Sur certains espaces vectoriels topologiques

Bourbaki, Nicolas

doi:10.5802/aif.16

Bourbaki, Nicolas

Annales de l'Institut Fourier, Tome 2 (1950), pp. 5-16.

Résumé

Cet article se rattache à un précédent travail paru dans ces mêmes annales (I, 1949, p. 61-101) : la dualité dans les espaces $F$ et $LF$ , par Dieudonné-Schwartz. La plupart des résultats énoncés antérieurement sont encore valables dans les espaces vectoriels appartenant à l’une ou l’autre des 2 catégories : bornologiques et tonnelés. Un espace vectoriel topologique localement convexe séparé est tonnelé si toute partie convexe, cerclée, fermée, engendrant l’espace, est un voisinage de $O$ . Tout espace vectoriel localement convexe séparé qui est un espace de Baire, en particulier tout espace $F$ , est tonnelé ; tout espace $LF$ est aussi tonnelé. La propriété fondamentale est la suivante : si $E$ est tonnelé, $F$ localement convexe, toute partie de $L (E, F)$ bornée pour la topologie de la convergence simple est équicontinue. D’où la généralisation du théorème classique de Banach-Steinhaus sur la limite simple d’une suite d’applications linéaires continues. Un espace vectoriel topologique localement convexe séparé est bornologique si toute partie convexe, cerclée, capable d’absorber toute partie bornée, est un voisinage de $O$ . Si $E$ est bornologique, $F$ localement convexe, toute application linéaire, qui transforme toute partie bornée de $E$ en une partie bornée de $F$ , est continue, et l’espace $L (E, F)$ , muni de la topologie de la convergence uniforme sur les parties bornées, est complet. Enfin la fin de l’article étudie les applications bilinéaires ; en dehors du cas des espaces de Banach, celles qu’on rencontre en général ne sont pas continues, mais possèdent une continuité partielle, qui permet de les appeler hypocontinues ; un théorème de prolongement d’une application bilinéaire hypocontinue définie sur des sous-ensembles denses existe comme pour les applications linéaires continues.

MR Zbl | 5 citations dans Numdam

DOI : 10.5802/aif.16

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Bourbaki, Nicolas. Sur certains espaces vectoriels topologiques. Annales de l'Institut Fourier, Tome 2 (1950), pp. 5-16. doi : 10.5802/aif.16. https://www.numdam.org/articles/10.5802/aif.16/

