ is within $\prod C$.

