TOPOLOGIES AND BORNOLOGIES DETERMINED BY OPERATOR IDEALS .2.

Let 21 be an operator ideal on LCS's. A continuous seminorm p of a LCS X is said to be 21-continuous if Q(p) is-an-element 21inj(X, X(p)), w here X(p) is the completion of the normed space X(p) = X/p-1(0) and Q( p) is the canonical map. p is said to be a Groth(21)-seminorm if there is a continuous seminorm q of X such that p greater-than-or-equal-to q and the canonical map Q(pq): X(p) --> X(p) belongs to 21(X(q), X(p)) . It is well known that when 21 is the ideal of absolutely summing (re sp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous semin orm p of X is 21-continuous if and only if every continuous seminorm p of X is a Groth(21)-seminorm. In this paper, we extend this equivalen ce to arbitrary operator ideals 21 and discuss several aspects of thes e constructions which were initiated by A. Grothendieck and D. Randtke , respectively. A bornological version of the theory is also obtained.