An Application of Gaussian Measures to Functional Analysis

  • Daniel W. StroockEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 34)


In a variety of settings, it is shown that all Borel measurable, linear maps from one locally convex topological vector space to another must be continuous. When the image space is Polish, this gives a proof of L. Schwartz’s Borel graph theorem. The proof is based on a simple probabilistic argument and, except for the application to Schwartz’s theorem, avoids the descriptive set theory used in previous treatments of such results.

Received 6/14/2011; Accepted 11/22/2011; Final 11/22/2011


  1. 1.
    Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes, Ecole d’Eté de Probabilités de Saint-Flour IV-1974. In: Hennequin, P.L. (ed.) Lecture Notes in Mathematics, vol. 480, pp. 1–97. Springer, Berlin (1975)Google Scholar
  2. 2.
    Kuratowski, K.: Topologie I. Academic, New York (1966)Google Scholar
  3. 3.
    Martineau, A.: Sur des théorèmes de S. Banach et L. Schwartz concernant le graphe fermé. Studia Mathematica XXX, 43–51 (1968)Google Scholar
  4. 4.
    Parthasarathy, K.P.: Probability Measures on Metric Spaces, vol. 276. AMS Chelsea Series, Providence (1967)zbMATHGoogle Scholar
  5. 5.
    Schwartz, L.: Sur le théorème du graphe. C. R. Acad. Sci. Paris 263 série A, 602–605 (1966)Google Scholar
  6. 6.
    Stroock, D.: On a Theorem of L. Schwartz. C. R. Acad. Sci. Paris Ser. I 349(1–2), 5–6 (2010)MathSciNetGoogle Scholar
  7. 7.
    Stroock, D.: Probability Theory, an Analytic View, 2nd edn. Cambridge University Press, Cambridge (2011)zbMATHGoogle Scholar
  8. 8.
    Stroock, D.: Essentials of integration theory for analysis. In: GTM series. Springer Heidelberg (2011)zbMATHCrossRefGoogle Scholar
  9. 9.
    Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, New York (1967)zbMATHGoogle Scholar
  10. 10.
    Yoshida, K.: Functional Analysis. In: Grundlehren Series # 123. Springer, Berlin (1965)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.M.I.T, 2-272CambridgeUSA

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