Functional Analysis: Surveys and Recent Results, Proceedings by Klaus-Dieter Bierstedt, Benno Duchssteiner

By Klaus-Dieter Bierstedt, Benno Duchssteiner

Those lawsuits from the Symposium on practical research discover advances within the frequently separate components of semigroups of operators and evolution equations, geometry of Banach areas and operator beliefs, and Frechet areas with functions in partial differential equations.

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S) = r. ,fn+l E G satisfying the ; which enables us to show admit a representation as in Theorem 3 G so that the corresponding matrices coincide. The argumentation for S Ao(S); and S i. Ao(S) and Poulsen simplices; is similar which yields Theorem 10 (ii). The above Lemma has some other consequences. It is not difficult to show that Lemma (i) is equivalent to Let E 2 I," ; F l i n e a r isometry n+ 1 ; E c F , so t h a t I_ i T : E -$ ex B(E) = @ . Then any T : F + G. n ex B(F) earl be extended t o an isometry G This strengthens the condition which defines a Gurarij space.

PROOF: (i) : If K c ex S is compact and int K # 0 , open non-empty U c S . Hence int K is dense in U then int K = U l ex S for some since ex S is dense in S. Thus the closed convex hull of K, F, would contain a non-empty open subset of S which is a contradiction because F is a face of S. (ii) is based on Proposition 9 and the fact that for any separable complete metric space H there is a metrizable simplex whose set of extreme points is homeomorphic to H. (iii) is a consequence of the homogeneity property of follows immediately from S, given below, and (iv) (iii).

Preliminaries Banach spaces We only consider real Banach spaces. To avoid trivialities we assume them to be nonzero. BY S(x,r) we mean the open ball with center x and radius r in the Banach space X (so S(f,r) is a ball in C(K,X) whereas S(f(k),r) is a ball in X(k E K ) ) . A l l norms (elements of X, functions in C(K,X), operators) are denoted by the same symbol I1 II Let ) i ' ( i=l,. ,n be a finite family of Banach spaces, lsplm. n i l p Xi is the direct product of the Xi provided with the norm i= 1 .

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