I suggest something different! $(H^1)^* = \text{BMO}$. –  Jonas Teuwen Jan 8 '12 at 18:15
How about giving a rigorous discussion of the Banach-Tarksi paradox and explaining why it doesn't violate conservation of area. Also might discuss why it is impossible to construct a set function on the power set of $\mathbb{R}$ satisfying the four properties: (i) It should be defined for all sets in the power set; (ii) for an interval it gives its length; (iii) it is countable additive; (iv) it is translation invariant. (It isn't known whether there is one satisfying (i)--(iii), but continuum hypothesis implies that there isn't). –  William Jan 8 '12 at 18:48