A setting for global analysis

Eells, James, Jr.

doi:10.1090/S0002-9904-1966-11558-6

A setting for global analysis
HTML articles powered by AMS MathViewer

by James Eells Jr. PDF

Bull. Amer. Math. Soc. 72 (1966), 751-807

References

Ralph Abraham, Transversality in manifolds of mappings, Bull. Amer. Math. Soc. 69 (1963), 470–474. MR 149495, DOI 10.1090/S0002-9904-1963-10969-6
V. I. Anosov, Critical points of periodic functionals, Soviet Math. Dokl. 1 (1960), 208–210. MR 0119115
Richard F. Arens and James Eells Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956), 397–403. MR 81458
Robert G. Bartle, On the openness and inversion of differentiable mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 257 (1958), 8. MR 104172
Andrée Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J. Analyse Math. 13 (1964), 1–114 (French). MR 177277, DOI 10.1007/BF02786619
C. Bessaga, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 27–31 (English, with Russian summary). MR 193646
Garrett Birkhoff, Analytical groups, Trans. Amer. Math. Soc. 43 (1938), no. 1, 61–101. MR 1501934, DOI 10.1090/S0002-9947-1938-1501934-4
Robert Bonic and John Frampton, Differentiable functions on certain Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 393–395. MR 172084, DOI 10.1090/S0002-9904-1965-11310-6
Robert Bonic and John Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. MR 0198492
H. J. Bremermann, Holomorphic functionals and complex convexity in Banach spaces, Pacific J. Math. 7 (1957), 811–831. MR 88709
Felix E. Browder, On a generalization of the Schauder fixed point theorem, Duke Math. J. 26 (1959), 291–303. MR 105629
Felix E. Browder, Infinite dimensional manifolds and non-linear elliptic eigenvalue problems, Ann. of Math. (2) 82 (1965), 459–477. MR 203249, DOI 10.2307/1970708
Felix E. Browder, Fixed point theorems on infinite dimensional manifolds, Trans. Amer. Math. Soc. 119 (1965), 179–194. MR 195082, DOI 10.1090/S0002-9947-1965-0195082-2
Ion Colojoară, On Whitney’s imbedding theorem, Rev. Roumaine Math. Pures Appl. 10 (1965), 291–296. MR 182018
P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
H. H. Corson, Collections of convex sets which cover a Banach space, Fund. Math. 49 (1960/61), 143–145. MR 125430, DOI 10.4064/fm-49-2-143-145
Jane Cronin, Fixed points and topological degree in nonlinear analysis, Mathematical Surveys, No. 11, American Mathematical Society, Providence, R.I., 1964. MR 0164101
J. A. Dieudonné, Recent developments in the theory of locally convex vector spaces, Bull. Amer. Math. Soc. 59 (1953), 495–512. MR 62334, DOI 10.1090/S0002-9904-1953-09752-X
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
Jacques Dixmier and Adrien Douady, Champs continus d’espaces hilbertiens et de $C^{\ast }$-algèbres, Bull. Soc. Math. France 91 (1963), 227–284 (French). MR 163182
J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116
Clifford J. Earle and James Eells Jr., Foliations and fibrations, J. Differential Geometry 1 (1967), no. 1, 33–41. MR 215320
James Eells Jr., On the geometry of function spaces, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 303–308. MR 0098419
James Eells Jr., On submanifolds of certain function spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1520–1522. MR 112148, DOI 10.1073/pnas.45.10.1520
James Eells Jr., A class of smooth bundles over a manifold, Pacific J. Math. 10 (1960), 525–538. MR 120656
James Eells Jr., Alexander-Pontrjagin duality in function spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, pp. 109–129. MR 0125580
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
James Eells Jr. and J. H. Sampson, Énergie et déformations en géométrie différentielle, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 1, 61–69 (French). MR 172310
Kazimierz Gęba, Sur les groupes de cohomotopie dans les espaces de Banach. Constructions générales, C. R. Acad. Sci. Paris 254 (1962), 3293–3295 (French). MR 139170
K. Gęba, Algebraic topology methods in the theory of compact fields in Banach spaces, Fund. Math. 54 (1964), 177–209. MR 162117, DOI 10.4064/fm-54-2-177-209
K. Gęba and A. Granas, Algebraic topology in linear normed spaces. I. Basic categories, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 287–290 (English, with Russian summary). MR 184235
Georges Glaeser, Étude de quelques algèbres tayloriennes, J. Analyse Math. 6 (1958), 1–124; erratum, insert to 6 (1958), no. 2 (French). MR 101294, DOI 10.1007/BF02790231
A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. MR 149253
