Skip to main content

The Krein–Milman Theorem and Hansen’s Variant of the Hansen–Pedersen Proof

  • Chapter
  • First Online:
Loewner's Theorem on Monotone Matrix Functions

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 354))

  • 1325 Accesses

Abstract

In this chapter, we will present a proof of Loewner’s theorem due to Hansen–Pedersen that relies on the Krein–Milman theorem; we follow a variant of Hansen [133].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. H. Armitage, The Riesz-Herglotz representation for positive harmonic functions via Choquet’s theorem, Potential Theory—ICPT 94 (Kouty, 1994), pp. 229–232, de Gruyter, Berlin, 1996.

    Google Scholar 

  2. R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, 169, Springer-Verlag, New York, 1997.

    Google Scholar 

  3. L. Boutet de Monvel, unpublished note, 1989; included as an appendix to this book.

    Google Scholar 

  4. G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1955), 131–295.

    Google Scholar 

  5. G. Choquet, Lectures on Analysis, Volume I: Integration and Topological Vector Spaces; Volume II: Representation Theory; Volume III: Infinite Dimensional Measures and Problem Solutions, W.A. Benjamin, New York-Amsterdam, 1969.

    Google Scholar 

  6. F. Hansen, The fast track to Löwner’s theorem, Linear Algebra Appl. 438 (2013), 4557–4571.

    Google Scholar 

  7. F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and Löwner’s theorem, Math. Ann. 258 (1982), 229–241.

    Google Scholar 

  8. F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer, 2014.

    Google Scholar 

  9. F. Holland, The extreme points of a class of functions with positive real part, Math. Ann. 202 (1973), 85–87.

    Google Scholar 

  10. J. L. Kelley, Note on a theorem of Krein and Milman, J. Osaka Inst. Sci. Tech. Part I, 3 (1951), 1–2.

    Google Scholar 

  11. M. Krein and D. Milman, On extreme points of regularly convex sets, Studia Math. 9 (1940), 133–138.

    Google Scholar 

  12. J. Lindenstrauss, G. H. Olsen, and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier Grenoble 28 (1978), 91–114.

    Google Scholar 

  13. R. R. Phelps, Integral representations for elements of convex sets, Studies in Functional Analysis, pp. 115–157, MAA Stud. Math., 21, Math. Assoc. America, Washington, DC, 1980.

    Google Scholar 

  14. E. T. Poulsen, A simplex with dense extreme boundary, Ann. Inst. Fourier Grenoble 11 (1961), 83–87.

    Google Scholar 

  15. B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. in Math. 137 (1998), 82–203.

    Google Scholar 

  16. B. Simon, Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics, 187, Cambridge University Press, Cambridge, 2011.

    Google Scholar 

  17. B. Simon A Comprehensive Course in Analysis, Part 1: Real Analysis, American Mathematical Society, Providence, RI, 2015.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Simon, B. (2019). The Krein–Milman Theorem and Hansen’s Variant of the Hansen–Pedersen Proof. In: Loewner's Theorem on Monotone Matrix Functions. Grundlehren der mathematischen Wissenschaften, vol 354. Springer, Cham. https://doi.org/10.1007/978-3-030-22422-6_28

Download citation

Publish with us

Policies and ethics