Bounded linear operators form a natural class of maps between normed linear spaces: they are, by definition, linear and continuous, so that, in other words, they preserve, at least to some extent, the linear and the topological structures of the normed spaces involved. As it turns out, bounded linear operators from one normed linear space to another form a normed linear space themselves. Moreover, if the range space is complete, the space of operators is complete also, and is thus a Banach space. We note that we have already encountered examples of bounded linear operators on the previous pages of this book and discuss a score of new ones. Nor do we refrain from calculating norms of some of them.

We consider linear maps between normed spaces. We define what it means for a linear map to be bounded and show that this is equivalent to continuity. We define the norm of a linear operator and show that the space of all linear maps from X to Y is a vector space, which is complete if Y is complete. We give a number of examples and then discuss inverses and invertibility in some detail.

This chapter bounds the condition numbers of thestiffness matrix of operator-adapted wavelets within each subband (scale). These resulting bounds are characterized through weak alignment conditions between measurement functions and eigensubspaces of the underlying operator. In Sobolev spaces, these alignment conditions translate into approximate error estimates associated with variational splines andscattered data approximation. These estimates are established for the three primary examples, subsampled Diracs, Haar prewavelets, and local polynomials,of hierarchies of measurement functions in Sobolev spaces.