Skip to main content

An Overview on Tools from Functional Analysis

  • 1025 Accesses

Part of the Geosystems Mathematics book series (GSMA)

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.

Keywords.

  • Functional analysis
  • metric spaces
  • normed spaces
  • function spaces
  • Sobolev spaces
  • reproducing kernel Hilbert spaces
  • basis systems
  • operators
  • convergence
  • weak derivatives
  • distributions
  • partial differential equations

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-57181-2_2
  • Chapter length: 35 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-57181-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)
Hardcover Book
USD   89.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Augustin, M., Eberle, S., Grothaus, M. (2018). An Overview on Tools from Functional Analysis. In: Freeden, W., Nashed, M. (eds) Handbook of Mathematical Geodesy. Geosystems Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-57181-2_2

Download citation