Some characterizations of the spaces 𝐿¹(𝜇)

Pothoven, Kenneth L.

doi:10.1090/S0002-9947-1971-0285894-1

Some characterizations of the spaces $L^{1}(\mu )$
HTML articles powered by AMS MathViewer

by Kenneth L. Pothoven PDF

Trans. Amer. Math. Soc. 162 (1971), 171-183 Request permission

Abstract:

Answers are given to the question of when the so-called hom and tensor functors in categories of Banach spaces preserve certain short exact sequences. The answers characterize the spaces of integrable, real-valued functions ${L^1}(\mu )$.

References

N. Bourbaki, Eléments de mathématique. XIII. Première partie: Les structures fondamentales de l’analyse. Livre VI: Intégration. Chapitre I: Inégalités de convexité. Chapitre II: Espaces de Riesz. Chapitre III: Mesures sur les espaces localement compacts. Chapitre IV: Prolongement d’une mesure; espaces $L^p$, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann & Cie, Paris, 1952 (French). MR 0054691
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
A. Grothendieck, Sur certains sous-espaces vectoriels de $L^p$, Canad. J. Math. 6 (1954), 158–160 (French). MR 58867, DOI 10.4153/cjm-1954-017-x
A. Grothendieck, Une caractérisation vectorielle-métrique des espaces $L^1$, Canadian J. Math. 7 (1955), 552–561 (French). MR 76301, DOI 10.4153/CJM-1955-060-6
Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
Joram Lindenstrauss, On the extension of operators with a finite-dimensional range, Illinois J. Math. 8 (1964), 488–499. MR 169062
B. S. Mitjagin and A. S. Švarc, Functors in categories of Banach spaces, Uspehi Mat. Nauk 19 (1964), no. 2 (116), 65–130 (Russian). MR 0166593
Leopoldo Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28–46. MR 32932, DOI 10.1090/S0002-9947-1950-0032932-3
Robert Schatten, On the direct product of Banach spaces, Trans. Amer. Math. Soc. 53 (1943), 195–217. MR 7568, DOI 10.1090/S0002-9947-1943-0007568-7
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234