Annali di Matematica Pura ed Applicata

, Volume 15, Issue 1, pp 197–219 | Cite as

Postulates for linear connections in abstract vector spaces

  • A. D. Michal


Vector Space Linear Connection Abstract Vector Abstract Vector Space 
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  1. (2).
    The recent literature onfinite dimensional differential geometry is extensive. We refer the reader to the following representative books on the subject.L. P. Einsenhart,Riemannian Geometry (1926),Non-Riemannian Geometry (« A. M. S. Coll. Publications », Vol. VIII, 1927),Continuous Groups of Transformations (1933);J. A. Schouten,Der Riccikalkul (1924);O. Veblen,Invariants of Quadratic Differential Forms (1927),Projective Relativitätstheorie (« Ergebnisse Der Mathematik Und Ihrer Grenzgebiete », 1933);T. Y. Thomas,Differential Invariants of Generalized Spaces (1934);Duschek-Mayer,Lehrbuch Der Differentialgeometrie (vol. I and II, 1930);T. Levi-Civita,The Absolute Differential Calculus (1927);E. Cartan,Leçons sur La Géométrie Des Espaces De Riemann (1928),La Théorie Des Groupes Finis Et Continus Et L'Analysis Situs (« Mémorial des Sciences Mathématiques », fasc. 42, 1930);G. Vitali,Geometria Nello Spazio Hilbertiano (1929);W. Blaschke,Differentialgeometrie (vol. II, 1923);H. Weyl,Space-Time-Matter (« English translation », 1921);O. Veblen andJ. H. C. Whitehead,The Foundations of Differential Geometry (1932);D. J. Struik,Theory of Linear Connections (« Ergebnisse Der Mathematik Und Ihrer Grenzgebiete », 1934);V. Hlavaty,Les Courbes De La Variété Générale à nDi mensions (« Mémorial des Sciences Mathématiques », Fasc. 63, 1934).Google Scholar
  2. (3).
    A. D. Michal, « American Journal of Math. », vol. 50 (1928), pp. 473–517. This paper was previously presented to the American Math. Soc. at the New York meeting, October 1927 (Cf. « Bulletin of A. M. S. », vol. 34, Jan. 1928, pp. 8–9 for the abstracts). Following the publication of this initial paper on the subject there appeared numerous papers on functional differential geometry and related topics byF. Conforto, A. Kawaguchi, M. Kerner, A. D. Michal, G. C. Moisil, T. S. Peterson andH. P. Thielman.zbMATHMathSciNetGoogle Scholar
  3. (4).
    A. D. Michal, « Bull. of American Math. Soc. », Vol. 39 (1933), pp. 879–881.zbMATHGoogle Scholar
  4. (1).
    Numerous references to papers on abstract space theory can be found inM. Fréchet,Les Espaces Abstraits (1928), and inS. Banach,Théorie Des Opérations Linéaires (1932).Google Scholar
  5. (2).
    M. H. Stone,Linear Transformations in Hilbert Space (« A. M. S. Colloquium publications », 1932), where references are given toHilbert's, Riesz's and v.Neumann's papers.Google Scholar
  6. (3).
    For the theory of functionals and related subjects seeG. C. Evans,Functionals And Their Applications (« A. M. S. colloquium publications », 1918);V. Volterra,Theory of Functionals (1930);P. Levy,Analyse Fonctionnelle (1922). See alsoFréchet's andBanach's books.Google Scholar
  7. (1).
    A. D. Michal, « Bull. of American Math. Soc. », vol. 39 (1933), abstracts 332 and 333, p. 879.zbMATHGoogle Scholar
  8. (4).
    « Annals of Math. », vol. 34 (1933), pp. 548–552.Google Scholar
  9. (1).
    It is well known that if theFréchet differential of a function exists then theGateaux differential also exists and the two are equal. Hence if a functionF(x, α) is linear in α, one can show readily with the aid of a theorem ofBanach (seeS. Banach, « Fundamenta Mathematicae », vol. 3 (1922), p. 157) that theGateaux differential\(\mathop {\lim }\limits_{\lambda \to 0} \frac{{F(x + \lambda \partial x,{\mathbf{ }}\alpha ) - F(x,{\mathbf{ }}\alpha )}}{\lambda }\) if it exists, is linear in α. Similarly for theFréchet differentialF(x, α; δx).Google Scholar
  10. (1).
    S. Banach, « Studia Mathematica », vol. 1 (1929), p. 238.Google Scholar
  11. (2).
    J. Schauder, « Studia Mathematica », vol. 2 (1930), pp. 1–6.zbMATHMathSciNetGoogle Scholar
  12. (3).
    A. D. Michal andV. Elconin, “ Bull. of American Math. Soc. », vol. 40 (1934), abstracts 228, p. 530 and 386, pp. 814–815. To be published in full elsewhere.Google Scholar
  13. (4).
    M. Fréchet, « Annales Sc. Ec. Normale », vol. 12 (1925). See alsoT. Hildebrandt andL. Graves, « Trans. of A. M. S. », vol. 29 (1927).Google Scholar
  14. (5).
    M. Kerner, « Annals of Math. », loc. cit.Google Scholar
  15. (2).
    A. D. Michal, « Amer. Journal of Math. », loc. cit., and « Proc. of the National Academy of Sciences », 1930–1931 (four papers), loc. cit.Google Scholar
  16. (1).
    A. D. Michal andV. Elconin, loc. cit.Google Scholar
  17. (2).
    L. Graves, « Trans. of Am. Math. Soc. », vol. 29 (1927);M. Kerner, « Prace Matematyczno-Fizyezne », vol. XL (1932).Google Scholar
  18. (1).
    S. Banach, « Fundamenta Math. », loc. cit.Google Scholar
  19. (2).
    A. D. Michal andV. Elconin, loc. cit.Google Scholar
  20. (4).
    M. Kerner, « Annals of Math. », loc. cit.Google Scholar
  21. (1).
    M. Fréchet, « Annales Sc. Ec. Normale Sup. », loc. cit.Google Scholar
  22. (1).
    M. Kerner, « Prace Matematyczno-Fizyczne », loc. cit.Google Scholar
  23. (1).
    A. D. Michal, loc. cit.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1936

Authors and Affiliations

  • A. D. Michal
    • 1
  1. 1.PasadenaU.S.A.

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