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# Closed Graph Theorems and Webbed Spaces (Research Notes in by Marc De Wilde

By Marc De Wilde

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E belongs to &~. Let 9 be an ul- trabornivorous disk of E (cf. def. 1). Define the Banach space E8 and the map j as in the preceding proof. To see that 9 23 is a neighborhood of zero, we have to verify that j is c ontinuous or, since E E- gt that G( j) is fast sequentially closed. fF ' I f (xm' jx ) is fast convergent to (x,y), there is a fast m compact disk K such that xm- x in EK (prop. 7)• Since 9 absorbs K, it follows that x m x for p 9 • Thus jx m - jx in E. But jxm - y in E • Thus y = jx and G(j) is fast sequentially 9 closed.

9. The space E is bornological (resp. ultrabornological) if and only if it is a Mackey space and E' me (resp. E£ 0 ) is complete. Prop. 9 and the main ideas in the proof of prop. 8 are due to Kothe in the bornological case[62, 63]. If E is ultrabornological, it is bornological. If it is bornological, since the neighborhoods of zero for the Mackey topology ~ of E are bornivorous (Mackey boundedness theorem), E is a Mackey space. By prop. 8, if E is ultrabornological, E£ 0 is complete, since (Eub)' 38 = E 1 • If E is bornological, E~ 0 is complete.

Such that vn+ 1 > v n for vn all n E JN is called a strand of the web. So a strand is a se- A sequence e quence e( ki i A n1, ••• ,nk. )(i EIN), where nk(k EJN) is arbitrary and ~ co. web is said to be completing if each of its strands is completing. It is strict if it is absolutely convex and if each of its strands is strictly completing. The space E is said to be webbed or strictly webbed if it admits a completing or a strict web. The following easy remarks are quoted for later reference. 5. s.