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Gudmundsdottir, Gudrun Global properties of differential operators of constant strength, Arkiv för Matematik, Volume 15 (1977), pp. 169-198 | DOI:10.1007/bf02386038 | Zbl:0374.35041
Castaing, Charles; Valadier, Michel Convex functions, Convex Analysis and Measurable Multifunctions, Volume 580 (1977), p. 1 | DOI:10.1007/bfb0087686
Dierolf, Peter Theorems of the Orlicz-Pettis-type for locally convex spaces, Manuscripta Mathematica, Volume 20 (1977), pp. 73-94 | DOI:10.1007/bf01181241 | Zbl:0339.46003
Pfister, Helmut A new characterization of barrelled spaces, Proceedings of the American Mathematical Society, Volume 65 (1977), pp. 103-104 | DOI:10.2307/2042001 | Zbl:0365.46005
Pfister, Helmut A new characterization of barrelled spaces, Proceedings of the American Mathematical Society, Volume 65 (1977) no. 1, p. 103 | DOI:10.1090/s0002-9939-1977-0440327-8
Blasco, J. L. On 𝜇-spaces and 𝑘_𝑅-spaces, Proceedings of the American Mathematical Society, Volume 67 (1977) no. 1, p. 179 | DOI:10.1090/s0002-9939-1977-0464152-7
Tolstonogov, A. A. Some properties of spaces of sublinear functionals, Siberian Mathematical Journal, Volume 18 (1977), pp. 308-317 | DOI:10.1007/bf00967166 | Zbl:0409.46006
References, Topological Algebras, Volume 24 (1977), p. 349 | DOI:10.1016/s0304-0208(08)71021-1
Petunin, Yu. I.; Pogrebnoj, V. D. Topological imbedding of semiordered topological vector spaces, Ukrainian Mathematical Journal, Volume 29 (1977), pp. 259-264 | DOI:10.1007/bf01104472 | Zbl:0422.46004
Metivier, M.; Pellaumail, J. Mesures stochastiques à valeurs dans des espaces $L_{0}$ , Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Volume 40 (1977), pp. 101-114 | DOI:10.1007/bf00532875 | Zbl:0354.60004
Dierieck, C. Simultaneous approximation of vector-valued functions, Journal of Approximation Theory, Volume 18 (1976), pp. 74-85 | DOI:10.1016/0021-9045(76)90123-4 | Zbl:0338.41030
Landsberg, M.; Schirotzek, W. Extremal and Invariant Extensions of Linear Functionals, Mathematische Nachrichten, Volume 71 (1976) no. 1, p. 191 | DOI:10.1002/mana.19760710115
Auberson, G.; Mahoux, G.; Simão, F.R.A. Semi-phenomenological bounds for the pion-pion coupling constant, Nuclear Physics B, Volume 113 (1976) no. 2, p. 285 | DOI:10.1016/0550-3213(76)90019-5
Markin, J. T. Existence of a fixed point for nonexpansive mappings with closed values, Proceedings of the American Mathematical Society, Volume 59 (1976), pp. 255-259 | DOI:10.2307/2041479 | Zbl:0349.47044
Krasichkov-Ternovskij, I. F. Methods of approximation of functions in invariant subspaces by Dirichlet polynomials, Siberian Mathematical Journal, Volume 16 (1976), pp. 779-787 | DOI:10.1007/bf00967109 | Zbl:0333.41015
Guaraldo, Francesco; Macri, Patrizia Categorie relative di indoggetti e di quozienti, ANNALI DELL UNIVERSITA DI FERRARA, Volume 21 (1975) no. 1, p. 37 | DOI:10.1007/bf02826779
Glorennec, P. Y.; Pellaumail, J. Théoreme de Riesz pour des processus réels, Annales de l'Institut Henri Poincaré. Nouvelle Série. Section B. Calcul des Probabilités et Statistique, Volume 10 (1975), pp. 355-367 | Zbl:0294.60047
Eberhardt, Volker Beispiele topologischer Vektorr�ume mit der Komplement�rraumeigenschaft, Archiv der Mathematik, Volume 26 (1975) no. 1, p. 627 | DOI:10.1007/bf01229792
Rejto, P. A. On a theorem of Titchmarsh-Neumark-Walter concerning absolutely continuous operators. II: The estimates, Letters in Mathematical Physics, Volume 1 (1975), pp. 57-66 | DOI:10.1007/bf00405587 | Zbl:0347.34021
Commichau, Michael Deformation kompakter komplexer Mannigfaltigkeiten, Mathematische Annalen, Volume 213 (1975), pp. 43-96 | DOI:10.1007/bf01883885 | Zbl:0285.32014
