CONSTRUCTION OF STANDARD EXACT SEQUENCES OF POWER-SERIES SPACES

The following result is proved: Let LAMBDA(R)p(alpha) denote a power s eries space of infinite or of finite type, and equip LAMBDA(R)p(alpha) with its canonical fundamental system of norms, R is-an-element-of {0 , infinity}, 1 less-than-or-equal-to p < infinity. Then a tamely exact sequence () 0 --> LAMBDA(R)p(alpha) --> LAMBDA(R)p(alpha) --> LAMBDA (R)p(alpha)N --> 0exists iff alpha is strongly stable, i.e. lim(n) alp ha2n/alpha(n) = 1, and a linear-tamely exact sequence () exists iff a lpha is uniformly stable, i.e. there is A such that lim sup(n) alpha(K n)/alpha(n) less-than-or-equal-to A < infinity for all K. This result extends a theorem of Vogt and Wagner which states that a topologically exact sequence () exists iff alpha is stable, i.e. sup(n) alpha2n/al pha(n) < infinity.