Abstract
For evolution equations of parabolic type in a Hilbert phase space 
, consideration is given to the problem of the effective parametrization (with a Lipschitzian estimate) of the sets 
by functionals 
in 
or, in other words, the problem of the linear Lipschitzian embedding of 
in 
. If 
is the global attractor for the equation, then this kind of parametrization turns out to be equivalent to the finite dimensionality of the dynamics on . Some tests are established for the parametrization (in various metrics) of subsets in and, in particular, of manifolds 
by linear functionals of different classes. We outline a range of physically significant parabolic problems with a fundamental domain 
that admit a parametrization of the elements 
by their values 
at a finite system of points 
.