Skip to main content

Introduction to Metric and Normed Spaces

  • Chapter
  • First Online:
Real Analysis
  • 5120 Accesses

Abstract

In this chapter, we extend the notion of distance and absolute value from the real and complex number systems to more general spaces, in particular, spaces of functions.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 69.99
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. Albeverio, S., Fenstad, J.E., Høegh–Krohn, R., Lindstrøm, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics. Academic, Orlando (1986)

    Google Scholar 

  2. Aldaz, J.M.: A general covering lemma for the real line. Real Anal. Exch. 17, 394–398 (1991/92)

    MathSciNet  MATH  Google Scholar 

  3. Anderson, R.M.: A nonstandard representation of Brownian motion and Itô integration. Isr. J. Math. 25, 15–46 (1976)

    Article  MATH  Google Scholar 

  4. Anderson, R.M., Rashid, S.: A nonstandard characterization of weak convergence. Proc. Am. Math. Soc. 69, 327–332 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arkeryd, L.: Loeb solutions of the Boltzmann equation. Arch. Ration. Mech. Anal. 86, 85–97 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bateman, P.T., Erdős, P.: Geometrical extrema suggested by a lemma of Besicovitch. Am. Math. Mon. 58(5), 306–314 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bergelson, V., Tao, T.: Multiple recurrence in quasirandom groups. Geom. Funct. Anal. 24, 1–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Besicovitch, A.S.: A general form of the covering principle and relative differentiation of additive functions (I), (II). Proc. Camb. Philos. Soc. 41, 103–110 (1945); 42, 1–10 (1946)

    Google Scholar 

  9. Bliedtner, J., Loeb, P.A.: A reduction technique for limit theorems in analysis and probability theory. Ark. Mat. 30, 25–43 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bliedtner, J., Loeb, P.A.: The optimal differentiation basis and liftings of L . Trans. Am. Math. Soc. 352, 4693–4710 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bliedner, J., Loeb, P.A.: A local maximal function simplifying measure differentiation. MAA Mon. 114, 532–536 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Constantinescu, C., Cornea, A.: Ideale Ränder Riemannscher Flächen. Springer, Heidelberg (1963)

    Book  MATH  Google Scholar 

  13. Cutland, N.J., Ng, S.-A.: The wiener sphere and wiener measure. Ann. Probab. 21(1), 1–13 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dacunha-Castelle, D., Krivine, J.L.: Application des ultraproducts a l’etude des espaces et des algebres de Banach. Stud. Math. 41, 315–334 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Davis, M.: Applied Nonstandard Analysis. Wiley, New York (1977)

    MATH  Google Scholar 

  16. Doob, J.L.: Stochastic processes depending on a continuous parameter. Trans. Am. Math. Soc. 42, 107–140 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dunford, N., Schwartz, J.: Linear Operators Part I. Interscience, New York (1958)

    MATH  Google Scholar 

  18. Egoroff, D.: Sur les suites des fonctions mesurables. C. R. Hebd. Séances Acad. Sci. 152, 244–246 (1911)

    MATH  Google Scholar 

  19. Füredi, Z., Loeb, P.A.: On the best constant for the Besicovitch covering theorem. Proc. Am. Math. Soc. 121(4), 1063–1073 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Henson, C.W., Moore, L.C.: Nonstandard analysis and the theory of Banach spaces. In: Hurd, A.E. (ed.) Nonstandard Analysis - Recent Developments, pp. 27–112. Springer, Berlin (1983)

    Chapter  Google Scholar 

  21. Insall, M., Loeb, P.A., Marciniak, M.: End compactifications and general compactifications. J. Log. Anal. 6(7), 1–16 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Keisler, H.J.: Elementary Calculus, An Infinitesimal Approach, 1st edn. Prindle, Weber & Smith, Boston (1976); 2nd edn. Prindle, Weber & Smith, Boston (1986)

