In several important and active fields of modern applied
mathematics, such as the numerical solution of PDE-constrained
control problems or various applications in image processing and
data fitting, the evaluation of (integer and real) Sobolev norms
constitutes a crucial ingredient. Different approaches exist for
varying ranges of smoothness indices and with varying properties
concerning exactness, equivalence and the computing time for the
numerical evaluation. These can usually be expressed in terms of
discrete Riesz operators.
We propose a collection of criteria which allow to compare different constructions. Then we develop a unified approach which is valid for non-negative real smoothness indices for standard finite elements, and for positive and negative real smoothness for biorthogonal wavelet bases. This construction delivers a wider range of exactness than the currently known constructions and is computable in linear time.