Let 𝔄 be an operator ideal on LCS's. A continuous seminorm p of a LCS X is said to be 𝔄- continuous if $Q̃_p ∈ 𝔄^{inj}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^{-1}(0)$ and $Q̃_p$ 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 $Q̃_{pq} : X̃_q → X̃_p$ belongs to $𝔄(X̃_q,X̃_p)$. 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 were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.
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