I read about Hahn-Banach theorem from the book "Introduction to topology and modern analysis" by Simmons. There it was stated like this:
Let $M$ be a linear subspace of a normed linear space $N$, and let $f$ be a functional defined on $M$. Then $f$ can be extended to a functional $f_0$ defined on the whole space $N$ such that $\|f_0\|$=$\|f\|$.
But I also came across the following version of the theorem See theorem 8, page 10
Let $S$ be a convex compact set in a finite dimensional Banach space. Let $ρ$ be a point in the space with $ρ \notin S$.Then there exists a hyperplane that separates $ρ$ from $S$.
I am unable to figure out the connection between the two statements. They seem unrelated to me and yet both of them have been stated as Hahn-Banach theorem. Can anyone please explain whether these two statements are equivalent? If they are then how do I obtain the latter from the former?
Explanations in simple language would be helpful as I am not a mathematics student.