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Closed Graph Theorems and Webbed Spaces by Marc De Wilde

By Marc De Wilde

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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. 4. 1. Let us denote by ~E the class of all spaces F such that every linear relation R of E into F such that ~(R) = E and G(R) is closed in E x F is continuous.

T& , hence the result. The condition is sufficient. Let (F,t 1 ) belong to & and T be a linear map of (F,t 1 ) into (E,t), such that G(T) is closed. Denote by U (reap. r) a base of neighborhoods of zero in E (resp. F). The topology t" which admits {TU+V : U f U, V f r} as a base of neighborhoods of zero is locally convex and Hausdorff. Indeed, if y f TU + V, ((o,y) + U x V] n G(T) ~ p. Thus y f TU + V, VU f U, VV f r l> (o,y) f l> y = To 'G""['T) = G(T) = o. It is clear that t" ~ t. Thus t~ = t& ~ t.

I E-m) Eo ~0 ~ F. ~ l£ ' F is the union of countably many subspa- ~ F Eo ~o· Let R be a linear relation of E into F, such that ~(R) is 00 R- 1 F. is non meagre non meagre in E. Since ~(R) = u R- 1 F. ~ ~ ' i= 1 for some i E-JN. Consider then the restriction R 1 of R defined by xR'y ~> xRy and y f Fi It is a new linear relation of E into F. • It is thus conti~ nuous. ) is a neighbor~ ~ hood of zero in E. Moreover ~(R) ) ~(R') =E. Hence the result. 7. A countabl• inductive limit of elements of belongs to ~ • ~ 0 Indeed, if F is the inductive limit of the spaces F.

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