Sum of Convex Sets in Vector Space is Convex/Corollary

Theorem

Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a vector space over $\Bbb F$.

Let $A$ and $B$ be convex subsets of $X$.

Let $\lambda, \mu \in \Bbb F$.

Then:

$\lambda A + \mu B$ is convex.

Proof

From Dilation of Convex Set in Vector Space is Convex, we have:

$\lambda A$ and $\mu B$ are convex.

From Sum of Convex Sets in Vector Space is Convex, we have:

$\lambda A + \mu B$ is convex.

$\blacksquare$