Functional Analysis

Part of the Encyclopedia of Physics / Handbuch der Physik book series (HDBPHYS, volume 1 / 2)


1. The Scope of the Article. The study of functional analysis, that is, that of numerically valued functions on an abstract space, has developed so rapidly and so extensively in the last twenty years that no textbook, however comprehensive, could hope to cover the entire subject at all adequately. The aim of the present article is a very modest one. It is to provide a brief survey of certain parts of the subject which are likely to be of most interest to, or to be used most frequently by, theoretical physicists, or others whose main interest lies in applying the theory. Even then, there are obvious omissions; for example, very little has been said about integral equations or the calculus of variations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


I. The theory of measure and integration

  1. Titchmarsh, E. C.: The Theory of Functions, Chapters X—XII. Oxford 1932.Google Scholar
  2. Saks, S.: Theory of the Integral. Warsaw 1937.Google Scholar
  3. Neumann, J. v.: Functional Operators, Vol.1: Measures and Integrals. Annals of Mathematics Studies, No. 22. Princeton 1950.Google Scholar
  4. Halmos, P. R.: Measure Theory. New York 1950.zbMATHGoogle Scholar
  5. Burkill, J.C.: The Lebesgue Integral. Cambridge 1951.zbMATHCrossRefGoogle Scholar
  6. Rogosinski, W. W.: Volume and Integral. Edinburgh 1951.Google Scholar
  7. Zaanen, A.C.: Linear Analysis, Chapters 1 to 5. Amsterdam 1953.Google Scholar
  8. Loomis, L. L.: An Introduction to Abstract Harmonic Analysis, Chapters I-III. New York 1953.zbMATHGoogle Scholar
  9. Munroe, M. E.: Introduction to Measure and Integration. Cambridge, Mass. 1953.zbMATHGoogle Scholar

II. Banach Space

  1. Frechet, M.: Les Espaces Abstraits. Paris 1928.zbMATHGoogle Scholar
  2. Banach, S.: Theorie des Operations Lineaires. Warsaw 1932.Google Scholar
  3. Hille, E.: Functional Analysis and Semi-Groups. New York 1948.zbMATHGoogle Scholar
  4. Zaanen, A.C.: Linear Analysis, Chapters 6–12. Amsterdam 1953.Google Scholar

III. Integral transforms

  1. Bochner, S.: Vorlesungen über Fouriersche Integrale. Leipzig 1932.Google Scholar
  2. Wiener, N.: The Fourier Integral and Certain of its Applications. Cambridge 1933.Google Scholar
  3. Titchmarsh, E. C.: Introduction to the Theory of Fourier Integrals. Oxford 1937.Google Scholar
  4. Widder, D. V.: The Laplace Transform. Princeton 1941.Google Scholar
  5. Carleman, T.: Integrale de Fourier. Uppsala 1944.zbMATHGoogle Scholar
  6. Bochner, S., and K. Chandrasekharan: Fourier Transforms. Annales of Mathematics Studies, No. 19. Princeton 1949.Google Scholar
  7. Jeffreys, H.: Operational Methods in Mathematical Physics. Cambridge 1931.zbMATHGoogle Scholar
  8. Doetsch, G.: Theorie und Anwendung der laplace-Transformation. Berlin 1937.Google Scholar
  9. Carslaw, H. S., and J. C. Jaeger: Operational Methods in Applied Mathematics. Oxford 1941.Google Scholar
  10. Churchill, R. V.: Modern Operational Mathematics in Engineering. New York 1944.zbMATHGoogle Scholar
  11. Parodi, M.: Applications Physiques de la Transformation de Laplace. Paris 1948.zbMATHGoogle Scholar
  12. Sneddon, I. N.: Fourier Transforms. New York 1951.Google Scholar
  13. Tranter, C. J.: Integral Transforms in Mathematical Physics. London 1951.zbMATHGoogle Scholar

IV. Hilbert Space

  1. Stone, M. H.: Linear Transformations in Hilbert Space. New York 1932.Google Scholar
  2. Murray, F. J.: An Introduction to Linear Transformations in Hilbert Space. Princeton 1941.zbMATHGoogle Scholar
  3. Cooke, R. G.: Infinite Matrices and Sequence Spaces. London 1950.zbMATHGoogle Scholar
  4. Halmos, P. R.: An Introduction to Hilbert Space. New York 1951.Google Scholar
  5. Cooke, R. G.: Linear Operators. London 1953.zbMATHGoogle Scholar

V. Theory of distributions

  1. Schwartz, L.: Theorie des Distributions, 2 tomes. Paris 1950/51.Google Scholar
  2. Halperin, I.: Introduction to the Theory of Distributions. Toronto 1952.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag OHG · Berlin, Göttingen and Heidelberg 1955

Authors and Affiliations

There are no affiliations available

Personalised recommendations