Liouville's theorem for vector-valued functions

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.