  This paper provides an overview of interpolation of Banach and Hilbert
spaces, with a focus on establishing when equivalence of norms is in fact
equality of norms in the key results of the theory. (In brief, our conclusion
for the Hilbert space case is that, with the right normalisations, all the key
results hold with equality of norms.) In the final section we apply the Hilbert
space results to the Sobolev spaces $H^s(\Omega)$ and
$\widetilde{H}^s(\Omega)$, for $s\in \mathbb{R}$ and an open $\Omega\subset
\mathbb{R}^n$. We exhibit examples in one and two dimensions of sets $\Omega$
for which these scales of Sobolev spaces are not interpolation scales. In the
cases when they are interpolation scales (in particular, if $\Omega$ is
Lipschitz) we exhibit examples that show that, in general, the interpolation
norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio
of these two norms can be arbitrarily large.
