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Banach Spaces of Analytic Functions by J. Baker, C. Cleaver, J. Diestel, G. Bennett, S.Y. Chang,

By J. Baker, C. Cleaver, J. Diestel, G. Bennett, S.Y. Chang, D.E. Marshall, J.A. Cima, W. Davis, W.J. Davis, W.B. Johnson, J.B. Garnett, J. Johnson, J. Wolfe, H.E. Lacey, D.R. Lewis, A.L. Matheson, P. Orno, J.W. Roberts

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Example text

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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