Alencar, Raymundo; Aron, Richard M.; Dineen, Seán A reflexive space of holomorphic functions in infinitely many variables. (English) Zbl 0536.46015 Proc. Am. Math. Soc. 90, 407-411 (1984). Let \(E=T^*\) be Tsirelson’s original Banach space, a reflexive Banach space with unconditional basis containing no copy of \(\ell_ p,c_ 0\), or \(\ell_ 1\). The authors prove that H(E), the space of holomorphic functions on E, endowed with the Nachbin topology \(\tau_{\omega}\) is reflexive. The method of proof is to show that for all n, the space \(P(^ nT^*)\) of continuous n-homogeneous polynomials is reflexive, and to accomplish this, the authors use a weak sequential continuity characterization of reflexity. Cited in 2 ReviewsCited in 41 Documents MSC: 46E10 Topological linear spaces of continuous, differentiable or analytic functions 32A30 Other generalizations of function theory of one complex variable 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory Keywords:space of continuous n-homogeneous polynomials; topology of uniform convergence on the closed unit ball; Tsirelson space; reflexive Banach space with unconditional basis; space of holomorphic functions; Nachbin topology; weak sequential continuity characterization of reflexity PDFBibTeX XMLCite \textit{R. Alencar} et al., Proc. Am. Math. Soc. 90, 407--411 (1984; Zbl 0536.46015) Full Text: DOI References: [1] R. M. Aron, C. Hervés, and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), no. 2, 189 – 204. · Zbl 0517.46019 · doi:10.1016/0022-1236(83)90081-2 [2] P. G. Casazza, Tsirelson’s space, Proceedings of research workshop on Banach space theory (Iowa City, Iowa, 1981) Univ. Iowa, Iowa City, IA, 1982, pp. 9 – 22. [3] Seán Dineen, Holomorphic functions on (\?\(_{0}\),\?_{\?})-modules, Math. Ann. 196 (1972), 106 – 116. · Zbl 0219.46021 · doi:10.1007/BF01419608 [4] Seán Dineen, Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on \cal\?(\?), Ann. Inst. Fourier (Grenoble) 23 (1973), no. 1, 19 – 54 (English, with French summary). · Zbl 0241.46022 [5] Seán Dineen, Complex analysis in locally convex spaces, North-Holland Mathematics Studies, vol. 57, North-Holland Publishing Co., Amsterdam-New York, 1981. Notas de Matemática [Mathematical Notes], 83. · Zbl 0484.46044 [6] J. L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), no. 1, 1 – 29. · Zbl 0329.46008 · doi:10.2307/1971054 [7] R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, Thesis, Trinity College, Dublin, 1980. [8] B. S. Tsirelson, Not every Banach space contains an imbedding of \( {l_p}\) or \( {c_0}\), Functional Anal. Appl. 8 (1974), 138-141. · Zbl 0296.46018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.