The Separation of Two Sets by a Linear Functional

# The Separation of Two Sets by a Linear Functional

 Definition: Let $X$ be a linear space. Two sets $A, B \subset X$ are said to be Separated or Separated by a Hyperplane if there exists a linear functional $f : X \to \mathbb{R}$ and a $c \in \mathbb{R}$ such that $f(a) < c < f(b)$ for all $a \in A$ and for all $b \in B$.

Note that in order to consider whether two sets $A$ and $B$ can be separated we must first assume that $A$ and $B$ are disjoint.

Also note that if $A = \{ a\}$ is a singleton set then $A$ and $B$ are separated if there exists a linear functional $f : X \to \mathbb{R}$ and a $c \in \mathbb{R}$ such that for all $b \in B$

(1)
\begin{align} \quad f(a) < c < f(b) \end{align}

Or equivalently:

(2)
\begin{align} \quad f(a) < \inf_{b \in B} f(b) \end{align}