Notation and Properties of Non-Uniform Spaces

Abstract

Time-varying linear systems can be compactly described by a recently developed notation in which the most important objects under consideration, sequences of vectors and the basic operators on them, are represented by simple symbols. Traditional time-varying system theory requires a clutter of indices to describe the precise interaction between signals and systems. The new notation helps to keep the number of indices in formulas at a minimum. Since in our case sequences of vectors may be of infinite length, we have to put them in a setting that can handle vectors of infinite dimensions. “Energy” also plays an important role, and since energy is measured by quadratic norms, we are naturally led to a Hilbert space setting, namely to Hilbert spaces of sequences of the ℓ₂-type. This should not be too big a step for engineers versed in finite vector space theory since most notions of Euclidean vector space theory carry over to Hilbert spaces. Additional care has to be exercised, however, with convergence of series and with properties of operators. The benefit of the approach is that matrix theory and system theory mesh in a natural way. To achieve that we must introduce a special additional flavor, namely that the dimensions of the entries of the vectors considered are not necessarily all equal.