no. 3
Properties of Orlicz-Pettis or Nikodym type and barrelledness conditions
Annales de l'Institut Fourier, Tome 28 (1978) no. 3, pp. 67-85.

On met en défaut des propriétés de type Orlicz-Pettis et de “bornitude uniforme” pour des mesures à valeurs dans certains espaces non localement convexes. On demande, en termes de conditions de tonnelage, quelles hypothèses de convexité généralisée assurent la validité de ces propriétés.

An Orlicz-Pettis type property for vector measures and also the “Uniform Boundedness Principle” are shown to fail without local convexity assumption. The author asks under which generalized convexity hypotheses these properties remain true. This problem is expressed in terms of barrelledness type conditions.

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     author = {Turpin, Philippe},
     title = {Properties of {Orlicz-Pettis} or {Nikodym} type and barrelledness conditions},
     journal = {Annales de l'Institut Fourier},
     pages = {67--85},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {3},
     year = {1978},
     doi = {10.5802/aif.701},
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     mrnumber = {80d:46080},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.701/}
}
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Turpin, Philippe. Properties of Orlicz-Pettis or Nikodym type and barrelledness conditions. Annales de l'Institut Fourier, Tome 28 (1978) no. 3, pp. 67-85. doi : 10.5802/aif.701. https://aif.centre-mersenne.org/articles/10.5802/aif.701/

[1] G. Bennett, N.J. Kalton, Inclusion theorems for K-spaces, Canad. J. Math., 25 (1973), 511-524. | MR | Zbl

[2] J.H.E. Cohn, On the value of determinants, Proc. Amer. Math. Soc., 14 (1963), 581-588. | MR | Zbl

[3] L. Drewnowski, Topological rings of sets, continuous set functions, integration, I, II and III, Bull. Acad. Polon. Sci., 20 (1972), 269-276, 277-286, 439-445. | Zbl

[4] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems, Bull. Acad. Polon. Sci., 20 (1972), 725-731. | MR | Zbl

[5] L. Drewnowski, Uniform boundedness principle for finitely additive vector measures, Bull. Acad. Polon. Sci., 21 (1973), 115-118. | MR | Zbl

[6] L. Drewnowski, On the Orlicz-Pettis type theorems of Kalton, Bull. Acad. Polon. Sci., 21 (1973), 515-518. | MR | Zbl

[7] L. Drewnowski, I. Labuda, Sur quelques théorèmes du type d'Orlicz-Pettis II, Bull. Acad. Polon. Sci., 21 (1973), 119-126. | MR | Zbl

[8] W. Fischer, U. Scholer, The range of vector measures into Orlicz spaces, Studia Math., 59 (1976), 53-61. | MR | Zbl

[9] W. Fischer, U. Scholer, Sur la bornitude d'une mesure vectorielle, C.R. Acad. Sci,., Paris, 282 A (1976), 519-522. | MR | Zbl

[10] A. Grothendieck, Espaces vectoriels topologiques, Instituto de Matematica Pura e Aplicada, Universidade de Sao Paulo, 3d ed., Sao Paulo, 1964. | Zbl

[11] T. Husain, The open mapping and closed graph theorems in topological vector spaces, Clarendon Press, Oxford, 1965. | MR | Zbl

[12] S.O. Iyahen, On certain classes of linear topological spaces, Proc. London Math. Soc., 18 (1968), 285-307. | MR | Zbl

[13] N.J. Kalton, Topologies on Riesz groups and applications to measure theory, Proc. London Math. Soc., (3) 28 (1974), 253-273. | MR | Zbl

[14] I. Labuda, Sur quelques généralisations des théorèmes de Nikodym et de Vitali-Hahn-Saks, Bull. Acad. Polon. Sci., 20 (1972), 447-456. | MR | Zbl

[15] I. Labuda, Sur quelques théorèmes du type d'Orlicz-Pettis I, Bull. Acad. Polon. Sci., 21 (1973), 127-132. | MR | Zbl

[16] I. Labuda, Ensembles convexes dans les espaces d'Orlicz, C.R. Acad. Sci., Paris, 281 (1975), 443-445. | MR | Zbl

[17] W. Robertson, Completions of topological vector spaces, Proc. London Math. Soc., (3) 8 (1958), 242-257. | MR | Zbl

[18] S. Rolewicz, Metric linear spaces, Monografie Matematyczne 56, PWN, Warsaw 1972. | MR | Zbl

[19] S. Rolewicz C. Ryll-Nardzewski, On unconditional convergence in linear metric spaces, Coll. Math., 17 (1967), 327-331. | MR | Zbl

[20] G.L. Seever, Measures on F-spaces, Trans. Amer. Math. Soc., 133 (1968), 267-280. | MR | Zbl

[21] E. Thomas, The Lebesgue-Nikodym theorem for vector valued Radon measures, Mem. Amer. Math. Soc., 139 (1974). | Zbl

[22] Ph. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math., 131, PWN, Warsaw, 1975. | Zbl

[23] Ph. Turpin, Conditions de bornitude et espaces de fonctions mesurables, Studia Math., 56 (1975), 69-91. | Zbl

[24] Ph. Turpin, Une mesure vectorielle non bornée, C.R. Acad. Sci., Paris, 280 A (1975), 509-511. | MR | Zbl

[25] Ph. Turpin, Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologique non supposé localement convexe, Colloque sur l'Intégration vectorielle et multivoque, Caen 22 et 23 mai 1975. | Zbl

[26] L. Waelbroeck, Topological vector spaces and algebras, Lecture Notes in Mathematics, 230, Springer, Berlin, 1971. | MR | Zbl

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