3-manifolds with(out) metrics of nonpositive curvature by Leeb B.

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By Leeb B.

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A) For any set A of indices let of indices in A < n}. {number = Suppose (bn) is any sequence of positive numbers which decreases monotonically to 0 and n a n is any series such that a n n = Show there is a set of indices A such that A' a (b) = °° and limn dn (A) /bn = 0. For S any space of sequences and (ba) a sequence of positive numbers which decreases monotonically to 0 let S(b ) consist of all s in S such that n — * lim Prove d (A )/b that = 0 where A (S (ba) a = (i: s(i) o}. and S (ba) is a normal sequence space.

And 11. SCHWARTZ AND NUCLEAR SEQUENCE SPACES In this section, we determine combinatorial properties of a Köthe space which are equivalent to its being a Schwartz space or a nuclear space in the normal topology. In the case of eschelon spaces these properties can be sharpened, and this becomes our next undertaking. Finally, we generalize the idea of normal to the u-normal topologies on a Kbthe space and note that the combinatorial theorems proven for the normal topology are still valid in the new setting.

Proposition. 2 If a set A of sequences is 1-separable then is an FK-space with its normal topology. Proof. Let B1 be a countable cofinal subset of A and let B denote B1 U fe11e21.. }. Then is an FK space with the topology generated by the seminorms = since uCB (2—1) this is the FK-topology on the intersection of the 29 Since countable family of FK—spaces u11 as U ranges over B. Aa has AK with FK-topology it follows that the dual of Aact is so that the topology is compatible with the duality of Thus, the FK-topology being a Mackey topology is to stronger than the normal topology.