La dualité dans les espaces () et ()
Annales de l'Institut Fourier, Tome 1 (1949), pp. 61-101.

L’objet du mémoire est l’extension des propriétés classiques des espaces de Banach à deux catégories d’espaces plus vastes, qui interviennent de manière essentielle dans la théorie des Distributions : les espaces (), qui sont définis comme les espaces localement convexes, métrisables et complets, et les espaces () qui s’obtiennent à partir des espaces () par un processus de “limite inductive” : un tel espace est réunion d’une suite croissante (Fn) d’espaces (), muni de la topologie la plus fine induisant sur chacun des Fn sa topologie propre. Pour un tel espace E, on définit son dual E comme l’espace vectoriel de toutes les formes linéaires x continues dans E, et on pose x,x=x(x) pour xE, xE. Pour développer la théorie de la dualité, on munit E d’une topologie (dite topologie forte) définie de la façon suivante : un ensemble B dans un espace vectoriel topologique étant dit borné quand tout voisinage de 0 contient un homothétique de B (dans un rapport >0 assez petit), la topologie forte sur E est la topologie de la convergence uniforme des formes linéaires dans tous les ensembles bornés de E. Il y a alors une dualité remarquable entre ensembles bornés et voisinages de 0 dans les deux espaces E et E : appelons polaire d’un ensemble AE (resp. AE) l’ensemble A0E (resp. A0E) formé des xE (resp. xE) tels que |x,x|1 pour tout xA (resp. tout xA) ; alors les polaires des ensembles bornés dans E (resp. E) sont les voisinages de 0 dans E (resp. E) et réciproquement. Grâce à cette propriété fondamentale, on peut développer, comme dans les espaces de Banach, les relations entre dualité forte et dualité faible, obtenir entre autres la condition pour qu’un espace () ou () soit réflexif (c’est-à-dire identique au dual de son dual fort), étudier la dualité des sous-espaces et espaces quotients, et les fonctions linéaires et bilinéaires définies dans des espaces () ou () ou dans leurs duals ; les résultats généralisant presque complètement les propriétés correspondantes des espaces de Banach.

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Dieudonné, Jean; Schwartz, Laurent. La dualité dans les espaces $({\mathcal {F}})$ et $({\mathcal {L}}{\mathcal {F}})$. Annales de l'Institut Fourier, Tome 1 (1949), pp. 61-101. doi : 10.5802/aif.8. http://www.numdam.org/articles/10.5802/aif.8/

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  • Revuz, André Fonctions croissantes et mesures sur les espaces topologiques ordonnes, Annales de l'Institut Fourier, Volume 6 (1956), pp. 187-269 | DOI:10.5802/aif.64 | Zbl:0074.28201
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  • Grothendieck, Alexander La théorie de Fredholm, Bulletin de la Société Mathématique de France, Volume 84 (1956), pp. 319-384 | DOI:10.24033/bsmf.1476 | Zbl:0073.10101
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  • Jacobs, Konrad Ergodentheorie und fastperiodische Funktionen auf Halbgruppen, Mathematische Zeitschrift, Volume 64 (1956), pp. 298-338 | DOI:10.1007/bf01166575 | Zbl:0070.11701
  • Robertson, Alex. P.; Robertson, Wendy On The Closed Graph Theorem, Proceedings of the Glasgow Mathematical Association, Volume 3 (1956) no. 1, p. 9 | DOI:10.1017/s2040618500033372
  • Grothendieck, A. Une Caracterisation Vectorielle-Metrique Des Espaces L1, Canadian Journal of Mathematics, Volume 7 (1955), p. 552 | DOI:10.4153/cjm-1955-060-6
  • SHIGA, Kôji Bounded Representations on a Topological Vector Space and Weak Almost Periodicity, Japanese journal of mathematics :transactions and abstracts, Volume 25 (1955) no. 0, p. 21 | DOI:10.4099/jjm1924.25.0_21
  • Schwartz, Laurent Espaces de fonctions différentiables à valeurs vectorielles, Journal d'Analyse Mathématique, Volume 4 (1955), pp. 88-148 | DOI:10.1007/bf02787718 | Zbl:0066.09601
  • Landsberg, Max Über das Spektrum der Endomorphismen eines linearen Raumes, Mathematische Nachrichten, Volume 13 (1955), pp. 1-8 | DOI:10.1002/mana.19550130102 | Zbl:0064.10903
  • Landsberg, Max Pseudonormen in der Theorie der linearen topologischen Räume, Mathematische Nachrichten, Volume 14 (1955), pp. 29-38 | DOI:10.1002/mana.19550140106 | Zbl:0065.34402
  • Collins, Heron Sherwood Completeness and compactness in linear topological spaces, Transactions of the American Mathematical Society, Volume 79 (1955), pp. 256-280 | DOI:10.2307/1992847 | Zbl:0064.35502
  • Edwards, R. E. A theory of Radon measures on locally compact spaces, Acta Mathematica, Volume 89 (1953), pp. 133-164 | DOI:10.1007/bf02393005 | Zbl:0050.05702
  • Grothendieck, A. Sur Les Applications Lineaires Faiblement Compactes D'Espaces Du Type C(K), Canadian Journal of Mathematics, Volume 5 (1953), p. 129 | DOI:10.4153/cjm-1953-017-4
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  • Tillmann, Heinz-G�nther Randverteilungen analytischer Funktionen und Distributionen, Mathematische Zeitschrift, Volume 59 (1953) no. 1, p. 61 | DOI:10.1007/bf01180242
  • On the representation of analytic functions by infinite series, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 245 (1953) no. 900, p. 429 | DOI:10.1098/rsta.1953.0003
  • Grothendieck, Alexander Summary of essential results in the theory of topological tensor products and nuclear spaces., Annales de l'Institut Fourier, Volume 4 (1952), pp. 73-112 | DOI:10.5802/aif.46 | Zbl:0055.09705
  • Dieudonné, Jean Sur un théorème de Šmulian, Archiv der Mathematik, Volume 3 (1952), pp. 436-440 | DOI:10.1007/bf01900559 | Zbl:0048.08604
  • Zeller, Karl �ber Stetigkeit von Integraltransformationen, Mathematische Zeitschrift, Volume 55 (1952) no. 2, p. 167 | DOI:10.1007/bf01268652
  • Köthe, Gottfried Die Randverteilungen analytischer Funktionen, Mathematische Zeitschrift, Volume 57 (1952), pp. 13-33 | DOI:10.1007/bf01192913 | Zbl:0047.35203
  • Donoghue, William F. jun.; Smith, Kennan T. On the symmetry and bounded closure of locally convex spaces, Transactions of the American Mathematical Society, Volume 73 (1952), pp. 321-344 | DOI:10.2307/1990672 | Zbl:0047.10601
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  • Köthe, Gottfried Neubegründung der Theorie der vollkommenen Räume, Mathematische Nachrichten, Volume 4 (1951), pp. 70-80 | DOI:10.1002/mana.19500040109 | Zbl:0042.11602
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  • Gottfried Köthe, Von Neubegründung der Theorie der vollkommenen Räume. Herrn Erhard Schmidt zuiri 75. Geburtstag gewidmet, Mathematische Nachrichten, Volume 4 (1950) no. 1-6, p. 70 | DOI:10.1002/mana.3210040109

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