Functional analysis - task on convexity and dual spaces

[TaPaKaH]

Homework Statement

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.

 

[micromass]

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.

 

[TaPaKaH]

The \subset[\itex] inclusion seems indeed simple as the right-hand-side is a closed set (intersection of closed sets) that contains C.<br /> <br /> I know the (two-set) separation version of HBT but still can&#039;t see how that can be used in this case.

 

[micromass]

Assume that x\notin \overline{C}, but assume that x is a member of the intersection. Try to find a contradiction. Start with applying the definition of the closure. This will give you an open set U disjoint from C. Does this give you ideas?

 

[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&gt;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!