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On the properties of analytic functions in abstract spaces. (English) Zbl 0018.36504


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[1] See the author’s paper,Analytic Functions in General Analysis, Annali della R. Scuola Normale Superiore di Pisa (in press), also Comptes Rendus 203 (1936), pp. 1228-1230. A theory of analytic functions defined by abstract power series was developed by R. S. Martin in his california Institute of Technology Thesis (1932). These results were first announced in 1931-1932 (see Bulletin of the American Mathematical Society37 abstract371, p. 824, and38 abstract138, p. 349) and were later used by Martin and Michal [see footnote 11)]. Professor Michal’s comments and criticisms concerning Martin’s work led me to establish the theory of analytic functions from a somewhat different approach, based on the work of Gateaux [footnote 3)]. At about the same time (1935) some of the essential features of the theory were announced by L. M. Graves (Bulletin of the American Mathematical Society41 (1935), pp. 641-662). My own work was independent, however.
[2] This was first noted by N. Wiener, Fundamenta Mathematicae4 (1923), pp. 136-143.
[3] This definition is due essentially to R. Gateaux, Bulletin de la Société Mathématique de France50 (1922), p. 1-21, especially p. 8. Its consequences have been fully developed by R. S. Martin in his thesis, California Institute of Technology, 1932. See also Mazur and Orlicz, Studia Mathematica 5 (1935).
[4] M. Fréchet, Annales de l’École Normale Supérieure42 (1925), pp. 293-323.
[5] A domain is connected if to each pairx 0,x 1 of points inD corresponds a continuous imagex(t) of the interval 0?t?1, withx(t) inD,x(0)=x 0,x(1)=x 1.
[6] We say, for brevity, that a regionR is interior toD if it is contained inD and if there is a positive lower bound to the distances between points ofR and the boundary ofd.
[7] This theorem was given by M. Fréchet forp=2, Rendiconti del Circolo Matematico di Palermo30 (1910), p. 18-19. The generalization entails no difficulty, and we shall omit the proof.
[8] M. Kerner, Annals of Mathematics34 (1933), p. 548, Hilfssatz 2.
[9] A. Wintner, American Journal of Mathematics53 (1921), p. 241-257. Wintner’s regular power series are analytic atx=0, for they are continuous and posses partial derivatives. · Zbl 0001.33601 · doi:10.2307/2370779
[10] A. D. Michal and R. S. Martin, Journal de Mathématiques Pures et Appliquées (9)13 (1934), p. 69. See also Bulletin of the American Mathematical Society38, abstract 235, p. 801.
[11] Rendiconti del Circolo Matematico di Palermo27 (1909), p. 70.
[12] It suffices to prove the continuity of the partial derivatives, and this may be accomplished by the usualiterated integral formula of Cauchy (see footnote 2) in § 1).
[13] W. F. Osgood,Lehrbuch der Funktionentheorie, Vol. II, 1 (1929), p. 227-244. F. Hartogs, Mathematische Annalen62 (1905), p. 1-88. Unfortunately the method of Hartogs is not well suited for application to the abstract theory, and no essentially different proof has ever been devised, even in the classical case.
[14] Banach,Opérations Linéaires (1932), p. 19. We recall that a complete space is of the second category.
[15] Sinceh n (x, ?) is a homogeneous polynomial of degreen inx it has a ?polar?, or unique symmetricn-linear functionh n (x 1, ...,x n , ?) which is equal toh n (x, ?) whenx 1= ... =x n =x. This function is continuous with respect to ?, and so, by a theorem of Kerner (Annals of Mathematics34 (1933), p. 548), continuous in the set (x 1, ...,x n , ?);h n (x, ?) is also continuous in (x, ?). For the details about the polar of a polynomial see the reference to Mazur and Orlicz, footnote 3), § 1.
[16] See, for instance,, p. 686.
[17] This is easily deduced from the theorem of bounded convergence for Lebesgue integrals.
[18] Fréchet, Comptes Rendus197 (1933), p. 1182; Minetti, Mémorial des Sciences Mathématiques, fasc. 79 (1936), p. 22-29.
[19] For the definition of a projection see M. H. Stone,Linear Transformations in Hilbert space (1932), p. 70. The relevant details will be found in Walsh,Interpolation and Approximation (1935), p. 141-145.
[20] Minetti, loc. cit. Minetti, Mémorial des Sciences Mathématiques, fasc. 79 (1936), p. 26.
[21] F. Hartogs,, p. 22.
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