Dupin, Jean-Claude; Coquet, Gérard À propos d'une question posée par V. Klee, Annales de l'Institut Fourier, Volume 24 (1974) no. 1, pp. 293-295 | DOI:10.5802/aif.500 | Zbl:0247.46026
Chevallier, D. P. Structures de variétés bimodelees et dynamique analytique des milieux continus, Annales de l'Institut Henri Poincaré. Nouvelle Série. Section A. Physique Théorique, Volume 21 (1974), pp. 43-76 | Zbl:0295.58004
Goullet de Rugy, Alain Simplex spaces, Foundations of Quantum Mechanics and Ordered Linear Spaces, Volume 29 (1974), p. 33 | DOI:10.1007/3-540-06725-6_6
Morgan-Sciarrino, J. Méthode directe de recherche du point de selle d'une fonctionnelle convexe-concave et application aux problèmes variationnels elliptiques avec deux contrôles antagonistes, International Journal of Computer Mathematics, Volume 4 (1974), pp. 143-175 | DOI:10.1080/00207167408803086 | Zbl:0296.65037
Grauert, Hans Der Satz von Kuranishi für kompakte komplexe Räume, Inventiones Mathematicae, Volume 25 (1974), pp. 107-142 | DOI:10.1007/bf01390171 | Zbl:0286.32015
Levine, P; Pomerol, J-Ch Infinite programming and duality in topological vector spaces, Journal of Mathematical Analysis and Applications, Volume 46 (1974) no. 1, p. 75 | DOI:10.1016/0022-247x(74)90282-0
Landsberg, M.; Schirotzek, W. General Extension Theorems for Linear Functionals, Mathematische Nachrichten, Volume 61 (1974) no. 1, p. 111 | DOI:10.1002/mana.19740610111
Hogbe-Nlend, Henri Topologies et bornologies nucléaires associées - applications, Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 89-104 | DOI:10.5802/aif.484 | Zbl:0251.46008
Bardos, C.; Brézis, David; Brézis, Haïm Perturbations singulières et prolongements maximaux d'opérateurs positifs, Archive for Rational Mechanics and Analysis, Volume 53 (1973), pp. 69-100 | DOI:10.1007/bf00735701 | Zbl:0281.47028
Michel, Philippe Problème des inégalités. Applications à la programmation et au contrôle optimal, Bulletin de la Société Mathématique de France, Volume 101 (1973), pp. 413-439 | DOI:10.24033/bsmf.1767 | Zbl:0295.90033
Snipes, Ray F. C-sequential and S-bornological topological vector spaces, Mathematische Annalen, Volume 202 (1973), pp. 273-283 | DOI:10.1007/bf01433457 | Zbl:0236.46004
Knauer, Ulrich; Mikhalev, Alexander V. Endomorphism monoids of acts over monoids, Semigroup Forum, Volume 6 (1973), pp. 50-58 | DOI:10.1007/bf02389107 | Zbl:0258.20059
Duistermaat, J. J.; Hörmander, L. Fourier integral operators. II, Acta Mathematica, Volume 128 (1972), pp. 183-269 | DOI:10.1007/bf02392165 | Zbl:0232.47055
Valdivia, Manuel On nonbornological barrelled spaces, Annales de l'Institut Fourier, Volume 22 (1972) no. 2, pp. 27-30 | DOI:10.5802/aif.410 | Zbl:0226.46006
Roelcke, W.; Lurje, P.; Dierolf, S.; Eberhardt, V. Einige Beispiele �ber Bairesche, normierte und ultrabornologische R�ume, Archiv der Mathematik, Volume 23 (1972) no. 1, p. 513 | DOI:10.1007/bf01304924
Castaing, Charles Quelques résultats de compacité lies à l'intégration, Bulletin de la Société Mathématique de France. Supplément. Mémoires, Volume 31-32 (1972), pp. 73-81 | Zbl:0245.46036
Craven, B. D. Linear mappings between topological vector spaces, Journal of the Australian Mathematical Society, Volume 14 (1972), pp. 105-118 | DOI:10.1017/s1446788700009691 | Zbl:0236.46038
Nickel, Paul A. The linear operator method and linear $S$ topologies, Kodai Mathematical Journal, Volume 24 (1972) no. 4 | DOI:10.2996/kmj/1138846632
Perdrizet, François Topologie et traces sur les $C^{*}$ -algèbres. (Topology and traces on $C^{*}$ -algebras), Bulletin de la Société Mathématique de France, Volume 99 (1971), pp. 193-239 | DOI:10.24033/bsmf.1716 | Zbl:0226.46061
