Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Completeness and compactness in linear topological spaces


Author: Heron Sherwood Collins
Journal: Trans. Amer. Math. Soc. 79 (1955), 256-280
MSC: Primary 46.0X
DOI: https://doi.org/10.1090/S0002-9947-1955-0069386-1
MathSciNet review: 0069386
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] Leonidas Alaoglu, Weak topologies of normed linear spaces, Ann. of Math. vol. 41 (1940) pp. 252-267. MR 0001455 (1:241e)
  • [2] R. F. Arens, Duality in linear spaces, Duke Math. J. vol. 14 (1947) pp. 787-794. MR 0022651 (9:241a)
  • [3] Stefan Banach, Théorie des opérations linéaires, Warsaw, 1932.
  • [4] N. Bourbaki, Espaces vectoriel topologiques, Actualités Scientifiques et Industriells, no. 1189, Paris, Hermann, 1953. MR 0054161 (14:880b)
  • [5] J. Dieudonné, Natural homomorphisms in Banach spaces, Proc. Amer. Math. Soc. vol. 1 (1950) pp. 54-59. MR 0033973 (11:524e)
  • [6] J. Dieudonné and L. Schwartz, La dualité dans les espaces $ F$ et $ LF$, Annales de L'Institut Fourier vol. 1 (1949) pp. 61-101. MR 0038553 (12:417d)
  • [7] J. Dieudonné, La dualité dans les espaces vectoriels topologiques, Annales de L'École Normale Superieure vol. 59 (1942) pp. 107-139. MR 0011532 (6:178g)
  • [8] W. F. Eberlein, Weak compactness in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. vol. 33 (1947) pp. 51-53. MR 0021239 (9:42a)
  • [9] S. Eilenberg and C. Kuratowski, Théoremes d'addition, Fund. Math. vol. 35 (1939) pp. 193-200.
  • [10] David Gale, Compact sets of functions and function rings, Proc. Amer. Math. Soc. vol. 1 (1950) pp. 303-308. MR 0036503 (12:119d)
  • [11] A. Grothendieck, Critéres de compacité dans les espaces fonctionnels généraux, Amer. J. Math. vol. 74 (1952) pp. 168-188. MR 0047313 (13:857e)
  • [12] -, Sur la complétion du dual d'un espace vectoriel localement convexe, C. R. Acad. Sci. Paris vol. 230 (1950) pp. 605-606. MR 0040573 (12:715b)
  • [13] Felix Hausdorff, Théorie der linearen metrischen Räume, J. Reine Angew. Math. vol. 167 (1932) pp. 294-311.
  • [14] Samuel Kaplan, Cartesian products of reals, Amer. J. Math. vol. 74 (1952). MR 0054944 (14:1002a)
  • [15] M. Katětov, Über normierte Vektorräume, Bull. Int. Acad. Tcheque vols. 43-44 (1942-1943).
  • [16] J. L. Kelley, Convergence in topology, Duke Math. J. vol. 17 (1950) pp. 277-283. MR 0036982 (12:194f)
  • [17] V. L. Klee, Convex sets in linear spaces, Duke Math. J. vol. 18 (1951) pp. 443-466. MR 0044014 (13:354f)
  • [18] G. Kothe, Die Quotientenräume eines linearen vollkommenen Räumes, Math. Zeit. vol. 51 (1947) pp. 17-35. MR 0023454 (9:358e)
  • [19] M. Krein and V. Šmulian, On regularly convex sets in the space conjugate to a Banach space, Ann. of Math. vol. 41 (1940) pp. 556-583. MR 0002009 (1:335e)
  • [20] G. W. Mackey, On infinite dimensional linear spaces, Trans. Amer. Math. Soc. vol. 57 (1945) pp. 153-207. MR 0012204 (6:274d)
  • [21] E. A. Michael, Locally multiplicatively-convex topological algebras, Memoirs of the American Mathematical Society, no. 11, 1952. MR 0051444 (14:482a)
  • [22] M. E. Munroe, Homomorphisms on Banach spaces, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 776-781. MR 0026245 (10:128c)
  • [23] R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. vol. 65 (1943) pp. 108-136. MR 0007938 (4:218f)
  • [24] Vlastimil Ptak, On complete topological linear spaces, Cehoslovack. Mat. Ž. (1953) pp. 301-364. MR 0064303 (16:262c)
  • [25] G. T. Roberts, The bounded-weak topology and completeness in vector spaces, Proc. Cambridge Philos. Soc. vol. 49 (1953) pp. 183-189. MR 0053387 (14:766b)
  • [26] V. Šmulian, Sur les ensembles régulierement fermés et faiblement compacts dans les espaces du type $ (B)$, C. R. (Doklady) Acad. Sci. URSS vol. 18 (1938) pp. 405-407.
  • [27] -, Über lineare topologische Räume, Mat. Sbornik N.S. vol. 7 (1941) pp. 425-448.
  • [28] A. Tychonoff, Über einen Funktionräum, Math. Ann. vol. 111 (1934) pp. 762-766.
  • [29] -, Ein Fixpunktsatz, Math. Ann. vol. 111 (1935) pp. 767-776. MR 1513031
  • [30] John von Neumann, On complete topological spaces, Trans. Amer. Math. Soc. vol. 37 (1935) pp. 1-20. MR 1501776

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46.0X

Retrieve articles in all journals with MSC: 46.0X


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1955-0069386-1
Article copyright: © Copyright 1955 American Mathematical Society

American Mathematical Society