Vector Spaces of Models and Data

Abstract

The mathematical theory of functional spaces plays a critical role in inversion theory. This chapter introduces the concept of a mathematical space and describes the different types of spaces, including multi-dimensional Euclidean, metric, linear vector, Hilbert, and Gramian spaces. The fundamental properties of all these spaces are discussed in detail. The definitions and properties of operators and functionals acting in mathematical spaces are also considered. The chapter concludes with a review of the major principles of variational calculus.