Portenier, Claude Espaces de Riesz, espaces de fonctions et espaces de sections, Commentarii Mathematici Helvetici, Volume 46 (1971) no. 1, p. 289 | DOI:10.1007/bf02566846
Karrer, G. Algebraische Operationen auf Faserbündeln, Mathematische Zeitschrift, Volume 122 (1971), pp. 177-188 | DOI:10.1007/bf01110091 | Zbl:0219.55014
Klee, Victor What Is a Convex Set?, The American Mathematical Monthly, Volume 78 (1971) no. 6, p. 616 | DOI:10.1080/00029890.1971.11992815
Metivier, M. Sur la construction de mesures aléatoires presque sûrement absolument continues par rapport à une mesure donnée, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Volume 20 (1971), pp. 332-344 | DOI:10.1007/bf00538379 | Zbl:0212.19303
Sörensen, Kay Topologische Inzidenzgruppen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Volume 35 (1970) no. 1-2, p. 75 | DOI:10.1007/bf02992476
Lindblad, G.; Nagel, B. Continuous bases for unitary irreducible representations of SU(1,1), Annales de l'Institut Henri Poincaré. Nouvelle Série. Section A. Physique Théorique, Volume 13 (1970), pp. 27-56 | Zbl:0203.57202
Vey, J. Sur les automorphismes affines des ouverts convexes saillants, Annali della Scuola Normale Superiore di Pisa. Scienze Fisiche e Matematiche. III. Ser, Volume 24 (1970), pp. 641-665 | Zbl:0206.51302
Pellaumail, J. Sur la dérivation d'une mesure vectorielle, Bulletin de la Société Mathématique de France, Volume 98 (1970), pp. 359-368 | DOI:10.24033/bsmf.1708 | Zbl:0193.43003
John, Kamil Differentiable manifolds as topological linear spaces, Mathematische Annalen, Volume 186 (1970), pp. 177-190 | DOI:10.1007/bf01433276 | Zbl:0184.15701
Horváth, John An Introduction to Distributions, The American Mathematical Monthly, Volume 77 (1970) no. 3, p. 227 | DOI:10.1080/00029890.1970.11992462
Avramescu, C. Sur l'existence des solutions convergentes des systèmes d'équations différentielles non linéaires, Annali di Matematica Pura ed Applicata. Serie Quarta, Volume 81 (1969), pp. 147-168 | DOI:10.1007/bf02413501 | Zbl:0196.10701
Floret, Klaus Zur Regularität kompakter induktiver Spektren, Archiv der Mathematik, Volume 20 (1969) no. 1, p. 54 | DOI:10.1007/bf01898991
Joly, J. L. Caractérisations d'espaces hilbertiens au moyen de la constante rectangle, Journal of Approximation Theory, Volume 2 (1969), pp. 301-311 | DOI:10.1016/0021-9045(69)90025-2 | Zbl:0177.40701
Dudley, R. M. Random linear functionals, Transactions of the American Mathematical Society, Volume 136 (1969), pp. 1-24 | DOI:10.2307/1994697 | Zbl:0186.21301
Avanissian, V.; Fernique, Xavier Sur l'analyticité des distributions harmoniques d'ordre infini, Annales de l'Institut Fourier, Volume 18 (1968) no. 2, pp. 1-9 | DOI:10.5802/aif.288 | Zbl:0181.13501
Temam, R. Sur la stabilité et la convergence de la méthode des pas fractionnaires, Annali di Matematica Pura ed Applicata. Serie Quarta, Volume 79 (1968), pp. 191-379 | DOI:10.1007/bf02415183 | Zbl:0174.45804
Sonner, Johann Cauchy conditions concerning differentiability relative to valued algebraic extensions, Archiv der Mathematik, Volume 19 (1968) no. 2, p. 171 | DOI:10.1007/bf01899279
Deutsch, N. Interpolation dans les espaces vectoriels topologiques localement convexes, Bulletin de la Société Mathématique de France. Supplément. Mémoires, Volume 13 (1968), p. 187 | Zbl:0172.39501
Floret, Klaus; Wloka, Joseph Nukleare Räume, Einführung in die Theorie der lokalkonvexen Räume, Volume 56 (1968), p. 148 | DOI:10.1007/bfb0098576
Loustaunau, J. On the state diagram of a linear operator and its adjoint in locally convex spaces. II, Mathematische Annalen, Volume 176 (1968), pp. 121-128 | DOI:10.1007/bf02056980 | Zbl:0155.46101
