Why are separation results important in analysis? I found in a book on real analysis in the part concerning linear functionals (and the Hahn-Banach theorem) that a relevant result – that relies on Hahn-Banach Theorem – is that for every pair of distinct vectors $x,y \in X$, with $X$ vector space, there are enough bounded linear functionals to separate the points of $X$.
Thus, I have the following intertwined questions:  

  
*
  
*Why all these separation results are important (beyond the fact that they are important in themselves)?  
  
*What do we get from them in terms of far-reaching conclusions?
  
*What do we miss when we work with generic spaces without this property?
  

The question goes beyond functional analysis, and it is quite general. I can see that for example separation results are important for optimization, but I am wondering if there is something more (way more) that I do not (cannot) see.
Any feedback is most welcome.
 A: You say you are autodidact in your presentation. This is the reason I give you here a HINT (not an answer) maybe of interest for you. Professor P.J. Laurent of University of Grenoble gave in his very good lessons these four consequences of Hahn-Banach theorem.
Let $E$ be a Banach space,
► If $x_0\in E$ then there is a bounded linear functional f defined on $E$ such that
(1º) $||f||=1$;
(2º) $f(x_0)=||x_0||$
► Let $G$ be a linear variety of $E$ and $y_0\notin G$ with $d=$ Inf$_{x\in G}||y_0 -x||\gt 0$. There is a bounded linear functional $f$ on $E$ such that 
(1º) $f(x)=0$ for $x\in G$;
(2º) $f(y_0)=1$;
(3º)$=||f||=\frac 1d$ 
► Same hypothesis of 2).
There is a bounded linear functional $u$, orthogonal to $G$ such that $||u||=1$ with $d=u(y_0)$. Besides $d=$ Sup $ u(y_0)$ for all $y_0\in E$ where the Supremun is take for the elements $u\in G$ with $||u||=1$ 
► Let $G$ a linear variety of $E$ having as generators {$x_1,x_2,x_3,...$} and $y_0\notin G$. The element $y_0$ can be approximated arbitrarily by elements of $G$ (i.e, $y_0$ is adherent to $G$) if and only if all bounded linear functional $f$ on $E$ that verify $f(x_i)=0$ for all index $i$ verifies also $f(y_0)=0$
You can see here properties of separation though maybe it is not what you are looking for. If you are interested in the proof write me to ortiz.silvio138@gmail.com
