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By Jorge Alberto Barroso

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Am (x»lbj(a\(x), ... , am (x» for x E X. 5) we get <8 J. Horvath I Life and Works of L. Nachbin 37 if 5 is sufficiently small. To conclude the proof it is enough to observe that p;(al , ••• , am)Wj E d'WC 'Wand so L;=,lj(a,(x), ... , am (x))wj E 'W. 2) simple sufficient conditions for 'W to be localizable under d, which he calls the 'analytic criterion' and the 'quasi-analytic criterion' ([49, Sections 28, 29]). A special case is that if W does not vanish anywhere on X and if for every a E A the series L:=,lIa n wr 11n diverges, then the module 'W = dw generated by W is localizable under d.

31]), let me return to the space ~(X; F). Consider a linear subspace Y of ~(X, F) and a seminorm p on Y.

For instance, the set of real numbers under the natural order is a Dedekindcomplete lattice but it is not complete in the earlier sense. Another consequence of the binary intersection property is that F is a Dedekindcomplete lattice. In particular F is an M-space in the sense of S. e. its norm satisfies the relation IIsup{x, y}/1 = sup{/Ix/l, Ilyll} for all x, y ;;a. O. Kakutani proved that such a space F is norm-isomorphic to a space ee(K) for some compact K. The fact that F is Dedekind-complete imposes an additional condition on K: the closure of any open set is open.

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