de Vito, C. L. A completeness theorem for locally convex spaces and some applications, Mathematische Annalen, Volume 177 (1968), pp. 221-229 | DOI:10.1007/bf01350865 | Zbl:0159.17702
BIBLIOGRAPHY, Spaces of Fundamental and Generalized Functions (1968), p. 257 | DOI:10.1016/b978-1-4832-2977-5.50010-7
Payen, R. Fonctions aléatoires du second ordre à valeurs dans un espace de Hilbert, Annales de l'Institut Henri Poincaré. Nouvelle Série. Section B. Calcul des Probabilités et Statistique, Volume 3 (1967), pp. 323-396 | Zbl:0159.45901
Martineau, A. Équations différentielles d'ordre infini, Bulletin de la Société Mathématique de France, Volume 95 (1967), pp. 109-154 | DOI:10.24033/bsmf.1650 | Zbl:0167.44202
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Umemura, Yasuo Measures on infinite dimensional vector spaces, Publications of the Research Institute for Mathematical Sciences, Series A, Volume 1 (1965), pp. 1-47 | DOI:10.2977/prims/1195196433 | Zbl:0181.41502
Falgas, Maurice Sur les séries de base de polynômes, Annales Scientifiques de l'École Normale Supérieure. Troisième Série, Volume 81 (1964), pp. 1-76 | DOI:10.24033/asens.1126 | Zbl:0137.04901
Saphar, P. Contribution à l'étude des applications linéaires dans un espace de Banach, Bulletin de la Société Mathématique de France, Volume 92 (1964), pp. 363-384 | DOI:10.24033/bsmf.1612 | Zbl:0139.08502
Fan, K. Invariant subspaces of certain linear operators, Bulletin of the American Mathematical Society, Volume 69 (1963), pp. 773-777 | DOI:10.1090/s0002-9904-1963-11028-9 | Zbl:0118.10803
Fischer, H. R. Über eine Klasse topologischer Tensorprodukte, Mathematische Annalen, Volume 150 (1963), pp. 242-258 | DOI:10.1007/bf01396993 | Zbl:0117.07902
Grauert, Hans Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Mathematische Zeitschrift, Volume 81 (1963), pp. 377-391 | DOI:10.1007/bf01111528 | Zbl:0151.09702
Edmunds, D. E. Σ-symmetric locally convex spaces, Proceedings of the American Mathematical Society, Volume 14 (1963) no. 5, p. 697 | DOI:10.1090/s0002-9939-1963-0154096-2
Akutowicz, Edwin J. Sur l'approximation par distributions à support discret, Annales Scientifiques de l'École Normale Supérieure. Troisième Série, Volume 79 (1962), pp. 71-91 | DOI:10.24033/asens.1107 | Zbl:0107.09104
Andreotti, Aldo; Vesentini, Edoardo Sopra un teorema di Kodaira, Annali della Scuola Normale Superiore di Pisa. Scienze Fisiche e Matematiche. III. Ser, Volume 15 (1961), pp. 283-309 | Zbl:0108.16604
Bruhat, François Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes $p$ -adiques, Bulletin de la Société Mathématique de France, Volume 89 (1961), pp. 43-75 | DOI:10.24033/bsmf.1559 | Zbl:0128.35701
Yamagata, Hideo Approximation of solutions of homogeneous differential equation, I, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 37 (1961) no. 10 | DOI:10.3792/pja/1195523539
Zervos, Spiros P. Aspects modernes de la localisation des zéros des polynômes d'une variable, Annales Scientifiques de l'École Normale Supérieure. Troisième Série, Volume 77 (1960), pp. 303-410 | DOI:10.24033/asens.1096 | Zbl:0096.01003
Roumieu, Charles Sur quelques extensions de la notion de distribution, Annales Scientifiques de l'École Normale Supérieure. Troisième Série, Volume 77 (1960), pp. 41-121 | DOI:10.24033/asens.1087 | Zbl:0104.33403
Tillmann, Heinz Günther Approximationssätze für Halbgruppen von Operatoren in topologischen Vektorräumen, Archiv der Mathematik, Volume 11 (1960), pp. 194-199 | DOI:10.1007/bf01236932 | Zbl:0102.10701
Grauert, Hans Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publications Mathématiques, Volume 5 (1960), pp. 233-292 | DOI:10.1007/bf02684746 | Zbl:0100.08001
