  A multiresolution analysis for a Hilbert space realizes the Hilbert space as
the direct limit of an increasing sequence of closed subspaces. In a previous
paper, we showed how, conversely, direct limits could be used to construct
Hilbert spaces which have multiresolution analyses with desired properties. In
this paper, we use direct limits, and in particular the universal property
which characterizes them, to construct wavelet bases in a variety of concrete
Hilbert spaces of functions. Our results apply to the classical situation
involving dilation matrices on $L^2(\R^n)$, the wavelets on fractals studied by
Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.
