Abstract
In [25] Taskinen shows that if <TEX>$\{E_n\}_n\;and\;\{F_n\}_n$</TEX> are two sequences of Frechet spaces such that (<TEX>$E_m,\;F_n$</TEX>) has the BB-property for all m and n then (<TEX>${\Pi}_m\;E_m,\;{\Pi}_n\;F_n$</TEX>) also has the ΒΒ-property. Here we investigate when this result extends to (i) arbitrary products of Frechet spaces, (ii) countable products of DFN spaces, (iii) countable direct sums of Frechet nuclear spaces. We also look at topologies properties of (<TEX>$H(U),\;\tau$</TEX>) for U balanced open in a product of Frechet spaces and <TEX>$\tau\;=\;{\tau}_o,\;{\tau}_{\omega}\;or\;{\tau}_{\delta}$</TEX>.
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