Near the bottom of <a href="http://ncatlab.org/nlab/show/Banach+space">the nlab page for Banach space</a> I see "To be described: duals (p+q=pq)".

Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of Banach spaces and linear contractions (with the tensor product described on that page)?