By Bor-Luh Lin

This quantity includes the lawsuits from a learn Workshop on Banach house thought held on the collage of Iowa in Iowa urban in July 1987. The workshop supplied contributors with a collaborative operating surroundings within which rules might be exchanged informally. numerous papers have been initiated throughout the workshop and are offered the following of their ultimate shape. additionally incorporated are contributions from numerous specialists who have been not able to wait the workshop. not one of the papers could be released somewhere else. in the course of the workshop, hours every day have been committed to seminars on present difficulties in such parts as vulnerable Hilbert areas, zonoids, analytic martingales, and operator conception, and those issues are mirrored in a few of the papers within the assortment

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**Additional info for Banach Space Theory: Proceedings of a Research Workshop Held July 5-25, 1987 With Support from the National Science Foundation**

**Sample text**

24) is satisfied. 3) The point zy is in s. 24), for a ·suitable choice of the constant () the surface Xo passes through the point M0 • On the other hand, it may be always assumed that the function f(z) is bo:unded in the domain D. In addition lwil < r 1 for all points of the set S. in common with the ·set S. Therefore, the family of surfaces {xe} must possess all the three properties stated in Lemma 5. This is impossible and thus Lemma 6 has been proved. n;, . ,, . COROLLARY 1. Let-the set 0' be decomposed into domains 0i, In each of these domains the functions R;(z 1) are logarithmically subharmonic.

V=O Here P 11 (z, f) are polynomials in zl' • • •, zn, fi (z), • • •, fN (z) ). 27) {,y (z). PROOF. +i:iv of the variables Zp ••• ' Zn' wl, ••• 'WN consider the set :E* = {z E~*. w 1 = f/z), j = 1, ... , N). By the preceding lemma, 7Tl (~*) = ~*. Therefore there exists a neighborhood ~* € rr 0 such that in its connected components ~ ::> /1 the function F(z) remains holomorphic. 27). REMARK. (z) belong to some complete family of functions I *, then, as is seen immediately from the definition· of the complete family, each §z. *

The Weil I) See, for example, I. I. Privalov, Introduction to the theory of functions of a com plex variable, Gostehizdat, Moscow-; I954, p. 420 (Russian). , §2. / (z) 41 are to be re- placed by polynomials. en -1 PROOF. We first suppose that the lemma is true for the space show that then it is also true for the space en. and Next we consider the case n = 1. 77 1 (I,)= S and hence the set s are bounded. If the set O' coincides wit~ the whole zl -plane, then the assertion of our lemma for the space is evident The set en from the Corollaries 1 and 2 of Lemma 6.