Liouville's theorem for vector-valued functions

Mati Abel

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2013), pp. 5-16.

Résumé

It is shown in [2] that any $X$-valued analytic map on $\mathbb{C}\cup \{\mathrm{\infty }\}$ is a constant map in case when $X$ is a strongly galbed Hausdorff space. In [3] this result is generalized to the case when $X$ is a topological linear Hausdorff space, the von Neumann bornology of which is strongly galbed. A new detailed proof for the last result is given in the present paper. Moreover, it is shown that for several topological linear spaces the von Neumann bornology is strongly galbed or pseudogalbed.

Bibliographie

[1] Abel Mart, Abel Mati, “On galbed algebras and galbed spaces”, Bull. Greek Math. Soc., 52 (2006), 9–23 | MR | Zbl

[2] Abel Mati, “Galbed Gelfand-Mazur algebras”, Topological algebras and their applications, Contemp. Math., 341, Amer. Math. Soc., Providence, RI, 2004, 17–24 | DOI | MR | Zbl

[3] Abel Mati, “Topological algebras with galbed von Neumann bornology”, International Conference on Topological Algebras and their Applications (ICTAA), Math. Stud. (Tartu), 4, Est. Math. Soc., Tartu, 2008, 18–28 | MR

[4] Allan G., Introduction to Banach spaces and algebras, Prepared for publication and with a preface by H. Garth Dales, Oxford Graduate Texts in Math., 20, Oxford University Press, Oxford, 2011 | MR | Zbl

[5] Akkar M., “Dualité entre la bornologie d'une algèbre topologique ou bornologique et certaines propriétés particulières de ses fonctions puissances”, C. R. Acad. Sci. Paris Sér. A–B, 285:7 (1977), A509–A512 | MR | Zbl

[6] Arens R., “Linear topological division algebras”, Bull. Amer. Math. Soc., 53 (1947), 623–630 | DOI | MR | Zbl

[7] Balachandran V. K., Topological Algebras, North-Holland Math. Studies, 185, Elsevier, Amsterdam, 2000 | MR | Zbl

[8] Hogbe-Nlend H., Bornologies and Functional Analysis, North-Holland Math. Stud., 26, North-Holland, Amsterdam, 1977 | MR | Zbl

[9] Jarchow H., Locally Convex Spaces, Mathematische Leitfäden, B. G. Teubner, Stuttgart, 1981 | DOI | MR | Zbl

[10] Köthe G., Topological vector spaces, v. I, Springer-Verlag New York Inc., New York, 1969 | MR | Zbl

[11] Turpin M. P., Sur une classe d'algèbres topologiques generalisant les algèbres localement bornees, Ph. D. Thesis, Faculty of sciences, University of Grenoble, 1966

[12] Turpin Ph., “Espaces et opérateurs exponentiellement galbés”, Séminaire Pierre Lelong (Analyse), Année–1974, Lecture Notes in Math., 474, Springer, Berlin, 1973, 48–62 | DOI | MR

[13] Turpin Ph., Waelbroeck L., “Intégration et fonctions holomorphes dans les espaces localement pseudo-convexes”, C. R. Acad. Sci. Paris Sér. A–B, 267 (1968), A160–A162 | MR | Zbl

[14] Waelbroeck L., “Continuous inverse locally pseudo-convex algebras”, Summer School Topol. Algebr. Theory, Bruges, 1966, 128–185 | Zbl

[15] Waelbroeck L., Topological vector spaces and algebras, Lecture Notes in Math., 230, Springer-Verlag, Berlin–New York, 1971 | MR | Zbl