Hilbert Spaces

Our previous discussions have been concerned with algebra. The representation of systems (quantities and their interrelations) by abstract symbols has forced us to distill out the most significant and fundamental properties of these systems. We have been able to carry our exploration much deeper for linear systems, in most cases decomposing the system models into sets of uncoupled scalar equations. Our attention now turns to the geometric notions of length and angle. These concepts, which are fundamental to measurement and comparison of vectors, complete the analogy between general vector spaces and the physical three-dimensional space with which we are familiar. Then our intuition concerning the size and shape of objects provides us with valuable insight. The definition of length gives rigorous meaning to our previous heuristic discussions of an infinite sequence of vectors as a basis for an infinite-dimensional space. Length is also one of the most widely used optimization criteria. We explore this application of the concept of length in Chapter 6. The definition of orthogonality (or angle) allows us to carry even further our discussion of system decomposition. To this point, determination of the coordinates of a vector relative to a particular basis has required solution of a set of simultaneous equations. With orthogonal bases, each coordinate can be obtained independently, a much simpler process conceptually and, in some instances, computationally.