Lawrence M. Graves, Riemann integration and Taylor’s theorem in general analysis, Trans. Amer. Math. Soc. 29 (1927), no. 1, 163–177. MR 1501382, DOI 10.1090/S0002-9947-1927-1501382-X
Lawrence M. Graves, Implicit functions and differential equations in general analysis, Trans. Amer. Math. Soc. 29 (1927), no. 3, 514–552. MR 1501403, DOI 10.1090/S0002-9947-1927-1501403-4
T. H. Hildebrandt and Lawrence M. Graves, Implicit functions and their differentials in general analysis, Trans. Amer. Math. Soc. 29 (1927), no. 1, 127–153. MR 1501380, DOI 10.1090/S0002-9947-1927-1501380-6
Magnus R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math. 1 (1951), 525–581. MR 46590
Magnus R. Hestenes, Quadratic variational theory and linear elliptic partial differential equations, Trans. Amer. Math. Soc. 101 (1961), 306–350. MR 133575, DOI 10.1090/S0002-9947-1961-0133575-0
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Lars Hörmander, Linear partial differential operators, Springer-Verlag, Berlin-New York, 1976. MR 0404822
Klaus Jänich, Vektorraumbündel und der Raum der Fredholm-Operatoren, Math. Ann. 161 (1965), 129–142. MR 190946, DOI 10.1007/BF01360851
Shizuo Kakutani, Topological properties of the unit sphere of a Hilbert space, Proc. Imp. Acad. Tokyo 19 (1943), 269–271. MR 14203
H. H. Keller, Über die Differentialgleichung erster Ordnung in normierten linearen Räumen, Rend. Circ. Mat. Palermo (2) 8 (1959), 117–144 (German). MR 121554, DOI 10.1007/BF02851024
Michael Kerner, Abstract differential geometry, Compositio Math. 4 (1937), 308–341. MR 1556977
Victor L. Klee Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc. 74 (1953), 10–43. MR 54850, DOI 10.1090/S0002-9947-1953-0054850-X
V. L. Klee Jr., Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30–45. MR 69388, DOI 10.1090/S0002-9947-1955-0069388-5
Wilhelm Klingenberg, The theorem of the three closed geodesics, Bull. Amer. Math. Soc. 71 (1965), 601–605. MR 177374, DOI 10.1090/S0002-9904-1965-11353-2
M. A. Krasnosel′skiĭ, Topologicheskie metody v teoriĭ nelineĭnykh integral′nykh uravneniĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian). MR 0096983
Nicolaas H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30. MR 179792, DOI 10.1016/0040-9383(65)90067-4
Ivan Kupka, Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds, Proc. Amer. Math. Soc. 16 (1965), 954–957. MR 182024, DOI 10.1090/S0002-9939-1965-0182024-4
J. Kurzweil, On approximation in real Banach spaces, Studia Math. 14 (1954), 214–231 (1955). MR 68732, DOI 10.4064/sm-14-2-214-231
Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257
Detlef Laugwitz, Über unendliche kontinuierliche Gruppen. I. Grundlagen der Theorie; Untergruppen, Math. Ann. 130 (1955), 337–350 (German). MR 75535, DOI 10.1007/BF01343900
Detlef Laugwitz, Grundlagen für die Geometrie der unendlichdimensionalen Finslerräume, Ann. Mat. Pura Appl. (4) 41 (1956), 21–41 (German). MR 78732, DOI 10.1007/BF02411658
Jacqueline Lelong-Ferrand, Sur les groupes à un paramètre de transformations des variétés différentiables, J. Math. Pures Appl. (9) 37 (1958), 269–278 (French). MR 98415
Jean Leray, Sur les équations et les transformations, J. Math. Pures Appl. (9) 24 (1946), 201–248 (French). MR 15788
Jean Leray, La théorie des points fixes et ses applications en analyse, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R.I., 1952, pp. 202–208 (French). MR 0047318
Jean Leray, Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233 (French). MR 143202
Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
Bernhard Maissen, Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math. 108 (1962), 229–270 (German). MR 142693, DOI 10.1007/BF02545768
A. D. Michal, General differential geometries and related topics, Bull. Amer. Math. Soc. 45 (1939), 529–563. MR 172, DOI 10.1090/S0002-9904-1939-07029-8
John Milnor, On spaces having the homotopy type of a $\textrm {CW}$-complex, Trans. Amer. Math. Soc. 90 (1959), 272–280. MR 100267, DOI 10.1090/S0002-9947-1959-0100267-4
Charles B. Morrey Jr. and James Eells Jr., A variational method in the theory of harmonic integrals. I, Ann. of Math. (2) 63 (1956), 91–128. MR 87764, DOI 10.2307/1969992
Marston Morse, Recent advances in variational theory in the large, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R.I., 1952, pp. 143–156. MR 0044758
Jürgen Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1824–1831. MR 132859, DOI 10.1073/pnas.47.11.1824
F. J. Murray, On complementary manifolds and projections in spaces $L_p$ and $l_p$, Trans. Amer. Math. Soc. 41 (1937), no. 1, 138–152. MR 1501894, DOI 10.1090/S0002-9947-1937-1501894-5
