  We prove that for every Banach space $Y$, the Besov spaces of functions from
the $n$-dimensional Euclidean space to $Y$ agree with suitable local
approximation spaces with equivalent norms. In addition, we prove that the
Sobolev spaces of type $q$ are continuously embedded in the Besov spaces of the
same type if and only if $Y$ has martingale cotype $q$. We interpret this as an
extension of earlier results of Xu (1998), and Mart\'inez, Torrea and Xu
(2006). These two results combined give the characterization that $Y$ admits an
equivalent norm with modulus of convexity of power type $q$ if and only if
weakly differentiable functions have good local approximations with
polynomials.
