Vol. 8, 1922 
MATHEMATICS: C. N. MOORE 
293 
We can now prove the equivalence of these two generalized limits by a 
method similar to that devised by Schur^ for proving the special theorem 
in the case of series. The proof breaks up into three lemmas and a theorem. 
Lemma 1. If we represent by E the identical functional operation, E6 = 
6(6), we have the identity 
(10) (C^M + l E) {C„n))M = (M(C„_i^)) (<r). 
LBmma 2. H lim (p{a) = o ■ | <p(a) | <0i (cr) • D • 
aiim [<P,nM]-K J<Pn){<r) = -, 
a n 
where 
(PQn (o") being defined by (2). 
Lkmma 3. If we set 
H lim w ((t) = a - D - H lim = a. 
Theorem. H lim (C„r7)(<r) = a • ^- H lim {Hni/]){a) = a. 
cr <r 
1 E. H. Moore, Introduction to a Form of General Analysis, The New Haven Mathe- 
matical Colloquium, Yale Univ. Press, 1910, p. 3; "On a F'orm of General Analysis with 
Applications to Linear Differential and Integral Equations," Atti IV Cong. Inter. Mat. 
(Roma, 1908), 2, p. 98; "On the Foundations of the Theory of Linear Integral Equa- 
tions," Bull. Amer. Math. Soc, Ser. 2, 18, p. 339; "Definition of Limit in General 
Integral Analysis," these Proce;:edings, 1, 1915, p. 628. 
2 K. Knopp, "Grenzwerte von Reihen bei der Annfiherung an die Konvergenzgrenze," 
Inauguraldissertation (Berlin, 1907); W. Schnee, "Die Identitat des Cesaroschen und 
Holderschen Grenzwertes," Math. Annalen, 67, 1909, pp. 110-125; W. B. Ford, "On the 
Relation between the Sum-Formulas of Holder and Cesaro," Amer. J. Math., 32, 1910, 
pp. 315-326. 
^ E. Landau, "Die Identitat des Cesaroschen und Holderschen Grenzwertes fiir Integ- 
ral," Leipzig Berichte, 65, 1913, pp. 131-138. 
^ Cf. the first three references in the first foot-note; also E. H. Moore, "On the Funda- 
mental Functional Operation of a General Theory of Linear Integral Equations," Proc. 
Fifth Inter. Congr Math. (Cambridge, 1912), 1, 1913, pp. 230-255. 
For the definition of these symbols see page 150 of the monograph referred to in the 
first footnote. 
I. Schur, "iiber die Aquivalenz der Cesaroschen und Holderschen Mittelwerte," 
Math. Annalen, 74, 1913, pp. 447-458. 