    Google Scholar 

  23. Keisler, H.J.: An infinitesimal approach to stochastic analysis. Mem. Am. Math. Soc. 48(297) (1984)

    Google Scholar 

  24. Kelley, J.L.: The Tychonoff product theorem implies the axiom of choice. Fundam. Math. 37, 75–76 (1950)

    MathSciNet  MATH  Google Scholar 

  25. Littlewood, J.E.: Lectures on the Theory of Functions. Oxford University Press, Oxford (1944)

    MATH  Google Scholar 

  26. Loeb, P.A.: A new proof of the Tychonoff Theorem. Am. Math. Mon. 72(7), 711–717 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. Loeb, P.A.: Compactifications of Hausdorff spaces. Proc. Am. Math. Soc. 22, 627–634 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  28. Loeb, P.A.: Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans. Am. Math. Soc. 211, 113–122 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. Loeb, P.A.: Applications of nonstandard analysis to ideal boundaries in potential theory. Isr. J. Math. 25, 154–187 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  30. Loeb, P.A.: Weak limits of measures and the standard part map. Proc. Am. Math. Soc. 77(1), 128–135 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  31. Loeb, P.A.: A construction of representing measures for elliptic and parabolic differential equations. Math. Ann. 260, 51–56 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Loeb, P.A.: A functional approach to nonstandard measure theory. In: Proceedings of the S. Kakutani Retirement Conference. Contemp. Math. 26, 251–261 (1984)

    MathSciNet  MATH  Google Scholar 

  33. Loeb, P.A., Talvila, E.: Covering lemmas and Lebesgue integration. Sci. Math. Jpn. 53, 209–221 (2001)

    MathSciNet  MATH  Google Scholar 

  34. Loeb, P.A., Talvila, E.: Lusin’s theorem and Bochner integration. Sci. Math. Jpn. 60, 113–120 (2004)

    MathSciNet  MATH  Google Scholar 

  35. Loeb, P.A., Wolff, M. (eds.): Nonstandard Analysis for the Working Mathematician, 2nd edn. Springer, Heidelberg (2015)

    MATH  Google Scholar 

  36. Lusin, N.: Sur les propriétés des fonctions mesurables. C. R. Acad. Sci. Paris 154, 1688–1690 (1912)

    MATH  Google Scholar 

  37. Luxemburg, W.A.J.: A general theory of monads. In: Luxemburg, W.A.J. (ed.) Applications of Model Theory to Algebra, Analysis, and Probability. Holt, Rinehart, and Winston, New York (1969)

    Google Scholar 

  38. Morse, A.P.: Perfect blankets. Trans. Am. Math. Soc. 61, 418–442 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nelson, E.: Internal set theory: a new approach to nonstandard analysis. Bull. Am. Math. Soc. 83, 1165–1198 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  40. Perkins, E.A.: A global intrinsic characterization of Brownian local time. Ann. Probab. 9, 800–817 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  41. Radó, T.: Sur un problème relatif à un théorème de Vitali”. Fundam. Math. 11, 228–229 (1928)

    MATH  Google Scholar 

  42. Raman-Sundström, M.: A pedagogical history of compactness. MAA Mon. 122, 619–635 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Reifenberg, E.F.: A problem on circles. Math. Gaz. 32, 290–292 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  44. Robinson, A.: Non-standard Analysis. North-Holland, Amsterdam (1966)

    MATH  Google Scholar 

  45. Royden, H.L.: Real Analysis, 3rd edn. Prentice Hall, New Jersey (1987)

    MATH  Google Scholar 

  46. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  47. Solovay, R.M.: A model of set-theory in which every set of reals is Lebesgue measurable. Ann. Math. 92(1), 1–56 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stroyan, K.D., Bayod, J.M.: Foundations of Infinitesimal Stochastic Analysis. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  49. Stroyan, K.D., Luxemburg, W.A.J.: Introduction to the Theory of Infinitesimals. Academic, New York (1976)

    MATH  Google Scholar 

  50. Sullivan, J.M.: An explicit bound for the Besicovitch covering theorem. Geom. Anal. 4(2), 219–231 (1993)

    Article  MATH  Google Scholar 

  51. Sun, Y.N.: A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN. J. Math. Econ. 29, 419–503 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tao, T.: Hilbert’s Fifth Problem and Related Topics. Graduate Studies in Mathematics, vol. 153. American Mathematical Society, Providence (2014)

    Google Scholar 

  53. Vakil, N.: Real Analysis Through Modern Infinitesimals. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Loeb, P.A. (2016). Introduction to Metric and Normed Spaces. In: Real Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30744-2_7

Download citation

Publish with us

Policies and ethics