By Marc De Wilde
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Additional info for Closed Graph Theorems and Webbed Spaces (Research Notes in Mathematics Series)
At this stage, we have two possibilities. The first is to characterize ~C for suitable c. This question and related pro- blems were solved for various classes of spaces, in terms of completeness or "Krein-Smulian" properties (cf. the theory of Ptak evoked in the comment of prop. 3). We do not intend to develop that theory in the present notes. Let us however mention an interesting result of Komura which underlies much of the current resear~ in this direction. B. s. s. e. the finest locally convex topology on a given space E which makes a suitable collection of maps ranging into E continuous).
Thus it converges for the topology of E to some x f x 1 + B C EB. 10. 1£ E is an f °B. Since B is sequentially -m 2 °B, hence x m - ~-space, x in EB. its compact subsets are fast compact and its convergent sequences are fast convergent. 8, there exists a bounded disk B such that the topologies induced 33 by tB and by E coincide on K. Thus K is compact in EB. By prop. 9, B is a Banach disk. Since tB ~ tB' K is still compact in EB. 11. s~ If a linear map T ££ E into F maps fast convergent sequences of E into bounded sequences of F, it maps bounded Banach disks into bounded Banach disk~ fast compact sets into fast compact sets and fast convergent sequences into fast convergent sequences.
Denote by ~g the intersection of ~E (Ef&). The next two natural questions are a) to find good permanence properties for ~E and ~g• b) to characterize ~g for important classes &. The following results about question a) are essentially due to Raikov . 2. s. and T a continuous li- near map of F ~ G. If F E- ~E' ~ G E- ~E. Let R be a linear relation of E into G. Then T- 1 o R is a linear relation of E into F. e. xaR(Ty )). Since Tis continuous, Ty - Ty, a a a a 1 thus xR(Ty) and x(T- oR)y. IfF E- ~E' T- 1 oR is thus continuous, hence R = ToT- 1 oR is continuous.