Topologies and bornologies

Let A be an operator ideal on LCS’s. A continuous seminorm p of a LCS X is said to be A–continuous if Q̃p ∈ Ainj(X, X̃p), where X̃p is the completion of the normed space Xp = X/p−1(0) and Q̃p is the canonical map. p is said to be a Groth(A)–seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map Q̃pq: X̃q − → X̃p belongs to A(X̃q, X̃p). It is well-known that when A 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 seminorm p of X is A–continuous if and only if every continuous seminorm p of X is a Groth(A)–seminorm. In this paper, we extend this equivalence to arbitrary operator ideals A and discuss several aspects of these constructions which are initiated by A. Grothendieck and D. Randkte, respectively. A bornological version of the theory is obtained, too.