Gordon, Hugh Topologies and projections on Riesz spaces, Transactions of the American Mathematical Society, Volume 94 (1960), pp. 529-551 | DOI:10.2307/1993438 | Zbl:0090.08802
Schwartz, Laurent Théorie des distributions à valeurs vectorielles. II, Annales de l'Institut Fourier, Volume 8 (1958), pp. 1-209 | DOI:10.5802/aif.77 | Zbl:0089.09801
Schwartz, Laurent Théorie des distributions à valeurs vectorielles, Annales de l'Institut Fourier, Volume 7 (1957), pp. 1-141 | DOI:10.5802/aif.68 | Zbl:0089.09601
Warner, Seth Weakly topologized algebras, Proceedings of the American Mathematical Society, Volume 8 (1957), pp. 314-316 | DOI:10.2307/2033734 | Zbl:0078.29202
Warner, Seth Weakly topologized algebras, Proceedings of the American Mathematical Society, Volume 8 (1957) no. 2, p. 314 | DOI:10.1090/s0002-9939-1957-0084727-4
Bruhat, François Sur les représentations induites des groupes de Lie, Bulletin de la Société Mathématique de France, Volume 84 (1956), pp. 97-205 | DOI:10.24033/bsmf.1469 | Zbl:0074.10303
Landsberg, Max Der Durchschnittsgrad hypercharakteristischer Filter, Mathematische Annalen, Volume 131 (1956), pp. 343-429 | DOI:10.1007/bf01350098 | Zbl:0074.17702
Kaplan, Samuel Biorthogonality and integration, Proceedings of the American Mathematical Society, Volume 7 (1956), pp. 109-114 | DOI:10.2307/2033254 | Zbl:0070.11203
Harrop, R.; Weston, J. D. An intersection property in locally convex spaces, Proceedings of the American Mathematical Society, Volume 7 (1956), pp. 535-538 | DOI:10.2307/2032767 | Zbl:0070.33404
Kaplan, Samuel Biorthogonality and integration, Proceedings of the American Mathematical Society, Volume 7 (1956) no. 1, p. 109 | DOI:10.1090/s0002-9939-1956-0076307-0
Harrop, R.; Weston, J. D. An intersection property in locally convex spaces, Proceedings of the American Mathematical Society, Volume 7 (1956) no. 3, p. 535 | DOI:10.1090/s0002-9939-1956-0080265-2
Sakai, Shôichirô The absolute value of $W^{*}$ -algebras of finite type, Tôhoku Mathematical Journal. Second Series, Volume 8 (1956), pp. 70-85 | DOI:10.2748/tmj/1178245010 | Zbl:0071.11402
Warner, Seth Inductive limits of normed algebras, Transactions of the American Mathematical Society, Volume 82 (1956), pp. 190-216 | DOI:10.2307/1992986 | Zbl:0070.34101
Warner, Seth Inductive limits of normed algebras, Transactions of the American Mathematical Society, Volume 82 (1956) no. 1, p. 190 | DOI:10.1090/s0002-9947-1956-0079226-3
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
Warner, Seth; Blair, Alexander On symmetry in convex topological vector spaces, Proceedings of the American Mathematical Society, Volume 6 (1955), pp. 301-304 | DOI:10.2307/2032362 | Zbl:0064.35603
Warner, Seth; Blair, Alexander On symmetry in convex topological vector spaces, Proceedings of the American Mathematical Society, Volume 6 (1955) no. 2, p. 301 | DOI:10.1090/s0002-9939-1955-0068728-6
Shirota, Taira On locally convex vector spaces of continuous functions, Proceedings of the Japan Academy, Volume 30 (1954), pp. 294-298 | DOI:10.3792/pja/1195526112 | Zbl:0057.33801
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
Dieudonné, Jean A. Recent developments in the theory of locally convex vector spaces, Bulletin of the American Mathematical Society, Volume 59 (1953), pp. 495-512 | DOI:10.1090/s0002-9904-1953-09752-x | Zbl:0053.25701
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
Donoghue, William F.; Smith, Kennan T. On the symmetry and bounded closure of locally convex spaces, Transactions of the American Mathematical Society, Volume 73 (1952) no. 2, p. 321 | DOI:10.1090/s0002-9947-1952-0049478-0

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