S. B. Myers, Algebras of differentiable functions, Proc. Amer. Math. Soc. 5 (1954), 917–922. MR 65823, DOI 10.1090/S0002-9939-1954-0065823-1
Mitio Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), 497–511. MR 42697, DOI 10.2307/2372304
I. Namioka, A duality in function spaces, Trans. Amer. Math. Soc. 115 (1965), 131–144. MR 202133, DOI 10.1090/S0002-9947-1965-0202133-5
Gerhard Neubauer, Über den Index abgeschlossener Operatoren in Banachrämen, Math. Ann. 160 (1965), 93–130 (German). MR 192351, DOI 10.1007/BF01361203
Richard S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340. MR 158410, DOI 10.1016/0040-9383(63)90013-2
Richard S. Palais, On the homotopy type of certain groups of operators, Topology 3 (1965), 271–279. MR 175130, DOI 10.1016/0040-9383(65)90057-1
Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1–16. MR 189028, DOI 10.1016/0040-9383(66)90002-4
Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165–172. MR 158411, DOI 10.1090/S0002-9904-1964-11062-4
M. E. Shanks and Lyle E. Pursell, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc. 5 (1954), 468–472. MR 64764, DOI 10.1090/S0002-9939-1954-0064764-3
Calvin R. Putnam and Aurel Wintner, The orthogonal group in Hilbert space, Amer. J. Math. 74 (1952), 52–78. MR 45121, DOI 10.2307/2372068
Guillermo Restrepo, Differentiable norms in Banach spaces, Bull. Amer. Math. Soc. 70 (1964), 413–414. MR 161129, DOI 10.1090/S0002-9904-1964-11121-6
Erich Rothe, Zur Theorie der topologischen Ordnung und der Vektorfelder in Banachschen Räumen, Compositio Math. 5 (1938), 177–197 (German). MR 1556993
E. H. Rothe, Critical points and gradient fields of scalars in Hilbert space, Acta Math. 85 (1951), 73–98. MR 43387, DOI 10.1007/BF02395742
E. H. Rothe, Leray-Schauder index and Morse type numbers in Hilbert space, Ann. of Math. (2) 55 (1952), 433–467. MR 49493, DOI 10.2307/1969643
E. H. Rothe, Gradient mappings, Bull. Amer. Math. Soc. 59 (1953), 5–19. MR 52681, DOI 10.1090/S0002-9904-1953-09649-5
E. H. Rothe, Critical point theory in Hilbert space under general boundary conditions, J. Math. Anal. Appl. 11 (1965), 357–409. MR 190951, DOI 10.1016/0022-247X(65)90092-2
H. L. Royden, Function algebras, Bull. Amer. Math. Soc. 69 (1963), 281–298. MR 149327, DOI 10.1090/S0002-9904-1963-10900-3
Arthur Sard, Hausdorff measure of critical images on Banach manifolds, Amer. J. Math. 87 (1965), 158–174. MR 173748, DOI 10.2307/2373229
J. Schauder, Über den Zusammenhang zwischen der Eindeutigkeit und Lösbarkeit partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Math. Ann. 106 (1932), no. 1, 661–721 (German). MR 1512780, DOI 10.1007/BF01455908
Jacob T. Schwartz, Generalizing the Lusternik-Schnirelman theory of critical points, Comm. Pure Appl. Math. 17 (1964), 307–315. MR 166796, DOI 10.1002/cpa.3160170304
Irving Segal, Differential operators in the manifold of solutions of a non-linear differential equation, J. Math. Pures Appl. (9) 44 (1965), 71–105; ibid. (9) 44 (1965), 107–132. MR 192369
S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. of Math. (2) 80 (1964), 382–396. MR 165539, DOI 10.2307/1970398
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
S. L. Sobolev, Applications of functional analysis in mathematical physics, Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by F. E. Browder. MR 0165337
E. H. Spanier, Duality and $\textrm {S}$-theory, Bull. Amer. Math. Soc. 62 (1956), 194–203. MR 85506, DOI 10.1090/S0002-9904-1956-10014-1
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
A. S. Švarc, On the homotopic topology of Banach spaces, Dokl. Akad. Nauk SSSR 154 (1964), 61–63 (Russian). MR 0160095
Jun Tomiyama and Masamichi Takesaki, Applications of fibre bundles to the certain class of $C^{\ast }$-algebras, Tohoku Math. J. (2) 13 (1961), 498–522. MR 139025, DOI 10.2748/tmj/1178244253
M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR 0176364
W. T. van Est and Th. J. Korthagen, Non-enlargible Lie algebras, Nederl. Akad. Wetensch. Proc. Ser. A 67=Indag. Math. 26 (1964), 15–31. MR 0160851
L. Waelbroeck, Le calcul symbolique dans les algèbres commutatives, J. Math. Pures Appl. (9) 33 (1954), 147–186 (French). MR 73953
Arthur Wasserman, Morse theory for $G$-manifolds, Bull. Amer. Math. Soc. 71 (1965), 384–388. MR 174064, DOI 10.1090/S0002-9904-1965-11306-4
R. Wood, Banach algebras and Bott periodicity, Topology 4 (1965/66), 371–389. MR 185598, DOI 10.1016/0040-9383(66)90035-8

Additional Information

Journal: Bull. Amer. Math. Soc. 72 (1966), 751-807
DOI: https://doi.org/10.1090/S0002-9904-1966-11558-6
MathSciNet review: 0203742