# Functional analysis - task on convexity and dual spaces

1. Dec 5, 2012

### TaPaKaH

1. The problem statement, all variables and given/known data
Let $C$ be a non-empty convex subset of a real normed space $(X,\|\cdot\|)$.
Denote $H(f,a):=\{x\in X: f(x)\leq a\}$ for $f\in X^*$ (dual space) and $a\in\mathbb{R}$.
Show that the closure $\bar{C}$ of $C$ satisfies $\bar{C}=\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)$.

Give an example of a real normed space $(X,\|\cdot\|)$ and a non-convex set $C$ for which the equality above does not hold.

2. Relevant information
This task comes in a homeworkset which relates to the application of the Hahn-Banach (extension) theorem, but I just can't see how one could apply it to the exercise above.

2. Dec 5, 2012

### micromass

Staff Emeritus
One inclusion is trivial (= doesn't require Hahn-Banach). Do you see which one that is?

Also, do you know the geometric Hahn-Banach theorem?? This is an analogous formulation of Hahn-Banach that does not deal with extensions of functionals, but rather with separation of subsets with hyperplanes. This version might be handy here.

3. Dec 5, 2012

5. Dec 5, 2012

### TaPaKaH

Yes, your hint makes perfect sense now.

If $x\in\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)$ but $x\notin\bar{C}$ then $d(x,\bar{C})=2\delta>0$ so we can separate $C$ and $U_\delta(x)$ with a hyperplane $\{f=a\}$ (for some f and a) and get a contradiction with $x\in\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)$.

I got the idea I was looking for, thank you very much!

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