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Let E and F be Koethe sequence spaces. I am interested in properties of the space $\mathcal{L}_{B}(E,F)$ of continuous linear operators between E and F, endowed with the topology of uniform convergence on bounded subsets of E, i.e. whether this space is for exampled barrelled or bornological. Particularly I am interested in the case where E and F are nuclear sequence spaces. So far I did not succeed in finding any papers dealing with this question and would be grateful for any references/answers.

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up vote 1 down vote accepted

It seems that this question is related to functor $Ext^{1}(E,F)$ and there are several papers by Vogt dealing with this question. In Vogt's 1984 paper "Some results on continuous linear maps between Fr\'{e}chet spaces" one can find for example the following theorem (p.369/ proposition 4.5) "Let E and F be nuclear Fr{e}chet spaces. If $E$ has property $(DN)$, and if $F$ has property $(\Omega)$, then $\mathcal{L}_{B}\left(E,F\right)$ is bornological."

Furthermore, as $\mathcal{L}_{B}\left(E,F\right)$ is complete one can also conclude that $\mathcal{L}_{B}\left(E,F\right)$ is ultrabornological.

It is also quite simple to conclude that $\mathcal{L}_{B}\left(E,F\right)$ is a semi-Montel space if $E$ is assumed to be a Montel space and $F$ is assumed to be semi-Montel.

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