Operations on subsets of a vector space

Let $X$ be a vector space and let $M,N\subset X$ be nonempty subsets, with $\alpha\in K$.

Define:

1. Set sum $M+N:=\{m+n \mid m\in M,\ n\in N\}.$
2. Scalar multiple of a set $\alpha M := \{\alpha x \mid x\in M\}.$
3. Negation and difference $(-1)M=: -M,\qquad M-N := M+(-N).$

These operations produce subsets of $X$, and one always has $M+N=N+M$.

These constructions are used throughout linear and convex analysis; for instance, sums of subspaces are defined using $M+N$, and Minkowski sums of sets are central in convexity.

Examples:

• In $X=\mathbb{R}$, if $M=[0,1]$ and $N=[2,4]$, then $M+N=[2,5]$.
• In $X=\mathbb{R}^2$, if $M=\{(0,0)\}$ and $N$ is any set, then $M+N=N$.
• If $M$ is a linear subspace, then $-M=M$ and $M-M=M$.