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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.

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  • $\begingroup$ To the admin: I hope it is not too broad. If there is a need for additional tags such as "softquestion", please let me know (even if I thought it is not actually a soft question, but rather a concrete one). $\endgroup$
    – Kolmin
    Nov 23, 2015 at 16:06
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    $\begingroup$ A particular application: in the study of the Dirichlet problem for the Laplace equation on relatively irregular domains, it can happen that no classical solution exists. One way of handling this is to introduce a particular kind of generalized solution, called the Perron solution. The Perron solution always satisfies the Laplace equation, but it may have the wrong boundary values. $\endgroup$
    – Ian
    Nov 23, 2015 at 16:10
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    $\begingroup$ (Cont.) The boundary functions for which the Perron solution has the right boundary values are called resolutive. By checking the hypotheses of the Stone-Weierstrass theorem, one can show that under certain conditions, all continuous functions are resolutive. A key hypothesis of Stone-Weierstrass is separation of points. $\endgroup$
    – Ian
    Nov 23, 2015 at 16:14

1 Answer 1

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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 [email protected]

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