An Overview on Tools from Functional Analysis

Abstract.

Many modern mathematical methods treat geodetic problems in terms of functions from certain spaces, proving convergence properties of such functions and regard the evaluation of such functions or their derivatives at given points as operators. In doing so, knowingly or unknowingly, they use the language of functional analysis.

This contribution aims at summarizing some fundamental concepts from functional analysis which are used throughout this book. In this way, it tries to add a layer of self-sufficiency and to act as supplement to other contributions for those readers who are not familiar with functional analytic tools. For this purpose, we introduce, among others, the general ideas of vector spaces, norms, metrics, inner products, orthogonality, completeness, Banach spaces, Hilbert spaces, functionals, linear operators, different notions of convergence. Then we show how functions can be interpreted as vectors in different kind of function spaces, e.g., spaces of continuous functions, Lebesgue spaces, or Sobolev spaces and how the more general concepts come into play here. Moreover, we have a brief glimpse at differential equations and how functional analytic tools provide the necessary background to discuss them, and at the idea of reproducing kernels and the corresponding reproducing kernel Hilbert spaces.