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## Homework Statement

Let [itex]C[/itex] be a non-empty convex subset of a real normed space [itex](X,\|\cdot\|)[/itex].

Denote [itex]H(f,a):=\{x\in X: f(x)\leq a\}[/itex] for [itex]f\in X^*[/itex] (dual space) and [itex]a\in\mathbb{R}[/itex].

Show that the closure [itex]\bar{C}[/itex] of [itex]C[/itex] satisfies [itex]\bar{C}=\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)[/itex].

Give an example of a real normed space [itex](X,\|\cdot\|)[/itex] and a non-convex set [itex]C[/itex] 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.