  We consider fractional Sobolev spaces $H^\theta$, $\theta\in (0,1)$, on 2D
domains and $H^1$-conforming discretizations by globally continuous piecewise
polynomials on a mesh consisting of shape-regular triangles and quadrilaterals.
We prove that the norm obtained from interpolating between the discrete space
equipped with the $L^2$-norm on the one hand and the $H^1$-norm on the other
hand is equivalent to the corresponding continuous interpolation Sobolev norm,
and the norm-equivalence constants are independent of meshsize and polynomial
degree. This characterization of the Sobolev norm is then used to show an
inverse inequality between $H^1$ and $H^\theta$.
