�ber die Regularitatsbegriffe induktiver lokalkonvexer Sequenzen

A Projective Description of Weighted Inductive Limits

“…On the other hand, Neus [25] proved that a compactly regular inductive limit E -ind"^ En of normed spaces En is already boundedly retractive whence clearly also strongly boundedly retractive. It then follows that a compactly regular inductive limit of a sequence of Banach spaces is a quasi-complete (DF)-space and hence complete.…”

“…Let X denote a locally compact Hausdorff space, and let °V= {vn)neN oe a decreasing sequence of weights vn on X such that inf{ü"(x); x G K] > 0 for each compact subset K of X, n = 1,2,_If E = ind"^ En is a boundedly retractive (or, equivalently, compactly regular; see the result of Neus [25] recalled in §0.3) injective inductive limit of normed spaces, then (a) 6Ù'YC(X, E) = TC( A, £) = CF(A", £) algebraically, the three spaces have the same bounded sets, and both indn-Ct>"(A, £") a«iZind"^ Cvn(X, E) are regular; (b) 6ÙCVC(X, E) is the bornological space associated with CV(X, E) (or with "(C(X, £)), and also the ultrabornological space associated with CV(X, E) if E is complete ( which would follow if En is complete for each « = 1,2,...).…”

A projective description of weighted inductive limits

“…Boundedly retractive inductive limits preserve quasi-completeness. On the other hand, Neus [25] proved that a compactly regular inductive limit E -ind"^ En of normed spaces En is already boundedly retractive whence clearly also strongly boundedly retractive. It then follows that a compactly regular inductive limit of a sequence of Banach spaces is a quasi-complete (DF)-space and hence complete.…”

Inductive limits and ε-tensor products.

“…We shall now consider inductive limits F=indF n of an increasing sequence of n -» oo locally convex spaces F n with F= (j F n such that the canonical embeddings F n c> F n+1 Every strongly boundedly retractive inductive limit is compact-regular and every compact-regular inductive limit is regul r. If F=mdF n is a compact-regular inductive n-» limit of normed spaces F H , then F is strongly boundedly retractive by H. Neus [28]. We shall further use the result that every strongly boundedly retractive inductive limit of quasi-complete locally convex spaces is again quasi-complete.…”