Journal of the
Korean Mathematical Society JKMS

In [25] Taskinen shows that if $\{E_n\}_n$ and $\{F_n\}_n$ are two sequences of Fr\'echet spaces such that $(E_m,F_n)$ has the $BB$-property for all $m$ and $n$ then $\left(\prod_m E_m,\prod_n F_n\right)$ also has the $BB$-property. Here we investigate when this result extends to
(i) arbitrary products of Fr\'echet spaces,
(ii) countable products of $\mathcal DFN$ spaces,
(iii) countable direct sums of Fr\'echet nuclear spaces.
We also look at topologies properties of $({\mathcal H}(U),\tau)$ for $U$ balanced open in a product of Fr\'echet spaces and $\tau=\tau_o,\tau_\omega$ or $\tau_ \delta$.

Keywords: $BB$-property, products, direct sums

MSC numbers: Primary 46G20, 46G25; Secondary 46A12