Construction of standard exact sequences of power series spaces

The following result is proved: Let ${\displaystyle {\Lambda }_R^p(\alpha )}$ denote a power series space of infinite or of finite type, and equip ${\displaystyle {\Lambda }_R^p(\alpha )}$ with its canonical fundamental system of norms, R ∈ {0,∞}, 1 ≤ p < ∞. Then a tamely exact sequence (⁎) ${\displaystyle 0\to {\Lambda }_R^p(\alpha )\to {\Lambda }_R^p(\alpha )\to {\Lambda }_R^p{(\alpha )}^{\mathbb{N}}\to 0}$ exists iff α is strongly stable, i.e. ${\displaystyle \underset{n}{lim}\frac{{\alpha }_{2n}}{{\alpha }_n}=1}$, and a linear-tamely exact sequence (*) exists iff α is uniformly stable, i.e. there is A such that ${\displaystyle lim{\supset }_n\frac{{\alpha }_{Kn}}{{\alpha }_n}\le A<\infty }$ for all K. This result extends a theorem of Vogt and Wagner which states that a topologically exact sequence (*) exists iff α is stable, i.e. ${\displaystyle {\supset }_n\frac{{\alpha }_{2n}}{{\alpha }_n}<\infty }$.