By Marc De Wilde
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Extra info for Closed Graph Theorems and Webbed Spaces (Research Notes in Mathematics Series)
E,tub)]. 4. The space Eb (resp. Eub) is the inductive limit of the spaces EB (B~~) where ~ is the class of all bounded disks, or of the closed absolutely convex hulls of all sequences ' which are Mackey convergent to zero (resp. where ~ is the class of all bounded Banach disks, or of all fast compact disks of E), In particular, E ~ Eb (resp. Eub) have the same bounded subsets (resp. bounded Banach disks). Thus the set of all bornivorous (resp. ultrabornivorous disks of (E,t) is a base of neighborhoods of zero of (E,tb) (resp.
We had- however to change a little bit Nakamura's condition to preserve the stability with respect to products and projective limits. l. e ll v (instea4 of being equal toe ). 5 would then proll duce another web verifying (c) in its initial form. Various other types of webs are also introduced in . M. Nakamura has proved the equivalence of several of them in . 50 The next elementary properties of completing sequences will be useful later. 7. ~ Bn(nE-lN) be a completing sequence of E, A (n E-JN) an associated sequence of numbers and U a neighborn hood of zero in E.
Vn. 28 III Bomological and ultrabomological spaces 1. FAST CONVERGENT SEQUENCES To substantiate the idea that the class of ultrabornological spaces is adequatly comprehensive for the needs of applications, this chapter is devoted to their study. The strong analogy with bornological spaces makes it reasonable to handle both cases together. 1. Let E be a locally convex Hausdorff space. An absolutely convex subset. A of E is also called a disk. A : r. ) 0} is a base of neighborhoods of x for ePch x E E.