We give conditions under which the functor projective limit is exact on the category of quotients of Fréchet spaces of L. Waelbroeck [18].

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.

Let be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be - continuous if $Q{̃}_p{\in }^{inj}(X,X{̃}_p)$, where $X{̃}_p$ is the completion of the normed space $X_p=X/p^{-1}\left(0\right)$ and $Q{̃}_p$ is the canonical map. p is said to be a Groth()- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q{̃}_{pq}:X{̃}_q\to X{̃}_p$ belongs to $(X{̃}_q,X{̃}_p)$. It is well known that when is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous...