Smooth partitions of unity on manifolds

Lloyd, John

doi:10.1090/S0002-9947-1974-0375374-X

Smooth partitions of unity on manifolds
HTML articles powered by AMS MathViewer

by John Lloyd PDF

Trans. Amer. Math. Soc. 187 (1974), 249-259 Request permission

Abstract:

This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces.

References

E. Asplund and R. T. Rockafellar, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467. MR 240621, DOI 10.1090/S0002-9947-1969-0240621-X
V. I. Averbuh and O. G. Smoljanov, Differentiation theory in linear topological spaces, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 201–260 (Russian). MR 0223886
V. I. Averbuh and O. G. Smoljanov, Different definitions of derivative in linear topological spaces, Uspehi Mat. Nauk 23 (1968), no. 4 (142), 67–116 (Russian). MR 0246118

Smooth partitions of unity in some infinitely dimensional manifolds

Robert Bonic and John Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. MR 0198492
N. Bourbaki, Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1333, Hermann, Paris, 1967 (French). MR 0219078
J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969. Enlarged and corrected printing. MR 0349288
Jean Dieudonné and Laurent Schwartz, La dualité dans les espaces $\scr F$ et $(\scr L\scr F)$, Ann. Inst. Fourier (Grenoble) 1 (1949), 61–101 (1950) (French). MR 38553
A. Frölicher and W. Bucher, Calculus in vector spaces without norm, Lecture Notes in Mathematics, No. 30, Springer-Verlag, Berlin-New York, 1966. MR 0213869, DOI 10.1007/BFb0068432

Topologische linear Räume

24

40

Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257

Jauges différentiables et partitions de l’unité

John W. Lloyd, Inductive and projective limits of smooth topological vector spaces, Bull. Austral. Math. Soc. 6 (1972), 227–240. MR 295031, DOI 10.1017/S0004972700044452
John Lloyd, Differentiable mappings on topological vector spaces, Studia Math. 45 (1973), 147–160; corrigenda, ibid. 49 (1973), 99–100. MR 333724, DOI 10.4064/sm-45-2-147-160

Two topics in the differential calculus on topological linear spaces

Jean-Paul Penot, Calcul différentiel dans les espaces vectoriels topologiques, Studia Math. 47 (1973), 1–23 (French). MR 336334, DOI 10.4064/sm-47-1-1-23
Daniel J. Randtke, A structure theorem for Schwartz spaces, Math. Ann. 201 (1973), 171–176. MR 326347, DOI 10.1007/BF01359795
A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. 54 (1948), 977–982. MR 26802, DOI 10.1090/S0002-9904-1948-09118-2

Lecture notes on differential calculus