Abstract
In this paper, we will prove the following : Let E be a real separable Banach space. Then every probability measure on E has a Hilbertian support if and only if E is isomorphic to a Hilbert space. In the case of l p (1 ≤ p < 2) we will give an explicit construction of probability measures without Hilbertian support.
Keywords
- Hilbert Space
- Banach Space
- Probability Measure
- Bilinear Form
- Gaussian Measure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
Baxendale, P. (1976) Gaussian measures on function spaces. Amer. J. Math. Vol. 98, pp891–952.
Chobanjan, S.A. and Trieladze, V.I. (1977) Gaussian characterization of certain Banach spaces. J. Mult. Anal. Vol. 7, pp183–203.
Diallo, B. (1977) Correction and supplement to the article "On the Hilbert subspace of full measure of the space l p.¹ Zapiski Nauchnix Seminaroff LOMI. Vol. 72, pp213–214. (in Russian)
Dudley, R.M. (1967) The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Func. Anal. Vol. 1, pp290–330.
Mandrekar, V. (1978) Characterization of Banach space through validity of Bochner theorem. Vector space measures and applications I, Lecture Note in Math. Vol. 644, Springer pp314–326.
Maurey, B. (1972) Espaces de cotype p, 0 < p <-2. Seminaire Maurey-Schwartz 1972–1973, Exposé 7.
Mouchtari, D. (1976) Sur l’éxistence d’une topologie du type de Sazonov sur un espace de Banach. Seminaire Maurey-Schwartz 1975–1976, Exposé 17.
Parthasarathy, K.R. (1967) Probability measures on metric spaces. Academic Press, N.Y.
Sato, H. (1969) Gaussian measure on a Banach space and abstract Wiener measure. Nagoya Math. J. Vol. 36, pp65–81.
Sazonov, V. (1958) A remark on characteristic functionals. Th. of prob. its Appl. Vol. 3, pp188–192 in English translation.
Smolyanov, O.G. and Uglanov, A.V. (1973) Every Hilbert subspace of a Wiener space has measure zero. Matematicheskie Zametki. Vol. 14, pp772–774 in English translation.
Vakhania, N. (1965) Sur les répartitions de probabilités dans les espaces de suite numeriques. C.R.Acad.Sc.Paris. Vol. 260, pp1560–1562.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1979 Springer-Verlag
About this paper
Cite this paper
Sato, H. (1979). Hilbertian support of a probability measure on a banach space. In: Beck, A. (eds) Probability in Banach Spaces II. Lecture Notes in Mathematics, vol 709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071958
Download citation
DOI: https://doi.org/10.1007/BFb0071958
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09242-1
Online ISBN: 978-3-540-35341-6
eBook Packages: Springer Book Archive
