On Tight Upper Bounds on the Supremum Norm using Norms induced by Inner ProductsI,II

Both supremum norms and 2-norms have found a huge number of applications as fitting and approximation criteria for instance in system identification, ro-bust control, model order reduction, mobile communications, machine learning and statistics. By definition a general 2-norm is induced by an inner product and thus linked to rich theory and computationally efficient algorithms. The supremum norm is often a natural fitting and approximation criterion due to its interpretation as an operator norm. Yet this norm is also notorious in leading to complicated optimization problems. In general it is well known that for finitely generated linear spaces all norms induce the same topology, i.e., are equivalent in the sense that for any two disctinct norms there exists a constant such that the length of any vector in one of the norms cannot exceed its length in the other norm up to multiplication by the constant. This article shows how to compute this bounding constant, as small as possible, in order to bound the supremum norm from above by a general 2-norm. Special attention is paid to linear spaces consisting of transfer functions of linear discrete- or continuous time systems indicating potential applications in the fields of control and system identification