Liouville's theorem for vector-valued functions

Mati Abel

Institute of Pure Mathematics, University of Tartu, 2 J. Liivi Str., room 614, 50409 Tartu, Estonia

Abstract: It is shown in [2] that any $X$-valued analytic map on $\mathbb C\cup\{\infty\}$ 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.

Keywords and phrases: Liouville's theorem, vector-valued analytic function, metrizable linear space, galbed space, locally pseudoconvex space, $F$-space, von Neumann bornology, strictly galbed bornology, pseudogalbed bornology.

MSC: 16W80, 46H05

Received: 09.10.2012

Language: English