Fourier Series on Hilbert Spaces

  • Toru Maruyama
Part of the Monographs in Mathematical Economics book series (MOME, volume 2)


Let e1, e2, …, el be the standard basis of an l-dimensional Euclidean space consisting of l unit vectors. Then any vector x can be expressed as
$$\displaystyle x=\sum _{i=1}^l c_ie_i $$
and such an expression is determined uniquely. The coefficients c1, c2, ⋯, cl are computed as ci = 〈x, ei〉 (inner product).


  1. 1.
    Cartan, H.: Théorie élémentaires des fonctions analytiques d’une ou plusieurs variables complexes. Hermann, Paris (1961) (English edn.) Elementary Theory of Analytic Functions of One or Several Complex Variables. Addison Wesley, Reading (1963) Google Scholar
  2. 2.
    Dudley, R.M.: Real Analysis and Probability. Wadsworth and Brooks, Pacific Grove (1988) Google Scholar
  3. 3.
    Folland, G.B.: Fourier Analysis and its Applications. American Mathematical Society, Providence (1992) Google Scholar
  4. 4.
    Halmos, P.R.: Introduction to a Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York (1951) Google Scholar
  5. 5.
    Kawata, T.: Fourier Kaiseki (Fourier Analysis). Sangyo Tosho, Tokyo (1975) (Originally published in Japanese) Google Scholar
  6. 6.
    Lax, P.D.: Functional Analysis. Wiley, New York (2002) Google Scholar
  7. 7.
    Maruyama, T.: Kansu Kaisekigaku (Functional Analysis). Keio Tsushin, Tokyo (1980) (Originally published in Japanese) Google Scholar
  8. 8.
    Schwartz, L.: Analyse hilbertienne. Hermann, Paris (1979) Google Scholar
  9. 9.
    Takagi, T.: Kaiseki Gairon (Treatise on Analysis), 3rd edn. Iwanami Shoten, Tokyo (1961) (Originally published in Japanese) Google Scholar
  10. 10.
    Terasawa, K.: Shizen Kagakusha no tame no Sugaku Gairon (Treatise on Mathematics for Natural Scientists). Iwanami Shoten, Tokyo (1954) (Originally published in Japanese) Google Scholar
  11. 11.
    Yosida, K.: Lectures on Differential and Integral Equations. Interscience, New York (1960) Google Scholar
  12. 12.
    Yosida, K.: Functional Analysis, 3rd edn. Springer, New York (1971)CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Toru Maruyama
    • 1
  1. 1.Professor EmeritusKeio UniversityTokyoJapan

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